Mathematical Finance - · PDF fileMathematical Finance. Contents I Stochastic calculus7 ... De nition 1.1.3 (Stochastic process) . A stochastic pressco is a collection fX t: t 0gof

Mathematical Finance

Contents

I Stochastic calculus 7

1 Stochastic Calculus 91.1 Introduction to stochastic processes . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.1 Quadratic Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.1.2 Stochastic Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.3 Itô's lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2 Stochastic Dierential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.1 Linear stochastic dierential equations . . . . . . . . . . . . . . . . . . 201.2.2 Stochastic dierential equations with linear multiplicative noise . . . . 21

1.3 Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.3.1 Solutions of SDE as Markov processes . . . . . . . . . . . . . . . . . . 261.3.2 Toward Feynman-Kac formula . . . . . . . . . . . . . . . . . . . . . . 27

1.4 Change of measure and Girsanov theorem . . . . . . . . . . . . . . . . . . . . 311.4.1 Application of Girsanov to SDE . . . . . . . . . . . . . . . . . . . . . 39

1.5 Stochastic optimal control theory . . . . . . . . . . . . . . . . . . . . . . . . . 421.5.1 Open loop optimal controllers . . . . . . . . . . . . . . . . . . . . . . . 431.5.2 Optimal feedback controllers . . . . . . . . . . . . . . . . . . . . . . . 46

1.6 An introduction to Lévy processes . . . . . . . . . . . . . . . . . . . . . . . . 501.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511.6.2 Lévy processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541.6.3 Innitely divisible distributions and characteristic functions . . . . . . 561.6.4 Lévy-Khintchine formula . . . . . . . . . . . . . . . . . . . . . . . . . 581.6.5 Lévy-Ito decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 601.6.6 Lévy measure, nite variation, moments and jumps . . . . . . . . . . . 621.6.7 Semi-martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631.6.8 Girsanov theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641.6.9 Ito formula and Feynman-Kac theorem . . . . . . . . . . . . . . . . . 65

1.7 Introduction to innite dimensional analysis . . . . . . . . . . . . . . . . . . . 661.7.1 Gaussian measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671.7.2 Gaussian random variables and Brownian motion . . . . . . . . . . . . 721.7.3 Markov semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741.7.4 Heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

1.8 Stochastic partial dierential equations . . . . . . . . . . . . . . . . . . . . . . 771.9 Kolmogorov equation for the Ornstein-Uhlenbeck process . . . . . . . . . . . 79

1.9.1 Stochastic perturbations of linear equations . . . . . . . . . . . . . . . 79

3

4 CONTENTS

1.9.2 Stochastic dierential equations with non Lipschitz coecients . . . . 86

II Mathematical nance 93

2 The binomial asset model 952.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

2.1.1 History: Milestones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.1.2 Sketch of the course . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

2.2 The one-period binomial model . . . . . . . . . . . . . . . . . . . . . . . . . . 962.2.1 Completeness and arbitrage . . . . . . . . . . . . . . . . . . . . . . . . 982.2.2 The Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992.2.3 Towards Risk Neutral Measure (RNM) and Hedging . . . . . . . . . . 100

2.3 The multi-period binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . 1032.3.1 Pricing under the risk-neutral measure and the hedging portofolio . . 103

3 Continuous time models 1053.1 Towards the Black-Scholes formula . . . . . . . . . . . . . . . . . . . . . . . . 1053.2 Martingale Pricing Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.2.1 Hedging and replicating . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.3 Packages and Exotic options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

3.3.1 Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163.3.2 Exotic options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.4 Main nancial models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.4.1 Equity market models . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183.4.2 Siegel's paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.5 The Merton model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4 Interest Rates 1234.1 Allowing for stochastic interest rates in the Black-Scholes Model . . . . . . . 123

4.1.1 The hedging portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.1.2 Solving for the option price . . . . . . . . . . . . . . . . . . . . . . . . 125

4.2 Change of Numeraire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.2.1 Option pricing under stochastic interest rate . . . . . . . . . . . . . . 1284.2.2 Change of numeraire under multiple source of risk . . . . . . . . . . . 131

4.3 Interest rate option problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.3.1 Interest rate caps, oors and collars . . . . . . . . . . . . . . . . . . . 132

4.4 Modelling interest rate dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 1364.4.1 The relationship between interest rates, bond prices and forward rates 1364.4.2 Modelling the spot rate of interest . . . . . . . . . . . . . . . . . . . . 1394.4.3 Modelling forward rates . . . . . . . . . . . . . . . . . . . . . . . . . . 142

4.5 Interest rate derivatives: one factor spot rate . . . . . . . . . . . . . . . . . . 1464.5.1 Arbitrage models of the term structure . . . . . . . . . . . . . . . . . . 1464.5.2 The martingale representation . . . . . . . . . . . . . . . . . . . . . . 1484.5.3 On some specic model . . . . . . . . . . . . . . . . . . . . . . . . . . 1504.5.4 Pricing bond options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

4.6 The Heath-Jarrow-Morton framework . . . . . . . . . . . . . . . . . . . . . . 1564.6.1 The basic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

CONTENTS 5

4.6.2 The arbitrage pricing of bonds and bond options . . . . . . . . . . . . 1584.6.3 Forward risk adjusted measure . . . . . . . . . . . . . . . . . . . . . . 1614.6.4 Reduction to Markovian form . . . . . . . . . . . . . . . . . . . . . . . 1624.6.5 On some specic models . . . . . . . . . . . . . . . . . . . . . . . . . . 164

5 Appendix 1675.1 Recall on functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

5.1.1 Dual space and weaker topologies . . . . . . . . . . . . . . . . . . . . . 1685.1.2 Spectrum and resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5.2 On some particular classes of operators . . . . . . . . . . . . . . . . . . . . . 1715.2.1 Compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1715.2.2 Trace class and Hilbert-Schmidt operators . . . . . . . . . . . . . . . . 176

5.3 Introduction on semigroup theory . . . . . . . . . . . . . . . . . . . . . . . . . 1785.3.1 Strongly continuous semigroups . . . . . . . . . . . . . . . . . . . . . . 1815.3.2 Semigroups, generators and resolvents . . . . . . . . . . . . . . . . . . 187

Part I

Stochastic calculus

7

Chapter 1

Stochastic Calculus

Before we launch into a detailed description of continuous-time nancial model, we brieyintroduce some fundamental notion on stochastic calculus. For a deeper treatment of thetopic we refer to [Shr04, KS98, Pro90].

1.1 Introduction to stochastic processes

Let us rst recall the denition of probability space.

Denition 1.1.1 (Probability space). A probability space is a triple (Ω,F ,P) where Ω isan abstract set, F is a σalgebra of subsets of the set Ω and P is a probability measure onΩ, that is

(i) P (Ω) = 1 and P (∅) = 0;

(ii) P (⋃∞i=1Ai) =

∑∞i=1 P (Ai) if Ai ∩Aj = ∅ for i 6= j.

A set A ∈ F is called event, points ω ∈ Ω are called sample points and P(A) is theprobability of the event A to occur.

We are now to introduce the notion of random variable.

Denition 1.1.2 (Random variable). A mapping X : Ω → Rn, n ∈ N, is called ann−dimensional random variable if, for each Borelian set B ∈ B(B), we have that X−1(B) ∈F or, in other words, X is F−measurable.

Denition 1.1.3 (Stochastic process). A stochastic process is a collection Xt : t ≥ 0 ofrandom variables. In other words, a stochastic process X assigns to each time t a randomvariable Xt. In fact, X = X(t, ω =, t ∈ I ⊂ R+, ω ∈ Ω.

The stochastic process X is called continuous if the map t 7→ Xt(ω) is P−a.s. continuous,that is continuous with probability 1. It is called mean square continuous on [0, T ] if, foreach t0 ∈ [0, T ] we have that

limt→t0

E|Xt −Xt0 |2

= 0 .

9

10 1. Stochastic Calculus

A family (Ft)t≥0 ⊂ F is called a ltration if Fs ⊂ Ft for 0 ≤ s ≤ t, F0 contains allA ∈ F , P(A) = 0.

A stochastic process Xt is said to be (Ft)t≥0−adapted if X−1t (B) ⊂ Ft for any Borel set

B ∈ B.

Denition 1.1.4 (Conditional expectation). Let us consider the usual probability space(Ω,F , (Ft)t ,P) and let G ⊂ F be a sub-σ-algebra of F .

The conditional expectation of a random variable X, denoted by E [X| G] is the uniquerandom variable s.t.

(i) E [X| G] is G−measurable;

(ii) ∫A

E [X| G] (ω)P(dω) =

∫A

X(ω)P(dω) ∀A ∈ G .

We are now to state the denition of martingale.

Denition 1.1.5 (Martingale). A stochastic process X = Xtt≥0 is a Ft-martingale w.r.t.P if

(i) EP[|Xt|] <∞,∀t ≥ 0 ;

(ii) EP[Xt+s|Ft] = Xt,∀t, s ≥ 0 .

Heuristically speaking we have that the martingale condition means that the best previ-sion we can give about past values of a given stochastic process is the values it has a giventime t.

Theorem 1.1.6 (Martingale inequality). Let X = (Xt)t∈[0,T ] be a continous martingale,

then for any p ∈ (1,∞) we have

E max0≤s≤t

|Xs|p ≤(

p

p− 1

)pE |Xt|p .

Denition 1.1.7 (Brownian motion). A stochastic process X = Xt : t ∈ [0, T ]t≥0 iscalled a standard Brownian motion if the following hold

(i) X0 ≡ 0;

(ii) the function

X(ω) : [0, T ]→ R , t 7→ Xt(ω)

is a P−a.s. continuous function of t;

(iii) for every 0 ≤ t1 < t2 ≤ t3 < t4 ≤ T , we have

Xt2 −Xt1 |= Xt4 −Xt3 ;

1.1 Introduction to stochastic processes 11

(iv) for every s < t,Xt −Xs ∼ N(0, t− s) .

We will in what follows use the notation W in order to the denote a Brownian motion.In particular it holds that, for any t ≥ 0,

EWt = 0 , EW 2t = t ,

P (a ≤Wt ≤ b) =1√2π

∫ b

a

e−12x

2

dx .

In the same manner we can consider n copies of Brownian motions, namely a stochasticprocess with values in Rn such that

Wt(ω) = (W 1t (ω), . . . ,Wn

t (ω))

where the i−th component W it (ω) is a standard Brownian motion in the sense of Def. 1.1.7

and it holds W i |= W j for j 6= i.Nevertheless, it is often useful to consider also an n-dimensional Brownian motion such

that its components are correlated as

ρ = (ρij)i,j=1,...,n =

1 ρ12 . . . . . . ρ1n

ρ21 1 . . . . . . ρ2n

......

ρn1 . . . . . . 1

with ρ a positive and symmetric matrix such that ρij ∈ [−1, 1] for all i, j = 1, . . . , n.

Let us recall that by positiveness of matrix ρ, the following must hold

n∑i,j=1

ρijxixj ≥ 0 , for all x = (x1, . . . , xn) ∈ Rn .

Moreover it implies that all the eigenvalues of ρ are positive, hence there exist a matrixH ∈ GLn such that ρ = H ·HT . Then we can rewrite the Brownian motion W in terms ofn uncorrelated Brownian motions, namely

Wt = (W 1t , . . . ,W

nt ) = H · Zt , Zt = (Z1

t , . . . , Znt )

i.e. W it =

∑nj=1 hijZ

jt , where Ztt≥0 is an uncorrelated n-dimensional Brownian motion.

It follows that the n-dimensional Brownian motion Wt = (W 1t , . . . ,W

nt ) satises

(i) W0 = 0;

(ii) Wt(ω) is a t-continuous function for almost every ω ∈ Ω;

(iii) ∀0 ≤ t1 < t2 ≤ t3 < t4 Wt4 −Wt3 |= Wt2 −Wt1 ;

(iv) Wt −Ws ∼ N(0, (t− s)ρ) ∀s ≤ t.


2 4 6 8 10

-2

-1

1

2

3

4

5

(a) Standard Brownian motion

0.2 0.4 0.6 0.8 1.0

-1.0

-0.5

0.5

(b) Brownian motion with drift

In this sense we have that W is not a standard Brownian motion in the sense of Def.1.1.7 (see that point (iv) does not hold). We will refer to a Brownian motion that satises1.1 as Brownian motion.

Example 1.1.1. Some classical examples of martingale processes are

(i) the Brownian motion Wt;

(ii) W 2t − t;

(iii) exp

[θWt − θ2t

2

].

The last example is an exponential martingale, which is a particularly interesting class ofmartingale since they are positive, and then they can be used to dene probability measure,as it will be seen later on.

1.1.1 Quadratic Variation

It is of a certain interest to recall that a Brownian motion is a particular example of thewider class of Lévy processes, namely the class of càdlàg (right continuous with nite leftlimit) stochastic processes with independent, stationary increments. In this framework aBrownian Motion can be characterized (Lévy's Theorem) as follows: a P−a.c. martingaleXtt with quadratic variation [X,X]t = t is a Brownian motion.

Let Π = t0, t1, . . . , tn be a partition of the time interval [0, t], i.e. 0 = t0 < t1 <· · · < tn = t and |Π| = maxk|tk − tk−1|. The quadratic variation of a stochastic processX = Xt≥0 is given by

lim|Π|→0

n∑k=1

(X(tk)−X(tk−1))2 = [X,X]t = [X]t .

It is known from standard analysis that usually the quadratic variation of a functionis equal to 0. In the case of the Brownian motion we have that the quadratic variation[W,W ]t = t. This is due to the fact that the Brownian motion has no nite variation onbounded interval.


Exercise 1.1.1. Let us recall that similarly to the quadratic variation we can dene thevariation of a function X as

V (X) = lim|Π|→0

n∑k=1

(X(tk)−X(tk−1)) .

A process X is said to be of nite variation if V (X) <∞ w.p.1.We will now show that if V (X) <∞, then [X]t = 0.Indeed let us note that

n∑k=1

(X(tk)−X(tk−1))2 ≤ maxk|X(tk)−X(tk−1)|

n∑k=1

|X(tk)−X(tk−1)| ≤ maxk|X(tk)−X(tk−1)|V (f) = 0 .

since by continuity of X, limn→∞maxk|X(tk)−X(tk−1)| = 0 that implies V (Xt) <∞.

Since a Brownian motion is a continuous nite quadratic variation process, recall indeedthe Lévy theorem [W ]t = t, it has innite total variation. The latter consideration has aparticular relevance when we have to price option based on a multiple Brownian motion.

We can further dene the covariance operator between two stochastic processes X =Xtt≥0 and Y = Ytt≥0 as

[X,Y ]t = lim|Π|→0

n∑k=1

(X(tk)−X(tk−1))(Y (tk)− Y (tk−1)) =1

2([X + Y ]t − [X]t − [Y ]t) .

In general a càdlàg stochastic process X with nite variation has quadratic variationgiven by [X]t =

∑0<s≤t(∆Xs)

2, where ∆Xs = Xs −Xs−.

1.1.2 Stochastic Integral

From Sec. 1.1.1 we have that the Brownian motion has no bounded variation. Thereforeto build an integral in the sense of the standard Riemann-Stiljes integral is not possible.We have therefore develop a new integration theory, in particular we require to be able tointegrate also function of unbounded variation. The aim is far from being trivial and it hasbeen rst developed by Itô.

In order to introduce such type of integral we rst need several technical tools.

Denition 1.1.8 (Stopping time). A random variable τ taking values in [0, T ] is called astopping time if the event τ ≤ t ∈ Ft ∀t > 0.

Heuristically speaking a stopping time is a random variable whose value is known withthe information available at time t. In this sense if we consider a standard Brownian motionW we have that the random variable

”the rst time t the Brownian motion W has value A” = mint ∈ [0, T ] : Wt = A ,

for a xed A ∈ R, is a stopping time. In the same manner we have that

”the last time t the Brownian motion W has value A” = supt ∈ [0, T ] : Wt = A ,

is not a stopping time.


It is clear that τ is a random time and in particular, its value is a part of the wholeinformation we have up to time τ .

Besides stopping times, a fundamental notion used in the development of this new inte-gration theory is the one of elementary process.

Denition 1.1.9 (Elementary process). A stochastic process X = Xtt is elementary ifit is piecewise constant, i.e. if there exists a sequence of stopping time 0 = t0 < t1 < · · · <tn = T and a set of Fti-measurable functions e0, e1, . . . , en−1 such that

Xt(ω) =

n−1∑i=0

ei(ω)1[ti−1,ti)(t) P− a.e. (1.1)

Having introduced the notion of elementary process we can now dene the stochasticintegral for such type of processes.

Denition 1.1.10. Let X be an elementary process, then the stochastic integral for theprocess Xt with respect to a Brownian motion Wt is dened as follows∫ T

0

Xt(ω) dWt(ω) =

n−1∑i=0

ei(ω)[W (ti+1)−W (ti)] .

We would like to stress that a fundamental dierence from standard integration theoryand stochastic integration theory is that the integral in eq. (1.1) has to be evaluate in theleft point, namely ei(ω)[W (ti+1) −W (ti), whereas for the standard integral it is the sameto evaluate the integral in the left point or in the right one, namely

ei(ω)[W (ti+1)−W (ti) = ei+1(ω)[W (ti+1)−W (ti) ;

this is due to the fact that we require a sort of non anticipating condition and furthermorethe process dened in eq. (1.1) can be shown to be a martingale.

Since now we have that elementary process are dense in the space of square integrableprocesses, we can extend by density argument the integral dened in eq. (1.1).

In particular let X = Xtt∈[0,T ] be an adapted process, then it is possible to choose asequence of elementary processes Xn = Xnt∈[0,T ] converging to X, i.e.

limn→∞

|Xnt −Xt|2 dt = 0 ;

then the Itô integral for X is dened as∫ T

0

Xt dWt = limn→∞

Xnt dWt . (1.2)

where the limit is taken in L2−sense.

Denition 1.1.11. Let L2([0, T ]) dene the set of all adapted stochastic processes X =

Xtt∈[0,T ] such that E[ ∫ T

0(Xt(ω))2 dt

]<∞ .

The following is a fundamental result for the stochastic integral dened in eq. (1.2).


Theorem 1.1.12 (Itô isometry). Let X ∈ L2([0, T ]), then we have

E[(∫ T

0

Xt dWt

)2]= E

[ ∫ T

0

X2t dt

]Theorem 1.1.13. Let Xt ∈ L2([0, T ]), then the stochastic integral

Yt =

∫ T

0

Xs(ω) dWs(ω)

is a martingale process.

This new integration theory allows us to now dene a new wide class of stochasticprocesses starting from the Itô integral (1.2).

Denition 1.1.14. Itô process A n-dimensional stochastic process X = Xtt∈[0,T ] is ofItô type if it can be represented as

Xt = x0 +

∫ t

0

µs ds+

∫ t

0

σs dWs , (1.3)

where

• Ws is a m-dimensional Brownian motion;

• µs is a n-dimensional Ft-adapted stochastic process;

• σs is a n×m-dimensional Ft-adapted stochastic process.

Remark 1.1.15. Equation 1.3 is often written in dierential form asdXt = µ(t,Xt) dt+ σ(t,Xt) dWt ,

X0 = x0

1.1.3 Itô's lemma

"Itô's calculus is little more than repeated

use of Itô's formula in a variety of

situations"

S. Shreve

Itô's lemma is most likely the most important tool when one is to use stochastic integra-tion theory. It gives the extension of the chain rule to the case of stochastic integral.

Theorem 1.1.16 (Itô-Doeblin Lemma). Lat Xt be a 1-dimensional Itô process satisfying

dXt = µt dt+ σt dWt

and let f ∈ C1,2([0,∞]× R), then the process Zt = f(t,Xt) satises

dZt =∂f

∂t(t,Xt) dt+

∂f

∂x(t,Xt) dXt +

1

2

∂f2

∂x2(t,Xt) (dXt)

2

=

(∂f

∂t(t,Xt) +

∂f

∂x(t,Xt)µt +

1

2

∂f2

∂x2(t,Xt)σ

2t

)dt+

∂f

∂x(t,Xt)σt dWt .


Let us notice that in case Zt = f(Wt) with f ∈ C2(R), Itô's Lemma reduces to

f(Wt) = f(0) +

∫ t

0

f ′(Ws) dWs +1

2

∫ t

0

f ′′(Ws) ds .

Exercise 1.1.2. Compute the Itô integral∫ T

0

Wt dWt =1

2W 2(T )− 1

2T .

As said Itô's lemma is an extremely powerful tool, in particular via Itô's lemma severalSDE can be solved. In order to better clarify what said above and let the reader morecomfortable with Th. 1.1.16 we are now to give few examples of notable Itô processes.

Example 1.1.2. Geometric Brownian motion Let St be a stochastic process represent-ing the dynamic of a certain stock-price, given as the solution of the following SDE

dSt = µtSt dt+ σtSt dWt (1.4)

where Wtt≥0 is a standard Brownian motion and µt and σt some suitable processregular enough.

Let us consider Yt = f(t, St) = f(St) = lnSt. Then by Itô-Doeblin formula (1.1.16)we have,

dYt = d(ln[St]) =

(µt −

1

2σ2t

)dt+ σs dWt

and hence

ln[St] = ln[S0] +

(µt −

1

2σ2t

)t+ σWt .

It follows that the solution of (1.4) we are looking for is

St = S0 exp

[ ∫ t

0

(µs −

σ2s

2

)ds+

∫ t

0

σs dWs

].

Notice that in the particular case when µs and σs are both constant, i.e. µs ≡ µ andσs ≡ σ, then

St = S0 exp

[(µ− σ2

2

)t+ σWt

].

Moreover note that

Yt = ln[St] ∼ N(

ln[S0] +

(µ− σ2

2

)t, σ2t

)

Ornstein-Uhlenbeck process Let us consider the process

dXt =(− γ(Xt − µt) + µ

)dt+ σ dWt ; (1.5)

1.2 Stochastic Dierential Equation 17

by Itô-Doeblin formula applied to the function Yt = exp[γt]Xt we have that

dYt = exp[γt] dXt +Xtd(exp[γt])

= exp[γt]((− γ(Xt − µt) + µ

)dt+ σ dWt

)+ γXt exp[γt] dt

= exp[γt][(γµt+ µ) dt+ σ dWt

]and hence

Yt = Y0 + µ

∫ t

0

eγs(γs+ 1) ds+ σ

∫ t

0

eγs dWs .

The starting process, i.e. the solution of the initial equation (1.5), turns out to be

Xt = e−γtX0 + µt+ σe−γt∫ t

0

eγs dWs .

We can further compute the mean and the variance of process Xt exploiting Itô isom-etry.

E[Xt] = X0e−γt + µt

and

Var[Xt] = Var

[σe−γt

∫ t

0

eγs dWs

]= σ2e−2γtE

[(∫ t

0

eγs dWs

)2]= σ2e−2γt

∫ t

0

e2γs ds =σ2

2γ

(1− e−2γt

).

Exercise 1.1.3. Calculate

(i) dWm, m ∈ N;

(ii) d(eλWt−λ

2

2 t);

(iii) d (tWt);

(iv)∫ t

0WsdWs.

1.2 Stochastic Dierential Equation

Let Wt be an m−dimensional Brownian motion and X0 an n−dimensional random variablewhich is independent of the Brownian motion Wt. Let T > 0 and b : [0, T ]× Rn → Rn andσ : [0, T ]× Rn → Rn×m be two given functions. We also set

b =

b1...bn

, σ =

σ1,1 . . . σ1,m

.... . .

...σn,1 . . . σn,m

. (1.6)


Let us rst recall that we dene the space Lpn([0, T ]), p ∈ [1,∞), the set of Rn-valuedprogressively measurable processes Y such that

E∫ T

0

|Ys|pds <∞ .

Denition 1.2.1. Let us consider an SDE of the formdXt = b(t,Xt)dt+ σ(t,Xt)dWt , t ∈ [0, T ] ,

X0 = x ∈ Rd(1.7)

with b : [0, T ] × Rn → Rn, σ : [0, T ] × Rn → Rn×m some measurable given functions asin (1.6) and Wt an m−dimensional Brownian motion. We say that an Rn−dimensionalstochastic process X : [0, T ]→ Rn is a strong solution to eq. (1.7) if X is (Ft)t≥0−adapted,b(t,Xt) ∈ L1

n([0, T ]) and σ(t,Xt) ∈ L2n×m([0, T ]) and it holds P−a.s.

Xt = x+

∫ t

0

b(s,Xs)ds+

∫ t

0

σ(s,Xs)dWs , t ∈ [0, T ] ,

Example 1.2.1. Let us consider the SDEdXt = g(t)XtdWt , t ∈ [0, T ] ,

X0 = x ∈ R ,(1.8)

with g : [0, T ] → R a given continuous funtion. It can be seen that eq. (1.8) is of the form(1.7) with b ≡ 0 and σ(t,Xt) = g(t)Xt. A solution to eq. (1.8) is given by

Xt = x exp

−1

2

∫ t

0

g2(s)ds+

∫ t

0

g(s)dWs

.

We have in fact that the process Yt := − 12

∫ t0g2(s)ds+

∫ t0g(s)dWs satises the equation

dYt = − 12g

2(t)dt+ g(t)dWt ,

Y0 = 0 ,

an application of Itô formula to X = f(Y ) = eY thus yields

dXt = deYt = eYtdYt +1

2g2(t)eYtdt =

=

(−1

2g2(t)dt+ g(t)dWt

)dYt +

1

2g2(t)eYtdt = eYtg(t)dWt ,

so that X = eY satsies eq. (1.8).

Example 1.2.2 (Stock price). Let us consider the SDEdStSt

= µdt+ σdWt , t ∈ [0, T ] ,

S0 = s ,(1.9)


with µ > 0, resp. σ > 0, a positive real constant called drift, resp. volatility.Proceeding as in Example 1.2.1 we can show, via Itô's lemma that a solution to eq. (4.2)

is given by

St = s exp

(µ− 1

2σ2)t+ σWt

.

We can further show that

ESt = seµt , t ∈ [0, T ] .

Remark 1.2.2. Eq. (4.2) is typically used in mathematical nance to model the stock priceprice evolution, it is usually refer to as Black-Scholes equation.

Theorem 1.2.3 (Existence and uniqueness result). Let us consider eq. (1.7)dXt = b(t,Xt)dt+ σ(t,Xt)dWt , t ∈ [0, T ] ,

X0 = x ∈ Rd(1.10)

with b : [0, T ] × Rn → Rn, σ : [0, T ] × Rn → Rn×m some measurable given functions as in(1.6) and Wt an m−dimensional Brownian motion. Let us further assume that b and σ arecontinuous and Lipschitz functions w.r.t. the second variable, that is, for any t ∈ [0, T ] andfor any x1, x2 ∈ Rn it exists L > 0 such that

|b(t, x1)− b(t, x2)| ≤ L|x1 − x2| ,|σ(t, x1)− σ(t, x2)| ≤ L|x1 − x2| .

(1.11)

Then for any initial random variable x independent of W0 such that E|x|2 < ∞ it exists aunique strong solution, in the sense of Def. 1.2.1, X ∈ L2

n ([0, t]) to eq. (1.10). Furthermorewe have that X is (Ft)t≥0−adapted and has continous sample paths.

Proof. We will use the Picard iteration scheme. Let us then dene X0t = x and

Xn+1t = x+

∫ t

0

µ(s,Xns )ds+

∫ t

0

σ(s,Xns )dWs ,

let us also dene

ant = E|Xn+1t −Xn

t |2 .

We rst prove that for any n ∈ N it holds

ant ≤(Mt)n+1

(n+ 1)!M > 0 . (1.12)

In fact it holds for n = 0 that, from (1.11)

a0t = E

∣∣X1t −X0

t

∣∣2 ≤ E∣∣∣∣∫ t

0

µ(s,X0s )ds+

∫ t

0

σ(s,X0s )dWs

∣∣∣∣2 ≤≤ 2E

∣∣∣∣∫ t

0

L(1 + x)ds

∣∣∣∣2 + 2E∣∣∣∣∫ t

0

L2(1 + x2)ds

∣∣∣∣2 ds ≤Mt .


Let us now assume that (1.12) holds for n and prove it for n + 1. In fact we haveproceeding as above

ant = E|Xn+1t −Xn

t |2 ≤ E∣∣∣∣∫ t

0

µ(s,Xn+1s )− µ(s,Xn

s )ds+

∫ t

0

σ(s,Xn+1s )− σ(s,Xn

s )dWs

∣∣∣∣2 ≤≤ 2L2(1 + T )

∫ t

0

Mnsn

n!ds ≤ Mn+1tn+1

(n+ 1)!.

Then it follows from the martingale inequality that

maxt∈[0,T ]

|Xn+1t −Xn

t |2 ≤ 2TL2

∫ T

0

|Xns −Xn−1

s |2ds+ 2 maxt∈[0,T ]

|∫ T

0

σ(s,Xns )− σ(s,Xn−1

s )dWs|2 ≤

≤ CT∫ T

0

E|Xns −Xn−1

s |2ds ≤ C (MT )n

n!,

this implies that

Xn+1t = c+

n∑j=1

(Xjt −X

j−1t ) ,

is convergent in L2 (Ω, C ([0, T ];Rn)) to a process X that satises eq. (1.10).

Remark 1.2.4. It follows by Th. 1.2.3 that it also holds

E maxt∈[0,T ]

|Xx1t −X

x2t |

2 ≤ CE |x1 − x2|2 ,

where we have denoted by Xxt the soution to eq. (1.10) at time t ∈ [0, T ] with initial value

x.

1.2.1 Linear stochastic dierential equations

Let us consider equation (1.10) with b(t, x) = A(t)x+b0(t), with A : [0, T ]→ Rn×n is a n×nmatrix continuous in t and b0 ∈ L1 ([0, T ];Rn). Let us further assume that σ(t, x) ≡ σ0(t),with σ0 : C([0, T ])→ Rn×m. Then eq. 1.10 reads

dXt = (A(t)Xt + b0(t)) dt+ σ0(t)dWt ,

X0 = x .(1.13)

Let us then denote bu U(t) the fundamental matric of the system X ′ = A(t)X that isddtU(t) = A(t)U(t) , t ≥ 0 ,

U(o) = I .(1.14)

Theorem 1.2.5. It exists a unique solution to eq. (1.13) and it is given by

Xt = U(t)x+

∫ t

0

U(s)U−1(s)b0ds+

∫ t

0

U(s)U−1(s)σ0dWs ,

with U solution to eq. (1.14).If further A is independent of t, then

U(t) = eAt =

∞∑n=0

(At)n

n!.


1.2.2 Stochastic dierential equations with linear multiplicative noise

Let us consider the following SDEdXt = (A(t)Xt + b0(t)) dt+

∑Ni1 XtσidW

it ,

Xt = x .(1.15)

Let us set

Xt = e∑Ni1XtσiW

it y(t) = eWty(t) , (1.16)

so that by Itô's formula we have

dXt = e∑Ni1XtσiW

it dy(t) + e

∑Ni1XtσiW

it y(t)

(N∑i1

XtσidWit −

1

2

N∑i1

σ2i

).

Substituiting it then into eq. (1.15) we getddty(t) = e−WtA(eWty(t)) + b0e

Wty(t) + σy(t) ,

y(0) = x ,(1.17)

where

σ =

N∑i1

σ2i .

Thus we have shown that a SDE with linear multiplicative noise of the form (1.15)reduces via transformation (1.16) to a random dierential equation of the form (1.17).

In general, let us consider a SDE of the formdXt = b(Xt)dt+ σ(Xt)dWt , t ∈ [0, T ] ,

X0 = x ∈ R ,(1.18)

with b : R→ R and σ : R→ R some given coecients satisfying assumptions (1.11). Let usalso consider

dYt = g(Yt)dt+ dWt , t ∈ [0, T ] ,

Y0 = y ∈ R ,(1.19)

with g a suitable function to be chosen properly according the specic problem, let us thenset

X = ϕ(Y ) , x = ϕ(y) , (1.20)

with ϕ ∈ C2 also to be specied in a while. By Itô's formula we thus have

dWt = ϕ′(Yt)dYt +1

2ϕ′′(Yt)dt ,

and plugging it in eq. (1.18) yields

ϕ′(Yt)dYt +1

2ϕ′′(Yt)dt = b(ϕ(Yt))dt+ σ(ϕ(Yt))dWt ,


hence we have

ϕ′(Yt)g(Yt)dt+ ϕ′(Yt)dWt +1

2ϕ′′(Yt)dt = b(ϕ(Yt))dt+ σ(ϕ(Yt))dWt .

Let us now choose ϕ such that, for all r it holds,

ϕ′(r) = σ(ϕ(r)) , ϕ(0) = 0 . (1.21)

Then we have that X, via substitution (1.20) solves eq. (1.18) providedϕ′(Yt)g(Yt) + 1

2ϕ′′(Yt) = b(ϕ(Yt)) ,

y = ϕ−1(x) ,

with ϕ given in (1.21).

Exercise 1.2.1. (i) Solve dXt = X3

t dt+XtdWt , t ∈ [0, T ] ,

X0 = x .

(Hint): Use transformation (1.16).

(ii) Solve dXt = aXtdt+X3

t dWt , t ∈ [0, T ] ,

X0 = x .

(Hint): Use transformation (1.19) and (1.20), with g chosen proeprly.

(iii) Solve dXt = aX3

t dt+ dWt , t ∈ [0, T ] ,

X0 = x .

(Hint): Use transformation X −W = Y .

(iv) Solve the Brownian bridge equationdXt = − Xt

1−tdt+ dWt , t ∈ [0, 1] ,

X0 = x .

(v) Solve the random harmonic oscillator equationXt = λ2Xt − bXt + σdWt , t ∈ [0, T ] ,

X0 = x1 , X0 = x2 .(1.22)

(Hint): Reduce eq. (1.22) to eq. (1.7) asdX1

t = X2t dt ,

dX2t = −λ2X1

t dt− bX2t dt+ σdWt ,

X10 = x1 , X2

0 = x2 .

1.3 Markov Processes 23

In this case we have

b =

(0σ

), A =

(0 1−λ2 −b

).

In the special case X1 ≡ 0, b = 0 and σ = 0, we have

X(t) = x1 cos(λt) +1

λ

∫ t

0

sin(λ(t− s))dWs .

(vi) Solve dXt = − 1

2e−Xtdt+ e−XtdWt , t ∈ [0, T ] ,

X0 = x .

(Hint): Use transformation X = ϕ(W ).

(vi) Solve the system dX1

t = X2t dt+ dW 1

t , t ∈ [0, T ] ,

dX2t = X1

t dW2t ,

with W = (W 1,W 2) a two dimensional Brownian motion.

1.3 Markov Processes

Let us introduce now Markov processes.

Denition 1.3.1. Let Xt, t ∈ I, where I is a totally ordered set of times, I ⊂ R, be astochastic process on (Ω,F , (Ft)t ,P). Let Fot = σ(Xs, s ≤ t) be the canonical ltration ofXt and so Fot ⊂ Ft. Let Xt be Ft-adapted. Then Xt is a Markov process (relative to Ft) if

P(Xt ∈ B|FS) = P(Xt ∈ B|σ(Xs)) for B ∈ B(Rn), s < t (1.23)

Equation (1.23) is called elementary Markov property.

This denition expresses in a formal way the idea that the present state allows to predictfuture states as well as the whole history of past and present state does, that is, the processis memoryless.

It is useful to know if a solution is a Markov process or not.

Remark 1.3.2. i) For s = t the elementary Markov property holds always.

ii) Suppose that σ(Xt) and Fs, with s < t, are independent, e.g. if Xt = Wt and Fs = Fos .Then the Markov property holds.

iii) In general Markov property doesn't imply independence; e.g. solutions of SDE will beshown to be Markov but do not have independence.

iv) There exist many other equivalent description of Markov property, e.g

P(Xt ∈ B|σ(Xs1 , ...Xsn)) = P(Xt ∈ B|Xsn), s1 < s2 < ... < t (1.24)

orE[f+f−|Xs] = E[f+|Xs]E[f−|Xs]. (1.25)


Remark 1.3.3. If Px is a family of probability measure, x ∈ Rn and

Px(Xs+t ∈ B|Fs) = PXs(Xt ∈ B), B ∈ B(Rn) (1.26)

where PXs(·) = Py(Xt ∈ B) |y=Ws and ∀ s, t ∈ R+ x 7→ Px(B) is B(Rn)-measurable, then(Xt,Px) has the Markov property, i.e. (1.23) holds with P = Px.

Proof. Set Y := PXs(Xt ∈ B); Y is σ(Xs)-measurable by hypothesis. So Y is Fos -measurable,then, by assumption (1.26), Y is Fs-measurable. Thus

Y = Ex[Y |σ(Xs)] = Ex[Ex[χBXs+t|Fs]|σ(Xs)] = Px(Xs+t ∈ B|Xs)

So we have proven

Px(Xs+t ∈ B|Xs) = PXs(Xt ∈ B), (1.27)

but by assumption (1.26) and (1.27) we obtain

Px(Xs+t ∈ B|Xs) = Px(Xs+t ∈ B|Fs), a.s. for s < t

Replace then t by t+ s and its proved.

Remark 1.3.4. We call (Px, Xt) with the above property, Markov family. There exists a1− 1 correspondence between such Markov family and Markov processes constructed fromMarkov semigroups.

We begin now a short discussion on Markov semigroups.

Denition 1.3.5. A 1-parameter Markov semigroup of kernels is a function Pt(x, dy), witht ≥ 0, x ∈ Rn such that

1. the map x 7→ Pt(x,B) is Borel-measurable ∀B ∈ B(Rn);

2. the function B 7→ Pt(x,B) is a probability measure ∀x ∈ Rn;

3. it holds the semigroup property :

Ps+t(x,B) =

∫RnPs(x, dy)Pt(y,B) ∀t, s ≥ 0, x ∈ Rn, B ∈ B(Rn). (1.28)

Equation (1.28) is also called Chapman-Kolmogorov equation, and we can write itshortly as Ps+t = PsPt

Denition 1.3.6. The 1-parameter Markov semigroup of kernels Pt is called normal ifP0(x,B) = δx(B), for B ∈ B(Rn).

Theorem 1.3.7. Given a probability space (Ω,F ,Px), x ∈ Rn, and a Markov process(Xt)t≥0, then Pt(x,B) := Px(Xt ∈ B) is a normal Markov semigroup of kernels.

Denition 1.3.8. Given a measurable function f ≥ 0, the transition semigroup associatedto (Xt,Px) is given by

(Ptf)(x) :=

∫Pt(x, dy)f(y) (1.29)


Remark 1.3.9. We have that

Pt(x,B) = Px(Xt ∈ B)⇐⇒ E(f(Xt)) = (Ptf)(x),

where Ex indicates the expected value with respect to probability measure Px.The proof of this fact is immediate for f = χB , since

∫Pt(x, dy)χB(y) = Pt(x,B) and

E(χB(Xt)) = Px(Xt ∈ B). Then one can prove by approximation that the equivalence holdsfor every measurable function.

Proof of theorem 1.3.7. In order to prove that Pt(x,B) := Px(Xt ∈ B) is a normal Markovsemigroup of kernels, we have to prove the three properties given in denition 1.3.5.

By denition of probability measure, properties (1) and (2) are trivial: indeed, x 7→Px(Xt ∈ B) is a Borel-measurable function for every B ∈ B(Rn) and obviously Pt(x,Rn) =Px(Xt ∈ Rn) = 1.

So we have only to prove the semigroup property, and we can show that

Ps+t(x,B) := Px(Xs+t ∈ B) = Ex(χXs+t∈B)

= E[Ex(χXs+t∈B | Fs)] = Ex(PXs(Xt ∈ B))

= Ex(Pt(Xs, B)) =

∫Ω

Pt(Xs(ω), B)Px(dω)

=

∫RnPt(y,B)PxXs(dy) =

∫RnPt(y,B)Ps(x, dy),

where the fourth equality is true since by assumption (Xt,Px) is a Markov family and wehave used the transformation formula for measures in order to prove that∫

Ω

Pt(Xs(ω), B)Px(dω) =

∫RnPt(y,B)PxXs(dy).

Indeed, if T : Ω→ Ω′ is a function that transform a measure µ in another measure Tµ, thefollowing formula holds: ∫

Ω′f ′dTµ =

∫Ω

f ′ · Tdµ.

Note that PxXs(dy) is the image of Px under the map Xs : Ω→ Rn.

Remark 1.3.10. The theorem guarantees that if we have a Markov family, i.e. (Px, Xt) withXt Markov process on Rn, then we can construct a Markov semigroup.

Also the viceversa holds: starting from Markov semigroups on Rn one can obtain aMarkov family, and this goes by constructing a proper family on (Rn)N and then usingKolmogorov extension theorem. In particular, the family is given by∫

Rn· · ·∫RnχB(x1, ..., xn)Ptk−tk−1

(xk−1, dxk) · · · Pt2−t1(x1, dx2)Pt1(x0, dx1),

where B ∈ B(Rn) and 0 ≤ t1 < t2 < ... < tk.For the Kolmogorov theorem there exists a space Ω = (Rn)[0,+∞] and a probability

measure Px0 on Ω such that (Xt ∈ Rn, Px0) is a Markov family. In particular, for anyprobability measure µ on Rn,B(Rn) and for any A ∈ B(Rn),∫

Px0(Xt ∈ A | σ(Xs, s ≤ t)µ(dx0) =

∫Pt−s(x,A)µ(dx)


Remark 1.3.11. A very special case of Markov process is the Brownian motion, whose asso-ciated Markov semigroup of kernels is

Pt(x, dy) = N(x, t1)(dy) =e−

12t |x−y|

2

(2πt)n2

, t > 0, x, y ∈ Rn, (1.30)

and we can write Pt(x,B) = PBt+x(dy) = Px(Bt ∈ B) for any B ∈ B(Rn). For exercise,prove that the semigroup property holds for this particular Markov semigroup of kernel(hint: use Fourier transform).

Remark 1.3.12. If we consider a function ϕ ∈ C2(Rd) and dene

ut(x) := (Ptϕ)(x) := E[ϕ(Bt + x)] = E[ϕ(Bt)],

it will turn out that ut(x) solves the heat or diusion equation:∂∂tut(x) = 1

24Rnut(x) x ∈ Rnu0(x) = ϕ(x)

(1.31)

1.3.1 Solutions of SDE as Markov processes

Theorem 1.3.13. Let us consider an SDE of the formdXt = b(t,Xt)dt+ σ(t,Xt)dWt,

X0 = x ∈ Rd(1.32)

with b : [0, T ] × Rd → Rd, σ : [0, T ] × Rd → Rd × Rn some measurable functions and Wt

an n−dimensional Brownian motion. Let us further assume that all assumptions regardingthe existence and uniqueness of a solution are satised. Then the solution of eq. (1.32) is aMarkov process in the sense that

P(Xt ∈ B| FXs

)= P (Xt ∈ B|Xs)

Remark 1.3.14 (Transition probability). The quantity

ps,t(x,B) = p(s, t; t, B) := P (Xt ∈ B|Xs = x) , s ≤ t (1.33)

is called transition probability family. These quantities do not depend on the initial conditionbut they depend only on the drift b and on the diusion σ.

Kolmogorov equation, to be discussed later, gives a way to determine analytically p(s, t; t, B)just using information on b and σ.

In the case of the drift and the diusion to be time independent, we speak of timehomogeneous case and thus the solution X of eq. (1.32) has the property that

p(s, t;x,B) = p(0, t− s;x,B) = p(τ ;x,B) .

In particular one hasp(s+ a, t+ a;x,B) = p(s, t;x,B).

Thus the 1−parameter family

pτ (x,B) := p(0, τ ;x,B)

is a transition semigroup satisfying the Chapman-Kolmogorov equation

pτpτ ′ = pτ+τ ′ .


Proof. By denition we have that

ps,t(x,B) = P (Xt ∈ B|Xs = x) = P (Xs,xt ∈ B)

where the last inequality follows from the Markov property. Our aim is to show

PXs,xt = PXs+a,xt+a, ∀a . (1.34)

We then have

Y (t) := Xs+a,xt+a = x+

∫ t+a

s+a

b(Xs+a,xr )dr +

∫ t+a

s+a

σ(Xs+a,xr )dWr =

r−a=ρ= x+

∫ t

s

b(Y (ρ))dρ+

∫ t

s

σ(Y (ρ))d(Wa+ρ −Wa).

(1.35)

where we have called Y (ρ) = Xs+a,xr . Noticing that by the independence of increments in

the denition of a Brownian motion we have that Wρ := Wa+ρ −Wa is again a Brownianmotion. Thus Y (ρ) satises the same SDE as Xt with Wt replaced by a dierent Brownianmotion Wt. Therefore from the uniqueness of weak solution we have that

L(Y (ρ)) = L(Xt).

Remark 1.3.15. The two parameter family ps,t satises an analogue of the Chapman-Kolmogorov equation

pτ,sps,t = pτ,t, 0 ≤ τ ≤ s ≤ t .

The proof is similar to the one parameter case.

Also analogously to one parameter case, one can dene

(ps,tϕ)(x) =

∫ps,t(x, dy)ϕ(y)

and for the particular case of ϕ = χB , we have that

(ps,tχB)(x) = ps,t(x,B) ⇒ (ps,tϕ)(x) = Eϕ(Xs,xt ), s < t .

1.3.2 Toward Feynman-Kac formula

In the present section we will exploit the connection between SDE and PDE. Eventuallywe will state the well-known Feynman-Kac formula, see, e.g. [DPZ08] for an extensivetreatment of the topic.

Denition 1.3.16 (Kolmogorov operator). Let us consider a function ϕ ∈ C2b (Rd). The

Kolmogorov operator L associated with the SDE (1.32) is dened as

(L(s)ϕ) (x) :=1

2Tr [ϕxx(x)a(s, x)] + 〈b(s, x), ϕx(x)〉 ; (1.36)


where we have dened a(s, x) := σ(s, x)σT (s, x). Further we have denoted

Tr [ϕxx(x)a(s, x)] :=

d∑i,j=1

∂

∂xi

∂

∂xjϕ(x)aij(s, x);

〈b(s, x), ϕx(x)〉 :=

d∑i=1

bi(s, x)∂

∂xiϕ(x) .

Proposition 1.3.17 (Kolmogorov forward equation). Let us consider the SDE (1.32), understandard assumptions of existence and uniqueness, and let further ps,t(x,B) dened as in(1.33). Then

ddtps,tϕ(x) = (L(t)ps,tϕ)(x), t0 ≤ s ≤ t ,limt↓s(ps,tϕ)(x) = ϕ(x) .

(1.37)

Proof. We have that

(ps,tϕ)(x) =

∫ps,t(x, dy)︸︷︷︸

P(Xt∈dy|Xs=x)

ϕ(y) = Eϕ(Xs,xt ) . (1.38)

Appying Ito's formula it can be shown that

dϕ(Xs,xt ) = L(t)ϕ(Xs,x

t )dt+ 〈ϕx(Xs,xt ), b(t,Xs,x

t )〉dWt .

Thus, taking the average and exploiting the martingale property of the stochastic integral,we have

Eϕ(Xs,xt ) = ϕ(x) +

∫ t

s

EL(τ)ϕ(Xs,xτ )dτ

Ψ=L(τ)ϕ= ϕ(x) +

∫ t

s

(ps,tΨ)(x)dτ ,

where in the last equality we have used Fubini's theorem.Thus from eq. (1.38), for s ≤ τ ≤ t, we have that

d

dtps,tϕ(x) = (ps,tΨ)(x) = (ps,tL(t)ϕ)(x) .

Proposition 1.3.18 (Kolmogorov backward equation). Let us consider the SDE (1.32),under standard assumptions of existence and uniqueness, and let further ps,t(x,B) denedas in (1.33). Then

ddsps,Tϕ(x) = −(L(s)ps,Tϕ)(x), t ≤ s ≤ T ,

lims↑T (ps,Tϕ)(x) = ϕ(x) .(1.39)

Proof. The proof is analogue to the proof of Prop. 1.3.17.

We refer to [DPZ08] for an extensive treatment of the topic.Let us now look at the following backward PDE

ddsz(s, x) + (L(s)z(s, ·))(x) = 0, s ≤ T ,

z(t, x) = ϕ(x), ϕ ∈ C2b (Rd) .

(1.40)


Theorem 1.3.19. Let us assume b, σ and ϕ as in Prop. 1.3.17. Then ∃! a solution to theeq. (1.40) given by

z(s, x) := Eϕ(Xs,xT ), 0 ≤ s ≤ T . (1.41)

Eq. (1.41) is known as probabilistic expression of the solution to the eq. (1.40).

Sketch. By Prop. 1.3.18 we have that

ps,Tϕ(x) := Eϕ(Xs,xT )

satisesd

dsps,Tϕ(x) = −(L(s)ps,Tϕ)(x) .

Setting z(s, x) = ps,Tϕ(x), then we have that it is a solution of eq. (1.40). For the uniquenesswe refer to [DPZ08].

Remark 1.3.20. Let us consider the special case of a time independent SDE, namely b(t,Xt) =b(Xt) and σ(t,Xt) = σ(Xt). Then it is easy to show that, as proven before, p0,t(x,B) =

pt(x,B) = P(X0,xt ∈ B

).

Theorem 1.3.21. Under usual assumption on the SDE (1.32) we have that ∃! a solutionto the forward PDE

ddtu(t, x) = Lu(t, x),

u(0, x) = ϕ(x) .(1.42)

given by u(t, x) = Eϕ(X0,xt ) = (ptϕ)(x).

The operator L is called the innitesimal generator associated with the semigroup pt, inthe sense that

Lϕ(x) = limt↓0

u(x, t)|t=0 =d

dtpt ϕ(x)|t=0 =

= limt↓0

ptϕ(x)− ϕ(x)

t= lim

t↓0

Eϕ(Xt)− ϕ(x)

t.

We refer to [DPZ08, EN00] for an extensive treatment of semigroup theory and innitesimalgenerators.

Example 1.3.1 (The Dirichlet problem). Let us consider the deterministic problem

∆u = 0 , in O ⊂ Rn ,u = g , on ∂O,

(1.43)

whit O a bounded subset of Rn, ∂O its boundary, ∆ the Laplacian operator and g a suitablegiven function.

Theorem 1.3.22. A solution to the deterministic problem (1.44) is given by

u(x) = Eg (Xτx) ,

with Xt = x+Wt and τx is the rst time X hits the boundary ∂O.


Taking into account that the function u is bounded, we have that Eτx < ∞ and soτx <∞ a.s. for all x ∈ O. Clearly we have that τx is a stopping time

Proof. Let u ∈ C2, then by Itô formula we have

du(Xt, t) =

n∑i=1

∂

∂xiudXi

t +1

2∆u(Xt)dt ,

where Xt = x+Wt. Then we have

u(Xτ , τ)− u(X0, 0) =1

2

∫ τ

0

∆uds+

∫ τ

0

∇u · dWs ,

and taking the expectation we have

Eu(Xτx)− Ex =1

2

∫ τx

0

Du(Xs)ds = u(x) .

In general we have that for a general equation of the form

−1

2∆u+ cu = f , in O ⊂ Rn ,

u = 0 , on ∂O,(1.44)

its solution is given by

u(x) = E∫ τx

0

f(Xt)e−

∫ t0c(Xs)dsdt ,

where as above Xt = x + Wt and τx is the rst time X hits the boundary ∂O. The prooffollows as above exploiting Itô's formula with

dZt = −c(x)dt , Yt = eZt , Z0 = 0 .

Example 1.3.2. In the case of Xt being the standard Brownian motion, namely Xt = Wt,we have that the innitesimal generator is L = 1

2∆, and pt is called the heat semigroup. Inparticular we have that u(x, t) := Eϕ(Wt + x) solves the heat equation

ddtu(t, x) = 1

2∆u(x, t),

u(0, x) = ϕ(x).

Theorem 1.3.23 (Feynman-Kac formula). The following PDEddsv(x, s) + (L(s)v(s, ·))(x) + V (s, x)v(s, x) = 0,

v(T, x) = ϕ(x) ∈ C2b (Rd).

(1.45)

is solved probabilistically and in particular uniquely by

v(s, x) = Eϕ(Xs,xT )e

∫ TsV (u,Xs,xu du,

where V ∈ Cb is called the potential and Xs,xt is the solution of eq. (1.32).

1.4 Change of measure and Girsanov theorem 31

1.4 Change of measure and Girsanov theorem

We will rst of all introduce the idea that lies behind Girsanov theorem in a discreet set-ting. The main idea is that of changing the measure of the driving noise for the stochasticdierential equation so that under the new probability measure the solution is easily nd.

Let us then consider a family of i.i.d random variables Zii=1,...,d such that Zi ∼ N (0, 1)on some probability space (Ω,F ,P). Thus computing the joint probability law of the randomvariables Zi on (Rd,B(Rd)) we get the following

P(ω|Z1(ω) ∈ dz1, ..., Zn(ω) ∈ dzd) =1

(2π)d2

exp

−1

2

d∑i=1

z2i

dz1 · · · dzd (1.46)

Setting

P(dω) := exp

d∑i=1

µiZi(ω)− 1

2

d∑i=1

E[(µiZi(ω))2]

P(dω), ω ∈ (Ω,F), µi ∈ R. (1.47)

So we have that

P(ω|Z1(ω) ∈ dz1, ..., Zn(ω) ∈ dzd) =

= exp

n∑i=1

µizi −1

2

n∑i=1

µ2iE[z2

i ]

1

(2π)d2

exp

−1

2

d∑i=1

z2i

dz1 · · · dzd

=1

(2π)d2

exp

−1

2

d∑i=1

(zi − µi)2

dz1 · · · dzd,

where the last equality follows from the fact that E[Z2i ] = Var[Zi] = 1.

Let us further notice that after after having changed the measure, the random variablesZ1, ..., Zd are still independent on (Ω,F , P). What is changed is the expectation, in factunder the new probability measure P we have

EP(Zi) :=

∫Ω

Zi(ω)P(dω) = µi 6= 1 = EP [Zi]

Furthermore, if we dene Zi := Zi − µi, we observe that Zi has now P-mean 0 and

variance given by EP(Zi2) = 1. So we can say that Zi,P are as good as Zi,P.

Let us state a preliminary theorem, recalling briey the concept of Radon-Nikodymderivative. In what follows for brevity reason we will use the notation

EP =: E, EP =: E.


Theorem 1.4.1. Let (Ω,F , (Ft)t ,P) be a probability space and let be Z a positive measurablerandom variable, i.e. Z(ω) ≥ 0, P−a.e. ω s.t. EZ = 1. Then for A ∈ F let us deneP (A) :=

∫AZ(ω)P (dω). Therefore we have:

(i) P (·) is a probability measure;

(ii) if X ≥ 0 r.v., then

E [X] :=

∫Ω

X(ω)P (dω) = EXZ ; (1.48)

(iii) if Z > 0 P-a.s., then for Y ≥ 0 P−a.s. we have

EY = E[Y

Z

](1.49)

Proof. (i) we have

P (Ω) =

∫Ω

ZP (dω) = EZ = 1 .

We have thus show the the measure P sum up to 1. We are left to show that it isσ−additive

P

( ∞⋃i=1

Ai

)=

∫⋃∞i=1 Ai

ZP (dω)n→∞→

∫⋃∞i=1 Ai

ZP (dω) =

∞∑i=1

P (Ai) ;

(ii) let us take rst X(ω) := χA(ω) the characteristic function of set A, then it easily follows

E [χA] = P (A) :=

∫A

ZP (dω) = EχAZ ;

we then exploit the fact that we can approximate any positive function via character-istic function and further knowing that any function can be written as the sum of twopositive function namely f = f+ − f− we are done;

(iii) proceed as in point (ii).

Remark 1.4.2. Z := P(dω)P(dω) is called the Radon-Nikodym derivative of P w.r.t. P, see, e.g.

[Rud74] for an extensive treatment of measure theory.

Example 1.4.1. Let X be a standard normal random variable on (Ω,F , (Ft)t ,P). Let usdene

Z := exp

−ϑX − 1

2ϑ2

for some constant ϑ.Let us further consider Y := X + ϑ and let P s.t. dP

dP = Z. Then Y is a

standard normal random variable under the measure P.


Sketch. We have to show that the characteristic function of the random variable Y is thecharacteristic function of a normal random variable i.e. if X ∼ N (µ, σ2) then ϕX(u) =

eiuµ−u2σ2

2 . We thus have to show that ϕY (u) = e−u2

2 exploiting the fact that ϕx(u) =

e−u2

2 .

Let us consider the probability space (Ω,F , (Ft)t ,P) and be Z a positive random variable

Z ≥ 0 P− a.s. s.t. EZ = 1 and let P = ZP in the sense of Radon-Nikodym.

Denition 1.4.3. The quantity dened by

Z(t) := EZ| Ft

is called the Radon-Nikodym derivative process associated with Z and Ft.

Theorem 1.4.4 (Girsanov theorem). Let Wt, 0 ≤ t ≤ T be a standard Brownian motionon the space (Ω,F , (Ft)t ,P) with values in Rd and (Ft)t the natural ltration of the Brownianmotion Wt. Let ϑ(ω, t), 0 ≤ t ≤ T be an Ft− measurable process with values in Rd.

Let us suppose that

P

(∫ T

0

|ϑi(t, ·)|2dt < +∞

)= 1 for i = 1, ..., n

and dene

Zϑ(t) := exp

−∫ t

0

ϑ(u)dWu −1

2

∫ t

0

|ϑ(u)|2du.

Assume that Zϑ(t) is a P-martingale on [0, T ]. Then

Wt := Wt +

∫ t

0

ϑ(u)du

is an Ito process on Rd and (Wt, (Ft)t) is an d-dimensional Brownian motion on (Ω, (Ft)t , P),

where P is dened by

P(A) := PT (A) :=

∫A

Zϑ(T )(ω)P(dω) A ∈ FT ,

i.e. P(·) = Zϑ(T )P(·).

Remark 1.4.5. Let us notice that P is a positive measure, since P is a positive measure andZϑ ≥ 0. Moreover, we can see that P is normalized, so it is a probability measure; indeedwe have

P(Ω) = E(Zϑ(T )) = E(Zϑ(t)) = E(Zϑ(0)) = 1,

where the second equality is true because Zϑ(t) is assumed to be a Ft,P-martingale on [0, T ].


Remark 1.4.6. As briey mentioned before the idea that lies behind Girsanov theorem isthat of changing the drift of a given Brownian motion so that we are able to nd a solutionof a given stochastic dierential equation driven by the new Brownian motion Wt := Wt +∫ t

0ϑ(u)du .

We are going to prove Girsanov theorem 1.4.4 in its general form, but rst we look atthe special case in which ϑ(u, ω) = ϑ(u), i.e. ϑ is a deterministic function. This is calledCameron-Martin theorem and was given by Robert Cameron and William Martin in 1944.

Let us introduce a lemma that will be useful for the proof of Cameron-Martin theorem.

Lemma 1.4.7 (Characterization theorem for Brownian motion). Let Xt be a continuousstochastic process on (Ω,F ,P) with t ∈ [0, T ] and X0 = 0. Then the following are equivalent:

(i) Xt is a Brownian motion;

(ii) ∀ 0 = to < t1 < · · · < tN < T , ui ∈ C

Eexp

N∑k=1

uk(Xtk −Xtk−1

)= exp

1

2

N∑k=1

u2k (tk − tk−1)

; (1.50)

(iii)

Eexp

∫ T

0

g(s)dXs

= exp

1

2

∫ T

0

g(s)2ds

∀ g ∈ L2[0, T ] (1.51)

Remark 1.4.8. This lemma is related to Lévy characterization theorem of Brownian motion.We have stated it for d = 1, but it is similar for a general d ∈ N.

Sketch of Proof. (Lemma 1.4.7)

(i) ⇒ (ii) By (i) we have that Xt − Xs ∼ N (0, t − s), s ≤ t and independent. Witha simple computation for 1-dimensional Gaussian random variable with distributionN (0, t− s) we obtain

E[expu(Xt −Xs)] = exp

u2

2(t− s)

for u ∈ C.

Now, (ii) follows using the fact that expectation of product of independent randomvariables is equal to the product of expecations.

(ii)⇒ (i) Dene Yt = Xtk −Xtk−1. By (ii) we have

E[expukYk)] = exp

1

2u2k(tk − tk−1)

and since exp

12u

2k(tk − tk−1)

is the characteristic function of a Gaussian random

variable with variance tk − tk−1, we can say that Yk ∼ N (0, tk − tk−1); this is exactlythe law of the increments of a Brownian motion, so the rst part of the proof isaccomplished.


In order to conclude, we have also to prove the independence of the increments. Againby (ii), for i 6= k we have

E[expuiYi) expukYk)] = exp

1

2u2i (ti − ti−1)

exp

1

2u2k(tk − tk−1)

= E[expuiYi)]E[expukYk)],

which implies that Yi and Yk are independent random variables.

(iii)⇒ (ii) Take a suitable step function g.

(ii)⇒ (iii) By approximation in L2[0, T ] (with a step function g).

Proof. (Cameron-Martin theorem). As for the general case of Girsanov theorem, let usassume that ϑ ∈ L2[0, T ], i.e.

∫ T

0

ϑi(t)2dt < +∞ ∀ i = 1, ..., d.

With this assumption, since ϑ ∈ L2[0, t] ⊂ L1[0, t],∫ t

0ϑ(u)du is well-dened and continuous

in t. So Wt := Wt+∫ t

0ϑ(u)du is well-dened and continuous in t, and Wt

∣∣∣t=0

= Wt|t=0 = 0

(by denition of Wt). Moreover, assume that Zϑ(T ) is a martingale.

Now, if we verify that property (iii) of lemma 1.4.7 holds for stochastic process Wt andprobability P, we will immediately prove that it is a Brownian motion. In other words, wewould like to prove that

E

[exp

∫ T

0

g(s)dWs

]= exp

1

2

∫ T

0

g(s)2ds

∀ g ∈ L2[0, T ],

where E(·) =∫· dP = expectation w.r.t P. Using the denitions of Wt, P and Zϑ(T ) we can


write

E

[exp

∫ T

0

g(s)dWs

]=

= E

[exp

∫ T

0

g(s)dWs

exp

∫ T

0

g(s)ϑ(s)ds

]

= E

[Zϑ(T ) exp

∫ T

0

g(s)dWs

exp

∫ T

0

g(s)ϑ(s)ds

]

= E

[exp

∫ T

0

(ϑ− g)(s)dWs

exp

−1

2

∫ T

0

ϑ(s)2ds

exp

∫ T

0

g(s)ϑ(s)ds

]

= exp

−1

2

∫ T

0

ϑ(s)2ds

exp

∫ T

0

g(s)ϑ(s)ds

E

[exp

∫ T

0

(ϑ− g)(s)dWs

]

= exp

1

2

∫ T

0

g(s)2ds

where the last equality follows by observing that for (iii) of lemma 1.4.7 (applied to thefunction ϑ− g ∈ L2[0, T ] and w.r.t the Brownian motion Wt)

E

[exp

∫ T

0

(ϑ− g)(s)dWs

]= exp

1

2

∫ T

0

(ϑ− g)(s)2ds

.

So we have

E

[exp

∫ T

0

g(s)dWs

]= exp

1

2

∫ T

0

g(s)2ds

,

that is (iii) of lemma 1.4.7 holds for Wt.In other words, we proved that (W , P) is a Brownian motion.

Before being able to prove Girsanov theorem, we need some results.We recall rst of all a fundamental property of conditional expectation that will be useful

in what follows.

Theorem 1.4.9 (Tower property). Let G and H e two σ−algebras such that G ⊂ H. Then

E[X | G] = E[E[X | H] | G] .

Proof. See, e.g. [Shr04, Th. 2.3.2].

Lemma 1.4.10. Let Xt be an adapted stochastic process on (Ω,F , (Ft)t ,P) with value in

Rd. If eiλXt+ 12 |λ|

2t is a (Ft,P)-martingale ∀λ ∈ Rd, then

i) Xt −Xs ∼ N (0, (t− s)Id) and is Fs independent;

ii) If Xt is, in addition, continuous and X0 = 0, then Xt is a Brownian motion.


Proof. i) Set Yt := eiλXt+12 |λ|

2t. For hypothesis Yt is a martingale and so we have

E[Yt | Fs] = Ys, P− a.s.

Substituting the denition of Yt, we obtain

E[eiλXt+12 |λ|

2t | Fs] = eiλXs+12 |λ|

2s

E[eiλXt | Fs] = eiλXs−12 |λ|

2(t−s)

E[eiλXt | Fs]e−iλXs = e−12 |λ|

2(t−s)

E[eiλ(Xt−Xs) | Fs] = e−12 |λ|

2(t−s), (1.52)

where in the last step we use the fact that eiλXs is Fs-measurable.

If we take expectation on both sides of (1.52), thanks to Th. 1.4.9, we get

E[E[eiλ(Xt−Xs) | Fs]] = E[eiλ(Xt−Xs)] = e−12 |λ|

2(t−s). (1.53)

We observe now that equation (1.53) gives the characteristic function of Xt −Xs andso Xt −Xs ∼ N (0, (t− s)Id).To complete the proof we need to show the independence of Xt−Xs. Take any A ∈ Fs,then

E[χAseiλ(Xt−Xs)] = E[E[χAse

iλ(Xt−Xs) | Fs]] = E[χAsE[eiλ(Xt−Xs) | Fs]] =

= E[χAse− 1

2 |λ|2(t−s)] = e−

12 |λ|

2(t−s)E[χAs ],

where the third equality is due to (1.52).

Now, thanks to (1.53), we get

E[χAseiλ(Xt−Xs)] = E[eiλ(Xt−Xs)]E[χAs ].

Recalling that two random variables A and B are independent if E[AB] = E[A]E[B],we conclude the proof of i).

ii) It follows from i) and the denition of Brownian motion.

We state now a second Lemma that we will use in Girsanov theorem's proof.

Lemma 1.4.11. If Zt ≥ 0 P-a.s., t ∈ [0, T ] is a P-martingale and ZtVt is also a P-martingale, then Vt is a P-martingale, where P = ZtP.

Proof.

EP[Vt | Fs] =EP[ZtVt | Fs]EP[Zt | Fs]

=ZsVsZs

= Vs,

where the rst equality is due to Bayes rule and the second follows from hypothesis.

It remains to make a comment on Bayes rule.


Lemma 1.4.12 (Bayes Rule). Let µ and ν be two probability measures on (Ω,G) such thatν = fµ for some f ∈ L1(µ). Let X be a random variable on (Ω,G) such that X ∈ L1(ν).Let H ⊂ G be a sub-σ-algebra. Then

Eν [X | H]Eµ[f | H] = Eµ[fX | H] a.s.

Proof. Exploiting the denition of conditional expectation, see, e.g. [SK91, Øks03], it followsthat whenever H ∈ H we have∫

H

Eν [X | H]fdµ =

∫H

Eν [X | H] =

∫H

Xdν =

=

∫H

Xfdµ =

∫H

Eµ[fX | H]dµ .

(1.54)

On the other hand, exploiting the so called tower rule, or iterated conditioning, see, e.g.Remark 1.4.9 or [Shr04, Th. 2.3.2], we obtain∫

H

Eν [X | H]fdµ = Eµ[Eν [X | H]fχH ] = Eµ[Eµ[Eν [X | H]fχH | H]] =

= Eµ[χHEν [X | H]Eµ[f | H]] =

∫H

Eν [X | H]Eµ[f | H]dµ .

(1.55)

Putting now together equation (1.54) and equation (1.55) we get∫H

Eν [X | H]Eµ[f | H]dµ =

∫H

Eµ[fX | H]dµ . .

From the fact that it holds for all H ∈ H follows the thesis.

We have now all the necessary tool to prove Girsanov theorem 1.4.4.

Proof of Thm 1.4.4. To prove the theorem we want to apply Lemma 1.4.10 usingXt Wt := Wt +

∫ t0ϑudu

P P := Zϑ(t)P.

So we want to show that the assumption of Lemma 1.4.10 hold for P and Wt, i.e. show

that Yt := eiλWt+12 |λ|

2t is a (Ft, P)-martingale ∀ λ ∈ Rn.Using denitions of Wt and Zϑ we obtain

Yt = exp

iλWt + iλ

∫ t

0

ϑudu+1

2| λ |2 t

YtZϑ(t) = exp

iλWt + iλ

∫ t

0

ϑudu+1

2| λ |2 t

exp

−∫ t

0

ϑudWu− 1

2

∫ t

0

ϑ2udu

= exp

−∫ t

0

(ϑu − iλ)dWu− 1

2

∫ t

0

(ϑu − iλ)2du

= Zϑ−iλ(t).


If we can show that Zϑ−iλ is a P-martingale, with the further assumption that Zϑ is aP-martingale, we obtain by Lemma 1.4.11 that Yt is a P-martingale. Then by Lemma 1.4.10we have that Wt would be a P-Brownian motion.

A straightforward application of Ito's lemma leads to

dZϑ−iλ(t) = −Zϑ−iλ(t)(ϑt − iλ)dWt. (1.56)

being Zϑ−iλ a stochastic integral, it implies that it is a P-martingale.To proceed (rigorously) we have to show that −Zϑ−iλ(t)(ϑt − iλ) is progressively mea-

surable so that the stochastic integral in equation (1.56) exists, i.e.

EP

[∫ t

0

| Zϑ−iλ(u)(ϑu − iλ) |2 du]<∞ . (1.57)

Using the martingale inequality we can show equation (1.57) holds and we are done.

Remark 1.4.13. In Girsanov theorem's statement we assumed that

Zϑ(t) = exp

−∫ t

0

ϑudWu− 1

2

∫ t

0

ϑ2udu

is a P-martingale. There exist sucient conditions for this involving only ϑ, for instance theso called Novikov condition

EP

[exp

1

2

∫ t

0

ϑ2udu

]< +∞. (1.58)

1.4.1 Application of Girsanov to SDE

We will deal in the present section with the following SDEdXt = β (t,Xt) dt+ σ (t,Xt) dWt,

X0 = x ∈ Rd,, (1.59)

with 0 ≤ t ≤ T , β : [0, T ]×Rd → Rd, σ : [0, T ]×Rd → Rd ×Rn and Wt an n− dimensionalBrownian motion.

Denition 1.4.14 (Weak solution). A continuous and adapted stochastic process Xt, t ∈[0, T ] is a weak solution of SDE (1.59) if there exists a probability space (Ω, F , (Ft)t, P)and a Ft-Brownian motion Wt dened on it such that Xt satises (1.59).

Remark 1.4.15. One does not require that Ft = σ(Xs, 0 ≤ s < t).

Remark 1.4.16. Existence of strong solution ⇒ existence of weak solution. The oppositedoes not hold, as we can see for example in the so called Tanaka equation.Moreover, we have that strong uniqueness ≡ pathwise uniqueness.

We can also dene a concept of weak uniqueness: one says that (1.59) has a uniqueweak solution when, given X1 and X2 weak solutions with the same initial conditions (i.e.X1(0) = X2(0) = x ∈ Rn), X1(t) and X2(t) have same nite dimensional distribution.

Remark 1.4.17. Strong uniqueness ⇒ weak uniqueness. The converse does not hold.


Remark 1.4.18. One proves, see, e.g., [SK91], that

weak existence + strong uniqueness ⇒ strong existence.

Remark 1.4.19. Suppose that β, σ are bounded and continuous, and that initial conditionsare bounded. These assumptions are sucient to ensure the existence of weak solution,so in this case lipschitz conditions are not required. Observe that in dimension 1 theseassumptions are necessary and sucient.

Remark 1.4.20. Weak solution ⇒ strong Markov property.

Further applications of Girsanov theorem

We now see a Corollary of Girsanov theorem that provides a useful method to nd weaksolutions of SDE.

Corollary 1.4.21. Let Yt ∈ Rn be an Ito process, i.e.

dYt = β(t, ω)dt+ σ(t, ω)dWt, t ≤ T (1.60)

Suppose ∃ϑ(t, ω), with P(∫ t

0ϑ2du <∞) = 1 and ∃α(t, ω) also with this property, such that

σ(t, ω)ϑ(t, ω) = β(t, ω)− α(t, ω).

Consider Zϑ(t) = exp−∫ t

0ϑudWu− 1

2

∫ t0ϑ2udu

and P = Zϑ(t)P as before (Girsanov

theorem). Assume that Zϑ(t) is a P-martingale.Then P is a probability measure, Wt = Wt +

∫ t0ϑudu is a P-Brownian motion and

dYt = α(t, ω)dt+ σ(t, ω)dWt

Remark 1.4.22. If n = m and σ−1 exists, then ϑ = σ−1(β − α).

Proof. By Girsanov theorem, we know that Wt is a P-Brownian motion. Introducing Wt in(1.60) instead of Wt, we obtain

dYt = β(t, ω)dt+ σ(t, ω)(dWt − ϑ(t, ω)dt)

= (β(t, ω)− ϑ(t, ω)σ(t, ω))dt+ σ(t, ω)dWt

= α(t, ω)dt+ σ(t, ω)dWt

Remark 1.4.23. This application of Girsanov theorem is often use in nance, where, byimposing α = 0, we want to obtain a zero-drift SDE.

Corollary 1.4.24. Let Xx be the solution of the stochastic dierential equationdXt = βx(Xt)dt+ σx(Xt)dWt,

X0 = x ∈ Rd,


where βx and σx are assumed to satisfy standard assumption such that a strong solutionexist and it is unique. Let Y y be a solution to the following stochastic dierential equation

dYt = (γ(ω, t) + βy(Yt)) dt+ σy(Yt)dWt,

Y0 = y ∈ Rd,

Let us now suppose it exists ϑ(ω, t) s.t. σ(Yt)ϑ(t) = γ(t) and dene Zϑ, P and β as before.Let us further assume Zϑ to be a P−martingale. Thus Y y is the solution of

dYt = βy(Yt)dt+ σy(Yt)dWt .

In particular the P−law of Y is equal to the P−law of X.

Proof. The proof is just an application of corollary 1.4.21. Equality of laws from the as-sumption of existence and uniqueness of a strong solution of Xx.

We now apply Girsanov theorem to construct a weak solution ofdXt = a(Xt)dt+ dWt,

X0 = x ∈ Rd,, (1.61)

with a assumed to be a bounded and measurable function (not necessarily Lipschitz).Let us consider the equation

dYt = dWt, Y0 = y ∈ Rd, .

Apply then Cor. 1.4.21 with ϑ(t, ω) = −a(Xt), σ = 1, β = 0 and α = a. From assumptionon regularity of a and from a straightforward application of Ito's lemma, it follows that

Zϑ(t) = e−∫ t0a(Wu)dWu−

∫ t0a(Wu)2du (1.62)

is a martingale. Applying then Girsanov theorem we have that

Wt = Wt −∫ t

0

a(Ws)ds

is a P Brownian motion. We can thus write

Wt =

∫ t

0

a(Ws)ds+ Wt

and

Y xt = x+

∫ t

0

a(Y xs )ds+ Wt .

Therefore the couple(Y xt , Wt

)is a weak solution of the original equation (1.61).

It can be proved, see, e.g. [SK91, p.304], the weak uniqueness, therefore it follows thatthe P−law of Y xt is equal to the P−law of Xx

t . In particular if fi ∈ Cb(Rd) we have that

Ef1(Xxt1 , . . . , fk(Xx

tk= E

[f1(Y xt1 , . . . , fk(Y xtk

]= EZϑ(T )f1(Wt1 + x, . . . , fk(Wtk + x)

where Zϑ is dened as in (1.62).Therefore we have seen that Girsanov theorem is a powerful tool to make estimate of

complicated Xt in terms of a Brownian motion.


1.5 Stochastic optimal control theory

In the present section we are to develop the theory for stochastic optimal control, for anapplication to nance and to Merton problem we refer to Sec. ??.

Let us consider the optimization problem

Minimize ϕ(u) = E∫ T

0

L (t,Xt, u(t)) dt+ Eg(XT ) , (P)

subject to

dX(t) = f(t,Xt, u(t))dt+ σ(t,Xt, u(t))dWt ,

X0 = x ,

u ∈ U = v ∈ L2ad ([0, T ];Rn) : v(t) ∈ U0 , , a.et ∈ [0, T ] ,

, (1.63)

where L : [0, T ]×Rn×Rn → R, f : [0, T ]×Rn×Rm → R, g : Rn → R, σ : [0, T ]×Rn×Rm →Rn×d, W is d−dimensional standard Brownian motion and U0 is a closed convex subset ofRm. Also we have denoted by L2

ad ([0, T ];Rn) the space of (Ft)t≥0−adapted processes u

such that E∫ T

0|u(t)|2dt <∞. In the following we will denote by fX , σX and LX , resp. fu,

σu and Lu, the derivative w.r.t. the second variable X, resp. w.r.t. the third variable u.

Hypothesis 1.5.1. Throughout the section we will assume that:

(i) the functions f , L, g and σ are C1 w.r.t. X and u and continuous w.r.t. t;

(ii) it exists C > 0 such that

|fX |+ |fu|+ |σX |+ |σu|+ |gx| ≤ C <∞ ;

(iii) it exists K > 0 such that

|LX |+ |Lu| ≤ K(1 + |X|+ |u|) ,

Denition 1.5.2. A fucntion u ∈ U is called admissible control, it is called an optimalcontrol if

ϕ(u) = minvϕ(v) .

An optimal control is also called open loop optimal control, the corresponding solution X iscalled optimal state.

Example 1.5.1 (Production planning problem). Let us consider the optimization problem


0

|Xt −X0|2 + |u(t)|2dt+ aEXT , (1.64)

subject to

dX(t) = −Stdt+ Utdt+ σdWt , t ∈ [0, T ] ,

X0 = x ,, (1.65)

where Xt is the inventory at time t, Ut is the production rate at time t, St the constantdemand at time t, T the length of planning period, X0 the factory optimal inventory level(the target) and x the initial inventory level.

1.5 Stochastic optimal control theory 43

1.5.1 Open loop optimal controllers

The maximum principle represents a set of necessary conditions of optimality for a certaincontrol u∗ ∈ U . In order to formulate it we associate to the problem (4.124) the backwardstochastic system, see, e.g. [EKPQ97] to a complete introduction to backward stochasticdierential equations,

dpt = −fX(t,X∗t , u∗(t))ptdt− σX(t,X∗t , u

∗(t))qtdt+ LX(t,X∗t , u∗(t))dt+ qtdWt ,

pT = −gX(X∗T ) ,

(1.66)where we have denoted by (X∗, u∗) the optimal solution to problem (4.124). By existencetheory of backward stochastic equations, see, e.g. [EKPQ97], we know that system (1.66)has a unique solution

(p, q) : [0, T ]× Ω→ Rn × L(Rd;Rn) ,

such that

E∫ T

0

|p(t)|2 + |q(t)|2dt <∞ ,

p ∈ L2(Ω;C([0, T ];Rn)) .

We thus have the following stochastic maximum principle (rst order necessary conditionsof optimality) for problem (4.124). In the following we will denote by f∗u , resp. σ

∗u, the adjoint

of the matrix fu ∈ L(Rn;Rm), resp. σu.

Theorem 1.5.3 (The maximum principle). Let (X∗, u∗) be the optimal problem in (4.124).Then we have a.e. t ∈ [0, T ] and P−a.s.

〈f∗u(t,X∗, u∗)p− Lu(t,X∗, u∗) + σ∗u(t,X∗, u∗)q, u∗ − v〉 ≥ 0 , ∀ v ∈ U0 , (1.67)

where (p, q) is the unique solution to (1.66). Equivalently

u∗(t) = argmaxv∈U0

H(t,X∗, v, p, q) , (1.68)

where H is the Hamiltonian function

H(t,X∗, v, p, q) = f∗(t,X∗, v)p− L(t,X∗, v) + σ∗(t,X, v)q . (1.69)

Corollary 1.5.4. Assume that

L(t, x, v) = g0(t, x) + h(v) , σ(t, x, v) = σ0(t, x) ,

with h : Rm → R, then eq. (1.67) reduces to

〈f∗u(t,X∗, u∗)p− hu(u∗), u∗ − v〉 ≥ 0 , ∀ v ∈ U0 , (1.70)

that is (1.68)-(1.69) become

u∗(t) = arg minv∈U0

−f(t,X∗, v)p+ h(v) . (1.71)


U0

u∗

NU0(u∗)

u∗NU0

(u∗)

u∗y ∈ NU0

(u∗)

Figure 1.2: the normal cone to U0 at u∗, NU0(u∗)

If we assume U0 = Rm, then eq. (1.70) reduces to f∗up = hu(u∗), if we denote by NU0(u∗)the normal cone to U0 at u∗, that is

NU0(u∗) = η ∈ Rm : 〈η, u∗ − v〉 ≥ 0 , ∀ v ∈ U0 ,

we can rewrite eq. (1.67) as

f∗u(t,X∗, u∗)p− Lu(t,X∗, u∗) + σ∗u(t,X∗, u∗)q ∈ NU0(u∗) , ∀ t ∈ [0, T ] ,

and eq. (1.70) as

f∗u(t,X∗, u∗)p− hu(u∗) ∈ NU0(u∗) , ∀ t ∈ [0, T ] ,

proof of Th. 1.5.3. Let us rst assume that U0 = Rm and that fu, σu ∈ C(Rn×m).Since u∗ is optimal we have for all v ∈ L2

ad and λ > 0 such that u∗ + λv ∈ U , that isu∗ + λv ∈ U0,

1

λ(ϕ(u∗ + λv)− ϕ(u∗)) ≥ 0 .

We then can choose such a v of the form v = v − u∗, v ∈ U0 and λ ∈ (0, 1), such we get

E∫ T

0

L(t,X∗t , u∗)dt+ Eg(X∗T ) ≤ E

∫ T

0

L(t,Xu∗+λvt , u∗ + λv(t)) + Eg(Xu∗+λv

T ) .

This then yields, diving by λ and taking the limit as λ ↓ 0 and setting

Zt = limλ↓0

1

λ

(Xu∗+λv −Xu∗

),

we thus get for any v ∈ U ,

E∫ T

0

LX(t,X∗t , u∗)Zt + fu(t,X∗t , u

∗)v(t)dt+ EgX(X∗T )ZT ≥ 0 , (1.72)


wheredZt = fX(t,X∗t , u

∗)Zt + fu(t,X∗t , u∗)v(t)dt+ σX(t,X∗t , u

∗)Zt + σu(t,X∗t , u∗)v(t)dWt ,

Z0 = 0 .

Then by Itô's formula we have

d (Ztpt) = ptdZt + Ztdpt + (σX(t,X∗t , u∗)Zt + σuv(t)) qtdt .

Integrating then from 0 to T and taking the expectation we get

EgX(X∗T )ZT = E∫ T

0

ptfX(t,X∗t , u∗)Zt − fX(t,X∗t , u

∗)ptZt − σX(t,X∗t , u∗)qtZt+

+ σu(t,X∗t , u∗)v(t)qt + σX(t,X∗t , u

∗)qtZt + fu(t,X∗t , u∗)v(t)pt + LX(t,X∗t , u

∗)Ztdt =

= E∫ T

0

fu(t,X∗t , u∗)v(t)pt + σ(t,X∗t , u

∗)v(t)qt + LX(t,X∗t , u∗)Ztdt = EgX(X∗T )ZT .

Substituting then into eq. (1.72) we get

E∫ T

0

(fu(t,X∗t , u∗)pt − Lu(t,X∗t , u

∗) + σu(t,X∗t , u∗)qt) v(t)dt ≥ 0 , ∀ v ∈ U ,

for all v = −u∗ + v, v ∈ U0 a.e., v ∈ U , and thus we have

〈f∗u(t,X∗, u∗)p− Lu(t,X∗, u∗) + σ∗u(t,X∗, u∗)q, u∗ − v〉 ≥ 0 , ∀ v ∈ U0 , (1.73)

a.e. on [0, T ]× Ω.If we then assume that the function ϕ : v 7→ L(t,X, v) − σ(t,X, v)q − f(t,X, v)p is

coercive, that is ϕ(v)→∞ as |v| → ∞, it follows from eq. (1.73) that eq. (1.68) holds.If U0 6= Rm the proof is the same with the only dierence that we take u∗+λv are taken

such that u∗ + λv ∈ U for λ ∈ [0, 1]. The case of general fu and σu follows similarly, werefer to [] for details.

Example 1.5.2. Let us consider eq. (1.88) with n = d = 1, σ independent of x and u,L(X,u) = |X −X0|2 + |u|2, f = −S + u. Then the backward system (1.66) reads as

dpt = 2(Xt −X0)dt+ qtdWt ,

pT = −a ,

and then by (1.71) we have

u∗(t) =1

2pt , ∀ t ∈ [0, T ] .

Example 1.5.3. Let us consider the problem


0

|Xt −X0|2 + |u(t)|2dt , (1.74)

subject to

dX(t) = A(t)Xtdt+XtdWt + u(t)dt , t ∈ [0, T ] ,

X0 = x ,. (1.75)


We then have that (1.66) reads asdpt = −A∗(t)ptdt− qtdt+Xtdt+ qtdWt ,

pT = 0 ,

u∗ = p .

Example 1.5.4 (Optimal consuption problem). Let us consider the optimization problem

Minimize ϕ(u) = E1

γ

∫ T

0

e−ρtuγ(t)dt+XT , (1.76)

subject to

dX(t) = (µXt − u(t))dt+ σ(Xt)dWt , t ∈ [0, T ] ,

X0 = x ,, (1.77)

where u is the control that represents the consumptions andX is a certain economic quantity.The Th. 1.5.3 reads as

dpt = −µdt− σ′(Xt)qt + qtdWt ,

pT = −1 ,

so that we have e−ρtuγ−1 + pt, which yields u = (e−ρtpt)1

γ−1 . We can thus take q = 0and p = −e−µ(T−t) and

u =(peρt

) 1γ−1 .

1.5.2 Optimal feedback controllers

Let us consider problem (4.124)

Minimize ϕ(u) , u ∈ U , (P)

with U the set of all (Ft)t≥0−adapted controls u : [0, T ] → Rm such that u(t) ∈ U0 a.e.t ∈ [0, T ], U a closed convex subset in Rm.

Let us then consider the valued function

V (t, x) = minE∫ T

t

L(s,Xs, u(s))ds+ g(XT ) ,

subject to dXs = f(Xs, u(s))ds+ σ(s,Xs, u(s))dWs , s ∈ [t, T ] ,

Xt = x .(1.78)

An optimal feedback control (or a closed loop optimal control) u∗ for problem (P) is afeedback control u ∈ U of the form u(t) = Φ(t,Xt) which is optimal for (P), that is for any(t,X, u) satisfying eq. (1.78) it holds

E∫ T

t

L(s,X∗s ,Φ(s,X∗s ))ds+ g(X∗T ) ≤ E∫ T

t

L(s,Xs, u(s))ds+ g(XT ) .


An optimal control feedback can be characterized as the solution to a certain partialdierential equation, called Hamilton-Jacobi-Bellman equation (HJB), arising from the dy-namic programming principle. In particular let us assume that the m−dimensional Wienerprocess of the form W idi=1 such that

(σWt)i =

d∑j=1

σi,j(t,Xt, u)W jt ,

and let us consider the covariance matrix

ai,j =

m∑l=1

σi,lσl,j , i , j = 1, . . . , n ,

that is a(t,X, u) = σσ∗(t,X, u) being σ∗ the adjoint matrix of σ. Let us further set

A(t)ϕ =1

2

n∑i,j=1

ai,j(t, x, u)∂2

∂xi∂xjϕ+

n∑i=1

fi(t, x, u)∂

∂xiϕ ,

then we have that the HJB equation associated to problem (P) is∂∂tϕ(t, x) + minu∈U0

A(t)ϕ(t, x) + L(t, x, u) = 0 , t ∈ [0, T ] ,

$(T, x) = g(x) .(1.79)

Equivalently we can consider the equation∂∂tϕ(t, x) + 1

2

∑ni6=j=1 a

i,j(t, x,Φ(t, x)) ∂2

∂xi∂xjϕ(t, x) +

∑ni=1 fi(t, x,Φ(t, x)) · ∇ϕ(t, x) + L(t, x,Φ(t, x)) = 0 ,

$(T, x) = g(x) ,

(1.80)with

Φ(t, x) = arg minu∈U0

A(t)ϕ+ f(x, u) · ∇ϕ(s, x) + L(s, x, u) .

Let us consider the particular case of U = Rn, d = n, σ = I, L(s, x, u) = g1(s, x) + h(u)and f(x, u) = f0(x) +Bu with B a n×m matrix. The the HJB (1.79) becomes

∂∂tϕ(t, x) + 1

2∆ϕ(t, x) + f0(x) · ∇ϕ(t, x) = 0 , t ∈ [0, T ] ,

$(T, x) = g(x) ,

andΦ(t, x) = (∂h)

−1(−B∗∇ϕ(t, x)) .

Theorem 1.5.5. Let ϕ ∈ C2 ([0, T ]× Rn) be the solution to eq. (1.79), then

ϕ(t, x) = V (t, x) , ∀ t ∈ [0, T ] , x ∈ Rn ,

andu∗(t) = Φ(t,X∗t ) , (1.81)

is optimal feedback control for problem (P), that is the solution X∗ to the closed loop systemdX∗t = f(t,X∗,Φ(t,X∗t ))dt+ σ(t,X∗t ,Φ(t,X∗t ))dWt ,

X∗0 = x ,

is optimal for problem (P).


Proof. Let us set Y the solution todYt = f(t,X∗,Φ(t, Yt))dt+ σ(t, Yt,Φ(t, Yt))dWt ,

Y0 = y ,.

Then by Itô's formula we have that for ϕ solution to (1.79) it holds

dϕ(s, Ys) =∂

∂sϕϕ(s, Ys) +∇ϕ(s, Ys) (f(s, Ys,Φ(s, Ys))ds+ σ(s, Ys,Φ(s, Ys))dWs) +

+1

2

n∑i,j=1

ai,j∂

∂xi∂xjϕ(s, Ys)ds = −L(s, Ys,Φ(s, Ys))ds , s ∈ [t, T ] .

Integrating from t to T we get

ϕ(t, y) = E∫ T

t

L(s, Ys,Φ(s, Ys))ds+ Eg(YT ) ≥ V (t, y) .

On the other hand for any pairs (z, v) such thatdzs = f(zs, v)ds+ σ(s, zs, v)dWs , s ∈ [t, T ] ,

zt = y ,

we have

dϕ(s, zs) =∂

∂sϕds+∇ϕ · fds+ σdWs +

1

2

n∑i,j=1

ai,j(v)∂

∂xi∂xjϕ(s, z)ds ≥ −L(s, zs, v(s))ds .

This yield

ϕ(t, y) ≤ E∫ T

t

L(s, z, v)ds+ Eg(zT ) .

Hence ϕ(t, y) = V (t, y), for any t and y. Moreover we have that the feedback control (1.81)is optimal.

Example 1.5.5. Let us consider the problem


0

|Xt −X0|2 + |u(t)|2dt , (1.82)

subject to

dX(t) = aXtdt+ dWt + u(t)dt , t ∈ [0, T ] ,

X0 = x ,. (1.83)

Then the HJB equation (1.79) reads∂∂tϕ(t, x) + 1

2∂2

∂x2ϕ(t, x) +(ax− ∂

∂xϕ(t, x))∂∂xϕ(t, x) + 1

2 |x−X0|2 + 1

2∂∂xϕ

2(t, x) = 0 ,

$(T, x) = 0 , x ∈ R ,


and then we have

u∗(t, x) = − ∂

∂xϕ(t, x) , .

Let us look for a solution ϕ of the form

ϕ(t, x) =1

2r1(t)x2 + r2(t)x+ r3(t) .

Then we have

1

2r′1(t)x2 + 2r1(t) + r′2(t)x+ r′3(t) + ax(r1(t)x+ r2) +

1

2(x−X0)2 − 1

2(r1(t)− r2(t))2 = 0 .

This yields

r′1(t)− r21(t) + ar1(t) = 0 , s ∈ [t, T ] , r1(T ) = 0 ,

r′2(t)− ar2(t)− r1(t)r2(t) = X0 , s ∈ [t, T ] , r2(T ) = 0 ,

r′3(t) =1

2(X0)2 +

1

2r22(t) , s ∈ [t, T ] , r3(T ) = 0 ,

and the optimal feedback control is of the form

u∗(t, x) = −r1(t)x− r2(t) .

Exercise 1.5.1. (i) Solve

Minimize E∫ 1

0

X4t + |u(t)|2dt , (1.84)

subject to

dX(t) = dWt + u(t)dt , t ∈ [0, T ] ,

X0 = x ,. (1.85)

(ii) Solve

Minimize E∫ 1

0

X4t + |u(t)|2dt+ E|X1|2 , (1.86)

subject to

dX(t) = aXtdt+ dWt + bu(t)dt , t ∈ [0, T ] ,

X0 = x ,. (1.87)

(iii) SolveMinimize E|X1|2 , (1.88)

subject to

dX(t) = Xtdt+ dWt + u(t)dt , t ∈ [0, T ] ,

X0 = x ,

0 ≤ u(t) ≤ 1 .

. (1.89)


1.6 An introduction to Lévy processes

Let us rst try to give an intuitive understanding of the behaviour of a general Lévy process.We have rst of all introduced the Lévy-Khintchine formula. It is a fundamental tool

that give us a complete characterization of a Lévy process. It say that any Lévy processLt has a characteristic function given by The law µ of a random variable X is innitelydivisible i

ϕLt(u) = exp

ibut− u2σ2t

2+

∫Rt(eiux − 1− iux1|x|≤1

)ν(dx)

with b ∈ R, σ2 ∈ R and ν a Borel measure on R such that ν (0) = 0 and

∫R(1 ∧ x2

)ν(dx) <

∞ where we have denoted 1 ∧ x2 = min1, x2.Such a formula might at rst sight rather mysterious so let us try analyse it component

by component.If we rst take σ2 = ν = 0 then we simply have

ϕX(u) = exp ibu

so that our Lévy process is nothing but a straight line deterministic process. Physically itis the velocity at which the particle, whose motion is driven by Lt, is moving.

Let us now take the diusion σ 6= 0. Then we have

ϕX(u) = exp

ibu− u2σ2t

2

It is easy to see that this is the characteristic function for of a Gaussian random variablewith mean bt and covariance σ2t. It is then a Brownian motion with drift and comes withall the standard properties.

Since there is nothing that is much complicated. Let us then assume also the nalcomponent ν 6= 0. Then we have dierent possibilities. Let us take ourselves in the mostsimple possible case, i.e. when the measure ν is nite and therefore it admit rst momentand we have that condition (1.99) is satised. Thus we can rewrite the general form for aLévy process Lt with characteristic triplet (bt, σ2t, νt) to be equation (1.101)

ϕLt(u) = exp

ibut− u2σ2t

2+

∫R\0

t(eiux − 1

)ν(dx)

with b = b−∫|x|<1

xν(dx). In order to really understand what does this actually mean let us

consider an even simpler case, i.e. ν = λδy with δy a Dirac mass concentrated at y ∈ R\0.In this particular case we have a Lévy process of the form

Lt = bt+ σWt +Nt

where Nt is an independent Poisson process of intensity λ. Heuristically we can think atthe process Lt as a process that moves according to a Brownian motion with drift until arandom time τ1 and then at time t = τ1 it has a jump of size |y|. Then it keep movingaccording the Brownian motion with drift and then again at a second random time τ2 it hasa jump of size |y| and so on and so forth.

1.6 An introduction to Lévy processes 51

Since we have at this point a pretty clear understanding of the simplest case let uscomplicate the situation a bit more. Let us consider a measure ν =

∑ni=1 λiδyi with the

same notation of before. We can then rewrite the process Lt as

Lt = bt+ σWt +N1t + · · ·+Nn

t

with N1, . . . , Nn independent Poisson processes with intensity λi independent of the Brow-nian motion Wt as well. The physical interpretation is as before just this time the size ofthe jump might be any of the n numbers |y1|, . . . , |yn|.

Eventually, in the most general case, we can have jumps of any possible size. So theinterpretation is of a Brownian motion with drift and with jumps at random time of arbitrarysize.

We have seen the case of a nite measure ν. Let us now see what we can say when we arein the case of an innite activity Lévy process (according to notation introduced in section1.6.6).

From a mathematical point of view we have that eiux−1 is no longer ν integrable. Recallanyway that eiux − 1− iux1|x|<1 always is.

Loosely speaking we can say that the measure ν is now so ne that it does not distinguishanymore between the drift and small jumps. It is still possible to give previous interpretationof Brownian motion with drift and jumps of any size provided we bare in mind that at amicroscopic level drift burst and tiny jumps are treated as one.

Then we whenever ν <∞ we can decompose the Lévy process as

Lt = bt+ σWt +∑s≤t

∆Ls

where we have already dened the jumps ∆Ls. Notice that in the previous case of ν = δythen we will either have ∆Ls = 0 or ∆Ls = y. It is not anyway mathematically convenientto deal directly with jumps so we introduced the counting measure µL that counts the timesat which jumps occur. We have explained that it is a random measure. In fact if we x aset A s.t. ν(A) <∞, then µL(t, A) is nothing but a Poisson process with intensity ν(A).

Then in the present case we can rewrite the summation as∑s≤t

∆Ls =

∫R\0

xµL(t, dx)

In the more subtle case of innite measure ν the Lévy-Ito decomposition (1.93) gives usthe most general form for a Lévy process

Lt = bt+ σWt +

∫ t

0

∫|x|>1

xµL(ds, dx) +

∫ t

0

∫|x|≤1

x(µL(ds, dx)− ν(dx)ds

)1.6.1 Introduction

The standard Black&Scholes model assumes an asset evolves according to

St = eBt−ct

with a standard Brownian motion. Of course this process has many nice properties typicalof the gaussian distribution and in particular it has a solution in closed form. It has beenempirically proved anyway that many of these nice properties do not hold in reality. Inparticular features that has been mostly criticized are


(i) symmetry around its mean;

(ii) thin tails;

(iii) continuity.

It is the last one, in particular in application to electricity markets, that has lead to recenthuge develpment of Lévy processes in nance.

We have rst of all introduced the Lévy-Khintchine formula. It is a fundamental toolthat give us a complete characterization of a Lévy process. It say that any Lévy processLt has a characteristic function given by The law µ of a random variable X is innitelydivisible i

ϕLt(u) = exp

ibut− u2σ2t

2+

∫Rt(eiux − 1− iux1|x|≤1

)ν(dx)

with b ∈ R, σ2 ∈ R and ν a Borel measure on R such that ν (0) = 0 and

∫R(1 ∧ x2

)ν(dx) <

∞ where we have denoted 1 ∧ x2 = min1, x2.Such a formula might at rst sight rather mysterious so let us try analyse it component

by component.If we rst take σ2 = ν = 0 then we simply have

ϕX(u) = exp ibu

so that our Lévy process is nothing but a straight line deterministic process. Physically itis the velocity at which the particle, whose motion is driven by Lt, is moving.

Let us now take the diusion σ 6= 0. Then we have

ϕX(u) = exp

ibu− u2σ2t

2

It is easy to see that this is the characteristic function for of a Gaussian random variablewith mean bt and covariance σ2t. It is then a Brownian motion with drift and comes withall the standard properties.

Since there is nothing that is much complicated. Let us then assume also the nalcomponent ν 6= 0. Then we have dierent possibilities. Let us take ourselves in the mostsimple possible case, i.e. when the measure ν is nite and therefore it admit rst momentand we have that condition (1.99) is satised. Thus we can rewrite the general form for aLévy process Lt with characteristic triplet (bt, σ2t, νt) to be equation (1.101)

ϕLt(u) = exp

ibut− u2σ2t

2+

∫R\0

t(eiux − 1

)ν(dx)

with b = b−∫|x|<1

xν(dx). In order to really understand what does this actually mean let us

consider an even simpler case, i.e. ν = λδy with δy a Dirac mass concentrated at y ∈ R\0.In this particular case we have a Lévy process of the form

Lt = bt+ σWt +Nt


where Nt is an independent Poisson process of intensity λ. Heuristically we can think atthe process Lt as a process that moves according to a Brownian motion with drift until arandom time τ1 and then at time t = τ1 it has a jump of size |y|. Then it keep movingaccording the Brownian motion with drift and then again at a second random time τ2 it hasa jump of size |y| and so on and so forth.

Since we have at this point a pretty clear understanding of the simplest case let uscomplicate the situation a bit more. Let us consider a measure ν =

∑ni=1 λiδyi with the

same notation of before. We can then rewrite the process Lt as

Lt = bt+ σWt +N1t + · · ·+Nn

t

with N1, . . . , Nn independent Poisson processes with intensity λi independent of the Brow-nian motion Wt as well. The physical interpretation is as before just this time the size ofthe jump might be any of the n numbers |y1|, . . . , |yn|.

Eventually, in the most general case, we can have jumps of any possible size. So theinterpretation is of a Brownian motion with drift and with jumps at random time of arbitrarysize.

We have seen the case of a nite measure ν. Let us now see what we can say when we arein the case of an innite activity Lévy process (according to notation introduced in section1.6.6).

From a mathematical point of view we have that eiux−1 is no longer ν integrable. Recallanyway that eiux − 1− iux1|x|<1 always is.

Loosely speaking we can say that the measure ν is now so ne that it does not distinguishanymore between the drift and small jumps. It is still possible to give previous interpretationof Brownian motion with drift and jumps of any size provided we bare in mind that at amicroscopic level drift burst and tiny jumps are treated as one.

Then we whenever ν <∞ we can decompose the Lévy process as

Lt = bt+ σWt +∑s≤t

∆Ls

where we have already dened the jumps ∆Ls. Notice that in the previous case of ν = δythen we will either have ∆Ls = 0 or ∆Ls = y. It is not anyway mathematically convenientto deal directly with jumps so we introduced the counting measure µL that counts the timesat which jumps occur. We have explained that it is a random measure. In fact if we x aset A s.t. ν(A) <∞, then µL(t, A) is nothing but a Poisson process with intensity ν(A).

Then in the present case we can rewrite the summation as

∑s≤t

∆Ls =

∫R\0

xµL(t, dx)

In the more subtle case of innite measure ν the Lévy-Ito decomposition (1.93) gives usthe most general form for a Lévy process

Lt = bt+ σWt +

∫ t

0

∫|x|>1

xµL(ds, dx) +

∫ t

0

∫|x|≤1


)


1.6.2 Lévy processes

We will consider the probability space (Ω,F , (Ft)t ,P) to be complete and the ltration (Ft)tto be right continuous i.e.

Ft+ =⋂s>t

Fs = Ft

Further even if not specied t ∈ [0, T ], T > 0.

Denition 1.6.1. [Lévy process] Let us previous hypothesis to hold and let us consider aprocess Lt, t ∈ [0, T ] F-adapted with L0 = 0 Pa.s. and such that

(i) it has independent increments i.e. Lt − Ls is independent of Fs, 0 ≤ s < t <∞;

(ii) it has stationary increments i.e. ∀ t, h Lt+h−Lt has a distribution that is independentof t;

(iii) it is continuous in probability (stochastic continuous) i.e.

∀t, ∀ε lims→t

P (|Lt − Ls| > ε) = 0 ;

(iv) the process Lt has cadlag trajectories, i.e. right continuous and with left limit denedeverywhere.

Remark 1.6.2. Let us notice that property (iii) it is not pathwise continuity. It is in fact aweaker requirement. Of course a pathwise continuous process such as the Brownian motion isin fact continuous in probability. On the contrary there are many discontinuous processes,such as the Poisson process, that are continuous in probability. Loosely speaking we arerequiring that the jumps of the process Lt are not deterministic.

Remark 1.6.3. (i) point (iv) in denition 1.6.1 is not necessary in order to dened a Lévyprocess but it is fundamental in models;

(ii) cadlag comes from the French continue à droite, limite à gauche;

(iii) synonymous for cadlag are RCLL (right continuous with left limits) or corlol (con-tinuous on (the) right, limit on (the) left.

Theorem 1.6.4 (Th. 30, [Pro90]). Let Lt be a Lévy process as in denition 1.6.1, points(i)− (iii). There exists a unique modication Lt of Lt which is cadlag and which is also aLévy process.

Theorem 1.6.4 is the theorem that allows us to add point (iv) without loss of generality.For the sake of brevity we will always consider the cadlag modication denoting it with Lt.

Example 1.6.1. the drift term

Lt = bt, t ∈ [0, T ], b ∈ R

It is a linear deterministic function and thus it is also a Lévy process. Let us noticethat not all deterministic processes are in fact Lévy processes.


10 20 30 40 50

5

10

15

20

25

30

Figure 1.3: Poisson process, λ = .5

the Brownian motion

Lt = σWt, t ∈ [0, T ], σ ∈ R

with Wt the standard Brownian motion.

As already mentioned before the standard Brownian is a Lévy process. We refer to[SK91] for an extensive treatment of the theory of Brownian motion.

Remark 1.6.5. Examples 1 and 2 are the only Lévy processes that are also continuous.

the Poisson process

Lt = Nt :=∑x≥1

1t≥Tn, t ∈ [0, T ]

with (Tn)n≥0 a strictly increasing sequence of positive random variables with T0 = 0and

1t≥Tn :=

1 t ≥ Tn(ω)

0 t < Tn(ω)

Further Nt takes values in N is called the Poisson process or counting process. Fur-thermore we have

P (Nt = n) =e−λt(λt)n

n!, ENt = λt,

with λ > 0 the intensity of the Poisson process. We will denote such a process asLt ∼ Po(λt). Let us notice that all the jumps are of size 1.

Furthermore if we consider the Poisson process Nt ∼ Po(λt) we can dene the com-pensated Poisson process to be

Mt := Nt − λt

In particular Mt is a martingale. We refer to [Pro90, Sat99, Shr04] for an extensivetreatment.

the compound Poisson process

Lt =

Nt∑k=1

Jk, L0 :=

0∑k=1

Jk = 0


with Nt ∼ Po(λt) and J1, J2, . . . i.i.d random variables independent of Nt and withC := EJn <∞. Such an Lt is called compound Poisson process.

In particular the jumps in Lt occur at the same times as the jumps in Nt, but whereasthe jumps in Nt are always of size 1, the jumps in Lt are of random size, possiblynegative.

As done previously we can dene the compensated compound Poisson process to be

Mt := Lt − Cλt

Again Mt is a martingale. For an extensive treatment again we refer to [Pro90, Sat99,Shr04].

the general Lévy process

Lt = bt+ σWt +

(Nt∑k=1

Jk − Cλt

)(1.90)

This is the most general form for a Lévy process. We would like to stress that wecould have added the term Cλt in the drift term but we preferred this form since itplays the role of compensate the Poisson process so that(

Nt∑k=1

Jk − Cλt

)

is a martingale w.r.t. the ltration (Ft)t. This way, form the independence of all thesources of randomness, we ave that Lt is a martingale i b = 0.

1.6.3 Innitely divisible distributions and characteristic functions

For an extensive treatment see, e.g [Sat99], section 1.2 and chapter 2.

Denition 1.6.6. [Characteristic function] Let be t xed, we then dene the characteristicfunction of the distribution of the random variable Lt to be

ϕLt(u) := EeiuLt =

∫Ω

eiuLtP (dω) =

∫Reiuxµ(dx), u ∈ R (1.91)

with P the probability measure and µ the law of the random variable Lt.Under certain conditions, i.e. µ << L, i.e. µ is absolutely continuous w.r.t. the Lebesgue

measure, we have that the density of the random variable Lt exists and denoting fLt := dµdL

we have

ϕLt(u) := EeiuLt =

∫ReiuxfLt(x)dx, u ∈ R

We will drop the sux Lt if not cause of any confusion.

Analytically speaking the characteristic function is nothing but the Fourier transform ofthe law/density of Lt and thus it has all nice properties that come with it.

It s widely used in probability because there is ano-to-one correspondence between adistribution and its characteristic function. Thus knowing the characteristic function isenough to know the distribution of a random variable.


Let us now consider a Lévy process of the most general form (1.90), we thus have

ϕLt(u) = EeiuLt = eiubtEeiuσWtEexp

iu

(Nt∑k=1

Jk − Cλt

)=

= eiubte−12σ

2u2tEexp

iu

(Nt∑k=1

Jk − Cλt

)= eiubt︸︷︷︸

1

e−12σ

2u2t︸︷︷︸2

expλtEeiuJn − 1− iuJn

︸︷︷︸3

(1.92)

where we have exploited the fact that all the sources of randomness are independent, theexact form for a characteristic function of a gaussian random variable and the fact thatbeing all Jn i.i.d. we can use any of them as representative of the whole class. Elements1 − 2 − 3 are characteristic of any Lévy process. In particular there are two fundamentalcharacterization theorems for Lévy processes: the Lévy-Khintchine theorem and the Lévy-Itotheorem.

Denition 1.6.7 (Convolution). Let µ and ν two distribution on R. We denote the con-volution of µ with ν to be

µ ∗ ν(A) :=

∫Rµ(A− x)ν(dx) =

∫Rν(A− x)µ(dx)

We will use the notationµn∗ = µ ∗ · · · ∗ µ︸︷︷︸

n

Denition 1.6.8. [Innitely divisible distribution] Let µ be a probability measure on R.We say that µ is innitely divisible if for any n ∈ N positive, there is a probability measureµn on R such that µ = µn∗n .

Theorem 1.6.9. The following are equivalent:

(i) denition 1.6.8;

(ii) X ∼ µ, there exists Xn ∼ µn such that Xd= nXn;

(iii) X ∼ µ and Xn ∼ µn ⇒ ϕX(u) = (ϕXn(u))n.

where we have denoted byd= equal in distributions.

Example 1.6.2. 1.X sinN

(µ, σ2

)According to point (iii) Th. 1.6.9 we have

ϕX(u) = expiµu− σ2u2

2 = expn(iu

µ

n− σ2u2

2n) =

=

(expiuµ

n− σ2u2

2n)n

= (ϕXn(u))n

with Xn ∼ N(µn ,

σ2

n

). Therefore we have proved that a gaussian random variable is

innitely divisible.


2. the Cauchy random variable is innitely divisible.

3. the uniform and the binomial are not innite divisible.

1.6.4 Lévy-Khintchine formula

The following theorem gives a representation of characteristic functions of all innitely di-visible distributions.

Theorem 1.6.10. [Lévy-Khintchine theorem] The law µ of a random variable X is innitelydivisible i

ϕX(u) = exp

ibu− u2σ2

2+

∫R

(eiux − 1− iux1|x|≤1

)ν(dx)

(1.93)

with b ∈ R, σ2 ∈ R and ν a Borel measure on R such that ν (0) = 0 and∫R(1 ∧ x2

)ν(dx) <

∞ where we have denoted 1 ∧ x2 = min1, x2.

Denition 1.6.11 (Generating triplet). We call the triplet(b, σ2, ν

)in Th. 1.6.10 the

generating triplet of µ. The parameter b is called the drift term, σ2 is called the gaussiancovariance whereas ν is called the Lévy measure of µ. If σ2 = 0 then µ is called purely nongaussian.

Proposition 1.6.12. If µ has generating triplet(b, σ2, ν

), then µt has generating triplet(

bt, σ2t, νt).

sketch of the proof of Th. 1.6.10, (⇐). For the complete proof see, e.g [Sal08], Th. 8.1.

Let εn a sequence such that ε ↓ 0. Let us then dene

ϕXn(u) = exp

iub−

∫εn<|x|≤1

xν(dx)︸︷︷︸1

− u2σ2

2+

∫R

(eiux − 1

)ν(dx)︸︷︷︸

2

Then

ϕXn(u)→ ϕX(u), asn→∞

Heuristically the terms |x| in 1 and 2 are the size of the jumps. So integrals in 1 and 2 arerespectively measuring respectively jumps of size less than 1 and greater than 1.

We have now to prove that ϕX(u) is continuous at 0.

We will denote by Ψ(u) the characteristic exponential i.e. in equation (1.93) we haveϕX(u) = expΨ(u). We focus now in particular on the characteristic exponential linked tothe jump term. We denote then

Ψν(u) :=

∫R

(eiux − 1− iux1|x|≤1

)ν(dx) =

∫|x|≤1

(eiux − 1− iux

)ν(dx)︸︷︷︸

1

+

∫|x|>1

(eiux − 1

)ν(dx)︸︷︷︸

2


Exploiting now the denition as a convergent series of the exponential function i.e. ex =∑n≥0

xn

n! we have the rst that do not cancel in 1 is (iux)2 = −u2x2. Thus we have

|Ψν(u)| ≤∫|x|≤1

1

2|u2x2|ν(dx) +

∫|x|>1

∣∣eiux − 1∣∣ ν(dx)

Previous evaluation, together with the fact that∣∣eiux − 1

∣∣ ≤ 1, gives us an idea why we need

the condition∫R(1 ∧ x2

)ν(dx) <∞ in Th. 1.6.10.

Lemma 1.6.13. Let µkk be a sequence of innitely divisible laws such that µk → µ ask →∞, i.e. ∫

Rfµk(dx)→

∫Rfµ(dx) as k →∞, ∀f ∈ Cb(R) . (1.94)

Then µ is also innitely divisible.

Remark 1.6.14. It should be clear by now that there is a one-to-one correspondence betweenthe laws of a probability distribution and its characteristic function. Thus condition (1.94)is equivalent to ϕXk → ϕX .

Example 1.6.3. Let

Lt = bt+ σWt +

(Nt∑k=1

Jk − Cλt

)a Lévy process and let

ϕLt(u) = expΨ(u)

its characteristic function with characteristic exponent Ψ(u). Recalling equation (1.92) wehave

ΨLt(u) = iubt− 1

2σ2u2t+

∫R

(eiux − 1− iux

)ν(dx)t

with characteristic triplet given by (bt, σ2t, tν(dx)

)Theorem 1.6.15. Any Lévy process Lt, t ∈ [0, T ] has innitely divisible distribution withcharacteristic triplet given by (

bt, σ2t, ν(dx)t)

being (b, σ2, λν(dx)

)the characteristic triplet of L1.

Sketch of the proof. Let us dene

ϕLt(u) = EeiuLt = fu(t)

Let us notice that the original variable u is playing the role of a parameter in f being t thevariable. We can write then the Cauchy problem

fu(t+ h) = fu(t)fu(h),

fu(0) = 1(1.95)


By taking

fu(t+ h) = EeiuLt+h = Eeiu(Lt+h−Lt)EeiuLt = EeiuLhEeiuLt = fu(h)fu(t)

This is usually called semigroup approach. A totally equivalent approach is to exploit theFeynman-Kac theorem.

Basically we are saying that the distribution of a Lévy process is characterized by itscharacteristic function, which is given by the LévyKhinchine formula (1.6.10). If Lt is aLévy process, then its characteristic function is given by equation (1.93).

Therefore a Lévy process is characterized by three independent components, the char-acteristic triplet dened in 1.6.11 composed by a linear drift b, a diusion term σ2 and asuperposition of independent (centered) Poisson processes with dierent jump sizes. Herethe Lévy measure ν represents the intensity of the Poisson process with jump of size |x|.

1.6.5 Lévy-Ito decomposition

Denition 1.6.16 (Jump). Let us dene

∆Lt := Lt − Lt− , (1.96)

being Lt− := lims↑t Ls(ω) the limit from the right, the jump of the Lévy process at time t.Since we are considering the cadlag modication Lt− is well dened.

Recall that from Denition 1.6.1 (iii) we have that being a Lévy process stochasticcontinuous, for a xed t we have ∆Lt = 0 P-a.s.

Let us now consider t ∈ [0, T ], and a set A ∈ B (R \ 0) such that 0 6∈ A. We denethen

µL (ω; t, A) := ] s ∈ [0, t] : ∆Ls(ω) ∈ A =∑s≤t

1A (∆Ls) (1.97)

µL dened in (1.97) counts the jumps that are in the set A.Starting from this denition we can construct a random measure on B ([0, T ])⊗B (R \ 0).

Remark 1.6.17. We have called it random measure since if we x ω it is a measure whereasif we x t and A it is a random variable.

Let us then construct the measure µL (ω; [0, t], A). We will use in the following thenotation µL (ω; ds, dx).

If we x a time t, ∀A we have

1. µL(t, A) is Ft-measurable since we are considering s ≤ t;

2. let us consider u < t, thus

µL(t, A)− µL(u,A) =∑u<s<t

1A (∆Ls) 3 σ (Lb − La : u ≤ azb ≤ t)

where we have denoted by σ(·) the sigma algebra generated by ·. We thus have thatµL(t, A)− µL(u,A) is independent of Fs;


3.

µL(t, A)−µL(u,A)z:=s−u

=∑

0<z≤t−u

1A (∆ (Lz+u − Lu))d=

∑0<z≤t−u

1A (∆Lz) = µL(t−u,A)

Though 1 − 2 − 3 and together with equation (1.97) we have dened a stationary withindependents increments counting measure i.e. a Poisson measure. The intensity of such aPoisson measure is given by

EµL(t, A) = ν(A)t

Theorem 1.6.18. Let us consider

(i) ∫A

f(x)µL(t, dx) =

∫ t

0

∫A

f(x)µL(ds, dx), t ∈ [0, T ]

is a compound Poisson process with

ϕ·(u) = Eeiu∫Af(x)µL(t,dx) = exp

t

∫A

(eiuf(x) − 1

)ν(dx)

, u ∈ R ,

if f = 1 we have a Poisson process;

(ii) If f ∈ L1(A, ν)

E∫A

f(x)µL(t, dx) = t

∫A

f(x)ν(dx)

is the rst moment of the compounf Poisson process;

(iii) If f ∈ L2(A, ν)

Var[∣∣∣∣∫

A

f(x)µL(t, dx)

∣∣∣∣] = t

∫A

f(x)2ν(dx)

Theorem 1.6.19 (Lévy-Ito decomposition). Let(b, σ2, ν

)a the characteristic triplet such

that b ∈ R, σ2 ∈ R and ν a Borel measure on R such that ν (0) = 0 and∫R 1∧x2ν(dx) <∞.

Then it exists a probability space (Ω,F ,P) on which one can dene four independent processes

L(1) a deterministic linear drift;

L(2) a Brownian motion;

L(3) a compound Poisson process;

L(4) a square integrable martingale (known as pure jump process) with P-a.s. a countablenumber of jumps of size less than 1 on each nite interval.


Therefore the processL := L(1) + L(2) + L(3) + L(4)

is a Lévy process with characteristic exponent given by

Ψν(u) := iub︸︷︷︸L(1)

− σ2u2

2︸︷︷︸L(2)

+

∫|x|>1

(eiux − 1

)ν(dx)︸︷︷︸

L(3)

+

∫|x|≤1

(eiux − 1− iux

)ν(dx)︸︷︷︸

L(4)

Proof. See, e.g. [Sat99], Ch. 4.

The term L(1) and L(2) are the same of standard stochastic dierential equation drivenby a Brownian motion whereas L(3) is a compound Poisson process with jumps greater thanwithout compensator while L(4) is a compound Poisson process with jumps less than 1 withcompensator. Using the counting measure µL introduced in (1.97) we can dene a generalLévy process with characteristic triplet

(b, σ2, ν

)as

Lt = bt+ σWt +

∫ t

0

∫|x|>1

xµL(ds, dx) +

∫ t

0

∫|x|≤1


)(1.98)

1.6.6 Lévy measure, nite variation, moments and jumps

Finite activity Vs. inite activity

Let us rstly focus on the Lévy measure ν and on its property∫R 1 ∧ x2ν(dx) < ∞. In

particular for big jumps, i.e. bigger than 1, I have that the measure ν is nite. Therefore Iam sure ν(R) < ∞ and thus P-a.s. the trajectories have a nite numbers of jumps on anycompact interval. Problems arise for small jumps. In fact for jumps smaller that 1 I do notknow anymore whether ν is nite or innite. If it is nite then I reduce to the previous casead I do not have problems. Conversely if I am in the case ν(R) =∞ I have to check if thesum of all jumps diverges or converges. Thus a Lévy process whose jumps can explode iscalled innite activity Lévy process whereas if the sum is always convergent it is known asnite activity Lévy process. Whether we are in the case of nite activity of innite activityis decided by jumps of magnitude less than 1.

Bounded variation

We start by recalling the denition of variation

Denition 1.6.20 (Total variation). The total variation of a function f : [a, b]→ R is thequantity

V ba (f) := supπ

n−1∑i=0

|f(xi+1)− f(xi)|

where the supremum is taken over all the partitions π of the interval [a, b].

Denition 1.6.21 (Bounded variation). Let f : [a, b]→ R be a function and we will denoteby BV ([a, b]) the function of bounded variation on the interval [a, b]. Then

f ∈ BV ([a, b]) ⇔ V ba (f) <∞


Again, as for the previous case is the small jumps part L(4) that can make the variationunbounded. Of course we are talking about the case of σ2 = 0 otherwise we already knowLt is of unbounded variation due to the Brownian motion.

Therefore we require σ2 = 0 together with∫|x|≤1

|x|ν(dx) < ∞ in order to have P-a.s.nite variation trajectories.

Moments

For the moments of a Lévy process problems are caused by big jumps, i.e. it is the L(3) thatcan make the moments explode.

Proposition 1.6.22. (i)

∀t E|Lt|p <∞ ⇔∫|x|≥1

|x|pν(dx) <∞ ; (1.99)

(ii)

∀t EepLt <∞ ⇔∫|x|≥1

epxν(dx) <∞ ; (1.100)

Proof. See, e.g. [Sat99] Th. 21.3,21.9,25.3.

If a have a nite rst moment, i.e.∫|x|≥1

|x|ν(dx) <∞

I can modify L(3) by adding and subtracting∫|x|≥1

xν(dx) in order to rewrite equation (1.98)as

Lt = bt+ σWt +

∫ t

0

∫Rx(µL(ds, dx)− ν(dx)ds

), b := b+

∫|x|>1

xν(dx) (1.101)

1.6.7 Semi-martingales

See as a general reference [Pro90], chapter 2 or [?].We will in the following assume the existence of the rst moment of the process Lt. Let

Mt be a martingale, therefore we have that Mt −M2t is a submartingale. It is natural to

ask whether it exists a process At such that the process

Nt := M2t −At

is again a martingale. One can also turn the problem into looking for a process

M2t = At +Nt

Denition 1.6.23 (Semi-martingale). Let Xt be a process. Let us suppose that Xt can bewritten as

Xt = X0 +At +Mt

with At a process of nite variation s.t. A0 = 0 and Mt a martingale s.t. M0 = 0. Then Xt

is said to be a semi-martingale.Further if the process At is also predictable then Xt is called a special semi-martingale.


Of course a Lévy process is a semi-martingale in fact recalling equation (1.101) we candecompose a Lévy process Lt as

Lt = bt︸︷︷︸At

+σWt +

∫ t

0

∫Rx(µL(ds, dx)− ν(dx)ds

)︸︷︷︸

Mt

Proposition 1.6.24. A Lévy process Lt is

(i) a martingale ⇔ b = 0;

(ii) a sub-martingale ⇔ b > 0;

(iii) a super-martingale ⇔ b < 0;

Proposition 1.6.25. Let us suppose∫|x|≥1

euxν(dx) < ∞, u ∈ R. Then from proposition

1.6.22 (ii) we have that

Mt :=euLt

EeuLtis well dened and it is a martingale.

sketch of the proof. From the standard properties of the conditional expectation we can seethat since Lt |= Ft we have that EeuLt = EeuLt |Ft and exploiting the tower rule we have

EEeuX |Fs |Ft = EeuX |Fs = EEeuX |Ft |Fs , s < t

Exploiting then Levy-Ito decomposition and from standard properties of the conditionalexpectation we have

EMt |Fs = Eeu(At+Mt)

Eeu(At+Mt)|Fs =

eu(As+Ms)E∫ tsAudu

Eeu(As+Ms)E∫ tsAudu

=eu(As+Ms)

Eeu(As+Ms)= Ms

And we are done.

1.6.8 Girsanov theorem

When we change the measure we can have the problem that any property linked to thedistribution may fall. In particular recall that in denition 1.6.1, properties (i) − (iii)depends on the distribution. Thus if we change the measure P into Q we may not have aLévy process anymore.

Theorem 1.6.26 (Girsanov theorem). Let be Xt a semi-martingale with standard decompo-sition as in denition 1.6.23 Xt = At +Mt under the probability measure P. Let us furtherconsider the process

Zt = exp

∫ t

0

βsσdWs −1

2

∫ t

0

β2sσ

2ds+

∫ t

0

∫R

(Y (s, x)− 1)(µL(dsdx)− ν(dx)ds

)+

+

∫ t

0

∫R

(Y (s, x)− 1− log Y (s, x))µL(dsdx)

, t ∈ [0, T ]


with∫ t

0

σ2β2sds <∞ P− a.s. and

∫ t

0

∫R|x(Y (s, x)− 1)| ν(dx)ds <∞ P− a.s.

If Zt is a martingale then we have that dQ = ZT dP and

WQt := Wt −

∫ t

0

βsσds

is a Q Brownian motion, and

νQ(dxds) = Y (s, x)ν(dx)ds (1.102)

is the compensator under Q of µL. Then we have

Lt = bQt+ σWQt +

∫ t

0

∫Rx(µL − νQ)(dsdx)

with

bQt := bt+

∫ t

0

σβsds+

∫ t

0

∫Rx (Y (s, x)− 1) ν(dx)ds

The problem arise in equation (1.102) where we have νQ(dsdx). This means that thetime can inuence the size of the jump so that the process is not stationary anymore (evenif it is still of independents increments). I need the Y (s, x) to be independent of the time t,i.e. Y (s, x) = Y (x).

In the case of stochastic dierential equations driven by Lévy noise I cannot have com-pleteness and thus multiple possible measure Q. Therefore the choice of the "best" Q is farfrom being an easy task (and it depend of our goal as well!).

Example 1.6.4 (Essecher transform). Let

Zt = expβσWt+

∫ t

0

∫Rαx(µL(dsdx)− ν(dx)ds

)−σ

2β2

2−∫R

(eαx − 1− αx) ν(dx), β > 0, α ∈ R

Then

Zt =eβσWteα

∫ t0

∫R x(µ

L(dsdx)−ν(dx)ds)

EeβσWtEeα∫ t0

∫R x(µL(dsdx)−ν(dx)ds)

and it is a martingale.

Proof. For the proof of the martingale part see proposition 1.6.25.

1.6.9 Ito formula and Feynman-Kac theorem

As for the case of the Brownian motion we have Ito formula for a SDE driven by a Lévynoise.


Theorem 1.6.27 (Ito formula). Let Xt be a semi-martingale and f ∈ C2(R). Then wehave

f(Xt) = f(X0) +

∫ t

0

f ′(Xs−)dXs +1

2

∫ t

0

f ′′(Xs−)d 〈Xc〉s +∑s≤t

∆f(Xs)=f(Xs−+∆Xs)−f(Xs− )︷︸︸︷f(Xs)− f(Xs−) −f ′(Xs−)∆Xs

=

=f(X0) +

∫ t

0

f ′(Xs−)dXs +1

2

∫ t

0

f ′′(Xs−)d 〈Xc〉s +

∫ t

0

∫R

[f(Xs+x)− f(Xs−)− f ′(Xs−)x] ν(dx)ds

(1.103)

with

Xt = X0 +At +Mt = X0 +Xct +Xj

t

with Xct the continuous part and Xj

t the jump part.

As done previously in section ?? we can exploit Ito's lemma in order to retrieve a deter-ministic problem, i.e. the Feynman-Kac theorem. We will as said only consider the case ofnite rst moment.

Theorem 1.6.28 (Discounted Feynman-Kac formula for jump diusion problem). Let usconsider the process Xt satisfying

Xt = bt+ σ2Wt +

∫ t

0

∫Rx(µL(dsdx)− ν(dx)ds

)a general Lévy process and let Φ be a Borel measurable function. Dene then the function

u(t, x) := Ee−∫ Ttr(s,Xs)dsΦ (XT )

∣∣∣Xt = x, x ∈ R, t ∈ [0, T ] . (1.104)

Then u(t, x) is a solution of the following partial integral dierential equation (PIDE)∂tu(t, x) + b(t, x)∂xu(t, x) + σ2(t,x)

2 ∂xxu(t, x) +∫R [u(x+ y)− u(x)− yux(t, x)] ν(dy)− r(t, x)u(t, x) = 0 ,

u(T, x) = Φ(x) .

(1.105)

The proof of the Feynman-Kac theorem in the case of jump diusion processes proceedexactly as for the case of continuous diusion processes.

1.7 Introduction to innite dimensional analysis

We present in this section the theory of measure, and in particular the Gaussian measure,in a general innite dimensional separable Hilbert space.

The present section is so structured, following mainly the approach given in [DP06,DPZ02] we will develop the theory of Gaussian measure in nite dimension and then we willgeneralize it to the case of a innite dimension Hilbert space. Eventually we will state somepreliminary result on possible application to solve equations of the from (??).

1.7 Introduction to innite dimensional analysis 67

1.7.1 Gaussian measures

Throughout the section we will consider a real separable Hilbert space H endowed with thescalar product 〈·, ·〉 and the norm | · |. We will further denote by L(H) the space of linearand bounded operator from H into H equipped with the standard operator norm denotedby ‖ · ‖, by L1(H) the space of trace class operator and by L2(H) the Hilbert-Schmidtoperators, see Sec. 5.2 for details. Further let us denote L+(H) the set of all linear andbounded operator which are symmetric and non negative, i.e.

L+(H) := A ∈ L(H) : 〈Ax, y〉 = 〈x,Ay〉 , 〈Ax, x〉 ≥ 0, x, y ∈ H . (1.106)

Let us notice that A ∈ L+(H) implies A is self-adjoint.

One-dimensional Gaussian measure

Let us consider a couple of real numbers a ∈ R and λ ≥ 0. For any couple of (a, λ) we candene the measure Na,λ in (R,B(R)) as follows: if λ = 0 we dene

Na = Na,0 := δa ,

the Dirac mass at a; if λ > 0, we dene

Na,λ(B) :=1√2πλ

∫B

e−(x−a)2

2λ dx; .

then Na,λ is absolutely continuous w.r.t. the Lebesgue measure, see, e.g., [Rud74] for details,and it admits density given by

Na,λ(dx) :=1√2πλ

e−(x−a)2

2λ dx; .

It can be easily checked that Na,λ is in fact a probability measure since Na,λ(R) = 1.Therefore it makes sense to dene a mean, a covariance and a characteristic function.

Proposition 1.7.1 (Prop. 1.2 [DP06]). Let a ∈ R and λ > 0. Then we have∫RxNa,λ(dx) = a,∫

R(x− a)2Na,λ(dx) = λ,

ˆNa,λ(ξ) :=

∫ReiξxNa,λ(dx) = eiξa−

12λξ

2

, ξ ∈ R ,

(1.107)

where a is called the mean, λ the covariance and ˆNa,λ(ξ) the characteristic function on Na,λ.

These three quantities are fundamental because they dene uniquely a Gaussian measurein the sense that given the mean a and the covariance λ we can determine uniquely aGaussian measure. Furthermore any other measure µ whose characteristic function is givenin eq. (1.113), is a Gaussian measure.


d-dimensional Gaussian measure

Let now dimH = d < ∞ so that H ' Rd. Let a ∈ R, Q ∈ L+(H) and be (e1, . . . , ed) anorthonormal basis of H so that Qek = λkek, k = 1, . . . , d, for some λk ≥ 0. Being an Hilbertspace, dening any element x ∈ H admits the so called Fourier representation given by

x =

d∑k=1

xkek :=

d∑k=1

〈x, ek〉 ek ,

where we have dened xk := 〈x, ek〉. We can dene a probability measureNa,Q in(Rd,B(Rd)

)setting

Na,Q =d¡

k=1

Nak,λk , ak := 〈a, ek〉 . (1.108)

As for the one-dimensional case, we will characterize the Gaussian measure by means ofits mean, its covariance matrix and its characteristic function.

Proposition 1.7.2 (Prop. 1.3 [DP06]). Let a ∈ H and Q ∈ L+(H) and µ = Na,Q. Thenwe have ∫

H

xNa,Q(dx) = a,∫H

〈y, x− a〉〈y, x− a〉Na,Q(dx) = 〈Qy, z〉 , y, z ∈ H

ˆNa,Q(ξ) :=

∫H

ei〈x,ξ〉Na,Q(dx) = ei〈ξ,a〉−12 〈Qξ,ξ〉, ξ ∈ H ,

where a is called the mean, Q the covariance operator and ˆNa,λ(ξ) the characteristic functionon Na,λ.

Let us notice that requiring the covariance operator Q to be non negative does notguarantee us its eigenvalues to be strictly positive. If it is the case then we have that, beingthe determinant of Q dierent from 0, the Gaussian measure Na,Q is absolutely continuousw.r.t. the Lebesgue measure and its density is given by

Na,Q(dx) :=1√

(2π)ddetQe−

12 〈Q−1(x−a),x−a〉dx . (1.109)

In the case of Q has at least one null eigenvalue λk = 0, we have that the density (1.109)cannot be dened, then the corresponding k−th projected Gaussian measure Nak,λk in thedecomposition of the Gaussian measure Na,Q in eq. (1.108) has to be intended as the Diracmass centered at a, i.e.

Nak,λk = Na,0 =: δa . (1.110)

It can be easily that in the case of d = 1, Prop. 1.7.2 reduce to Prop. 1.7.1. Furthermorewe still have that these three quantity uniquely determine the Gaussian measure in the sensegiven before.


Innite dimension Gaussian measure

We are going in the present section to show that a Gaussian measure µ := Na,Q is naturalextension of Prop. 1.7.2 for d =∞ with a ∈ H and Q ∈ L+

1 (H) := L1(H)∩L+(H). We willfurther show that such a measure is unique. In particular, since we require Q ∈ L+

1 (H), itexists an orthonormal basis (ek)k on H and a sequence of non-negative numbers (λk)k suchthat

Qek = λkek .

Furthermore any element x ∈ H admits Fourier representation and it exists a naturalisomorphism between l2 and H so that we will identify H with l2. We shall thus dene theproduct measure

µ :=∞¡

k=1

Nak,λk , ak := 〈a, ek〉 . (1.111)

The measure µ is actually dened on

R∞ :=∞¡

k=1

R.

Let us rst focus on a general measure probability measure µ on (H,B(H)), H will be aninnite dimensional separable Hilbert space and (ek)k an orthonormal system in H. Asbriey mentioned before any element x ∈ H can be written as an innite series, calledFourier representation,

x =∑k

〈x, ek〉 ek ;

we can thus dene the following.

Denition 1.7.3 (Projection map). For any n ∈ N let us dene the projection map Pn :H → Pn(H) as

Pnx =

n∑k=1

〈x, ek〉 ek, x ∈ H .

It obviously holds that limn→∞ Pnx = x.

We are now to dene, as done before, the mean and the covariance of a general probabilitymeasure µ. Let us notice that being now H innite dimensional previous denition does nothold in general.

Denition 1.7.4 (Def. 1.1,1.2,1.5 [DP07]). Let µ be a probability measure on (H,B(H))such that ∫

H

|x|2µ(dx) <∞ , (1.112)

then we dene the mean a ∈ H of µ by

〈a, h〉 =

∫H

〈x, h〉µ(dx), h ∈ H , (1.113)

and the covariance operator Q ∈ L(H) of µ by

〈Qh, k〉 =

∫H

〈x− a, h〉〈x− a, k〉µ(dx), ∀h, k ∈ H . (1.114)


Furthermore we can dene the characteristic function µ of µ by

µ(h) :=

∫H

ei〈h,x〉µ(dx), h ∈ H . (1.115)

Remark 1.7.5. The mean and the characteristic function can be dened even for a lessrestrictive condition than eq. (1.112), it is in fact enough to require∫

H

xµ(dx) <∞ .

Furthermore it can be shown that, as in the nite dimensional case, µ uniquely determinesthe measure µ.

A priori Def. 1.7.4 or to be precise the mean a and the covariance operator Q, is notwell posed. Let us notice that the functional Ta : H → R dened by

H 〈Ta, h〉H :=

∫H

〈x, h〉µ(dx), h ∈ H

is linear and continuous; in fact it is an integral, thus the linearity holds, and furthermore

|H 〈Ta, h〉H | ≤∫H

|x|µ(dx)|h| <∞, h ∈ H .

Then by Riesz representation theorem it exists a unique element a ∈ H so that eq. (1.7.4)is well dened. The same holds for the covariance operator Q.

Proposition 1.7.6 (Prop. 1.8 [DP06]). Let be µ a probability measure such that condition(1.112) holds. Then Denition 1.7.4 apply so that we can dene both the mean a and thecovariance operator Q. Then Q ∈ L+

1 (H), i.e. Q is symmetric, positive and of trace class.

We are now ready to introduce the Gaussian measure on an innite dimensional Hilbertspace H and to show that the Gaussian measure dened in eq. (1.111) is in fact a Gaussianmeasure on H ' l2 with mean a and covariance Q.

Theorem 1.7.7 (Th. 1.2.1 [DPZ02]). Let be a ∈ H and Q ∈ L+1 (H). Then it exists a

unique probability measure µ on (H,B(H)) such that∫H

ei〈h,x〉µ(dx) = ei〈a,h〉−12 〈Qh,h〉. h ∈ H . (1.116)

Moreover µ is the restriction to l2 of the measure dened as in eq. (1.111) dened on(R∞,B(R∞)).

We thus set µ := Na,Q and as usual we will call a the mean and Q the covarianceoperator.

Proof. We recall that a standard result in probability theory, see, e.g. [Pro90], the charac-teristic function uniquely determines the measure. So we have only to prove the existenceof eq. (1.116).

Let us rst notice that being µk := Nak,λk a Gaussian measure on R, the product measureµ :=∞

k=1Nak,λk is a product measure dened on R∞, see, e.g. [DP06] for general resultson product probability measures.


We rst of all prove that µ is concentrated on l2, i.e. µ(l2) = 1.From the monotone convergence theorem it follows that∫

R∞|x|2l2µ(dx) =

∫R∞

∞∑k=1

x2kµ(dx) =

∞∑k=1

∫Rx2kNak,λk(dxk) =

=

∞∑k=1

(∫R

(xk − ak)2Nak,λk(dxk) + a2k

)=

∞∑k=1

(λk + a2k) = TrQ+ |a|2 <∞ .

Therefore we have thatµ(x ∈ R∞ : |x|2l2 <∞

)= 1 ,

and we are done.We can now consider the restriction of µ to l2 that we will still denote by µ. We want

now to prove the existence of eq. (1.116). In fact, for any h ∈ H we have∫l2

ei〈h,x〉µ(dx) = limn→∞

∫l2

ei〈Pnh,Pnx〉µ(dx) = limn→∞

n∏k=1

∫ReixkhkNak,λk(dxk) =

= limn→∞

n∏k=1

eiakhk−12λkh

2k = lim

n→∞ei〈Pna,h〉−

12 〈PnQh,h〉 = ei〈a,h〉−

12 〈Qh,h〉 .

A Gaussian measure Na,Q is said to be non degenerate if KerQ = 0. As for thed−dimensional case, since we do not require the Gaussian measure to be non degenerate,the covariance operator Q can admit null eigenvalues. If the k−th eigenvalue λk = 0, thenthe Gaussian measure Nak,λk in eq. (1.111) is dened to be the Dirac mass centered ata according to eq. (1.110). In fact let us notice that the characteristic function for theGaussian measure in eq. (1.116) makes sense even in the case of Q to admit some nulleigenvalues.

Let now (Ω,F ,P) a probability space and as usual H a separable innite dimensionalHilbert space. Let be X : (Ω,F) → (H,B(H)) a random variable. We will say that X is aGaussian random variable if the law of X is a Gaussian measure, see, e.g. [Pro90] for detailson random variables. According to Th. 1.7.7 we have that X is Gaussian if and only if forany h ∈ H the real valued random variable 〈h,X(ω)〉 is Gaussian.

Let us now dene by L2(Ω,F ,P;H) the space of square integrable random variables

L2(Ω,F ,P;H) :=

X : Ω→ H :

∫Ω

|X(ω)|2P(dω) <∞.

It is known from a standard result in probability theory that∫Ω

〈X(ω), h〉P(dω) =

∫H

〈x, h〉µ(x)

with x ∈ H and µ the law of the random variable X (sometimes the law is denoted byP#X).

Let us now consider the space L2(H,Na,Q), then from Th. 1.7.7 easily follows thefollowing characterization of mean and covariance operator for a random variable in H.


Proposition 1.7.8 (Prop. 1.2.4 [DPZ02]). Let a ∈ H and Q ∈ L+1 (H). Then we have∫

H

〈x, h〉Na,Q(dx) = 〈a, h〉 , ∀h ∈ H∫H

〈h, x− a〉〈k, x− a〉Na,Q(dx) = 〈Qh, k〉 , ∀h, k ∈ H ,∫H

|x− a|2Na,Q(dx) = TrQ ,

where a is called the mean and Q the covariance operator.

Proof. Use Th. 1.7.7 for the n − th projector Pn. Taking the limit we have the desiredresult.

1.7.2 Gaussian random variables and Brownian motion

Let (Ω,F , (Ft)t ,P) be a probability space and let us consider Q ∈ L+1 (H). We will further

consider the space L2(H,B(H), Na,Q) and two Hilbert spaces H and U . Any other notationis as before.

Let us dene the operator Q12 : H → H as

Q12x :=

∞∑k=1

λ12 〈x, ek〉 ek, x ∈ H . (1.117)

The subspace Q12 (H) is called the reproducing kernel or the Cameron-Martin space of NQ.

Proposition 1.7.9. Let be Q12 dened as in eq. (1.117). Let us assume further that KerQ =

0 Then Q 12 (H)

d⊂ H, where the inclusion is strict. Furthermore we have that NQ(Q

12 ) = 0

Proof. If x0 ∈ H is such that⟨Q

12x, x0

⟩= 0 for all x ∈ H, we have that Q

12x0 = 0 implies

Qx0 = 0 and therefore x0 = 0. The hypothesis follows from the fact that KerQ = 0.

In the nite dimensional case the following formula, known as the Cameron-Martinformula holds. Let us consider the measures NQ and Na,Q with a ∈ Q 1

2 . Then we can provethat NQ ∼ Na,Q with density given by

dNa,QdNQ

(x) = e− 1

2 |Q− 1

2 a|2+⟨Q−

12 a,Q−

12 x

⟩, x ∈ H ; . (1.118)

As said previous formula holds only in the nite dimensional case since⟨Q−

12 a,Q−

12x⟩

makes sense only for x ∈ Q 12 , which by Prop. 5.3.8 is a zero NQ−measure set. To give a

proper meaning to formula (1.118) we have to introduce the white noise mapping W .Let us assume from now on that KerQ = 0. We will now dene the isomorphism W

dened as follows.

Denition 1.7.10. Let be f ∈ H. Let us then dene the map W : H → L2(H,NQ),f 7→Wf , setting

Wf (x) :=⟨Q

12 f, x

⟩, x ∈ H . (1.119)


Let us notice that eq. (1.119) makes sense only for f ∈ Q 12 . Since we have proven in

5.3.8 Q12 (H)

d⊂ H, we can dene the map W : Q

12 → L2(H,NQ) as dened in eq. (1.119).

Exploiting then Prop. 1.7.8 we have that∫H

Wf (x)Wg(x)NQ(dx) = 〈f, g〉 , f, g,∈ Q 12 ,

and therefore W is an isometry so that it can be uniquely extended to all the space H. Wewill thus denote by W the extension to the whole space H. W is called the white noisemapping.

Proposition 1.7.11. Let be n ∈ N and let be f1, . . . , fn ∈ H. Let us set

W (x) := (Wf1(x), . . . ,Wfn(x)), x ∈ H .

Thus we have that W (x) is a Gaussian (vector) random variable in (Rn,B(Rn)) with 0 meanand covariance operator given by

QW = 〈fi, fj〉i,j=1,...,n .

Furthermore the real random variablesWf1 , . . . ,Wfn are independent if and only if f1, . . . , fnis orthogonal.

Proof. We will follow the proof given in [DP06]. Let us consider the projection

PNx =

N∑k=1

〈x, ek〉 ek, x ∈ H,N ∈ N ,

since we have already proven that W is an isomorphism we have

limN→∞

(WPNf1 , . . . ,WPNfn) = (Wf1 , . . . ,Wfn) .

Since we have that (WPNf1 , . . . ,WPNfn) ∼ NQ′N , with Q′N = (〈PNfi, PNfj〉)i,j=1,...,n, wehave that (Wf1 , . . . ,Wfn) ∼ NQ∗ with Q∗ = (〈fi, fj〉)i,j=1,...,n.

Proposition 1.7.12 (Prop. 1.30 [DP06]). The mapping

H → L2(H,NQ), f 7→ eWf

is continuous in f .

We can now generalize the Cameron-Martin formula (1.118) to the innite dimensionalcase as follows.

Theorem 1.7.13 (Th. 1.4 [DP04]). Let Q ∈ L+1 (H) and a ∈ Q 1

2 . Then the measures Na,Qand NQ are equivalent and

dNa,QdNQ

(x) = e− 1

2 |Q− 1

2 a|2+⟨Q−

12 a,Q−

12 x

⟩, x ∈ H ; .

If a 6∈ Q 12 , then Na,Q and NQ are singular.


Let us notice that the term⟨Q−

12 a,Q−

12x⟩should be intended as W

Q−12 a

(x).

We are now ready to nally construct the Brownian motion. Let be (Ω,F ,P) a probabilityspace and H and U two Hilbert spaces. Let furthermore be Q ∈mathcalL+

1 (U)

Denition 1.7.14 (Q-Wiener process). A family of U -valued random variables is calledQ-Wiener process if and only of

(i) W0 = 0 and Wt −Ws ∼ N ()0, (t− s)Q;

(ii) if 0 < t1 < · · · < tn, then Wt1 ,Wt2 −Wt1 , . . . ,Wtn −Wtn−1 are independent randomvariables;

(iii) for almost all ω ∈ Ω Wt(ω) is continuous.

In particular we have

E〈Wt, a〉UE〈Ws, b〉U = t ∧ s 〈Qa, b〉U , a, b ∈ U .

Let now ekk be a sequence of eigenfunctions corresponding to the eigenvalues λkk.If λk > 0, then

βk(t) =1√λk〈Wt, ek〉U , t ≥ 0 ,

are one-dimensional standard Wiener process mutually independent. It therefore holds that

Wt =

∞∑k=1

√λkβk(t)ek ,

see, e.g. [DPZ08] Ch. 4 fro more details on Q-Wiener processes.We are now going to prove the existence of a Brownian motion on the probability space

(L2(R+), NQ) where Q ∈ L+1 (H) non degenerate, i.e. KerQ = 0.

Theorem 1.7.15 (Th. 1.4.1 [DPZ02]). Let us set B0 = 0 and Bt = W1[0,t] , t ≥ 0 and1[0,t] the characteristic function of the interval [0, t]. Then B has a modication which isa real Brownian motion on (L2(R+), NQ).

Proof. See, e.g. [DPZ02].

1.7.3 Markov semigroup

We will now state some denition that will be used later on.

Denition 1.7.16 (Probability kernel). Let H be a Hilbert space, a probability kernel p onH is a mapping

[0,+∞]×H → [0, 1], (t, x) 7→ pt,x

such that

(i) pt+s,x(A) =∫Hps,y(A)pt,x(dy), for all t, s ≥ 0, x ∈ H, A ∈ B(H);

(ii) px(A) := p0,x(A) = 1A(x), for all x ∈ H, A ∈ B(H).


Given a probability kernel we can dene a semigroup of linear operators Pt on the spaceof Borel bounded functions Bb(H).

Denition 1.7.17 (Markov semigroup). Let be H a Hilbert space and Bb(H) the space ofbounded and Borel functions, then any probability kernel pt,x denes a markov semigroupPt by

Ptϕ(x) =

∫H

pt,x(dy)ϕ(dy), t ≥ 0, x ∈ H,ϕ ∈ Bb(H) ;

the semigroup Pt is called markov semigroup.A markov semigroup Pt is said to be:

Feller ϕ ∈ Cb(H) ⇒ Ptϕ ∈ Cb(H), for all t ≥ 0;

strong Feller ϕ ∈ Bb(H) ⇒ Ptϕ ∈ Cb(H), for all t ≥ 0;

Denition 1.7.18 (Invariant measure). A probability measure µ inH is said to be invariantfor the Markov semigroup Pt if∫

H

Ptϕdµ =

∫H

ϕdµ, ∀t > 0, ϕ ∈ Cb(H) . (1.120)

Denition 1.7.19 (Ergodic measure). Let be Pt a Markov semigroup and µ and invariantmeasure for Pt. Let us set

M(T )ϕ(x) =1

T

∫ T

0

Psϕ(x)ds, ϕ ∈ L2(H,µ), x ∈ H .

Then an invariant measure µ is said to be ergodic if

limT→∞

1

T

∫ T

0

Ptϕdt = M∞ϕ = ϕ, ϕ ∈ L2(H,µ) ,

where ϕ is the mean of ϕ

ϕ =

∫h

ϕ(x)µ(dx) .

1.7.4 Heat equation

We can now proper develop a theory for parabolic equation on Hilbert spaces. We willfocus in particular on the Heat equation, for a complete treatment we refer to [DPZ02]. Allnotation and assumptions on H are as before. Let us thus consider the following

Dtu(x, t) = 12Tr

[Q(x)D2u(x, t)

], x ∈ H, t ≥ 0,

u(x, 0) = u0(x), x ∈ H ,(1.121)

with, Q ∈ L+(H) and u0 ∈ UCb(H) the space of function F : H → R that are bounded anduniformly continuous. Furthermore D denotes the Frechet derivative.


It is a well know established result, that in the case of H being a nite dimensionalHilbert space, a solution to eq. (1.121) is given by

u(x, t) =

∫H

u0(x+ y)fNtQ(y)dy, x ∈ H, t ≥ 0 , (1.122)

where we have denoted by fNtQ the density of the Gaussian measure with mean equal to 0and covariance tQ.

Unfortunately solution (1.122) does hold in the general case of a innite dimensionalHilbert space since it does not exist a counterpart in innite dimensional of the Lebesguemeasure.

We will now state some results that claries why we assumed certain properties of theoperator Q.

Proposition 1.7.20 (Prop. 3.1.2 [DPZ02]). Assume that u0 ∈ Cb(H) and lim|y|→∞ u0(y) =0. If TrQ = ∞ and Qnn is a sequence of nite rank nonnegative operators convergingstrongly to Q, then limn→∞ un(t, x) = 0, for all t > 0 and x ∈ H.

The main idea here is to construct a sequence unn, solution toDtun(x, t) = 1

2Tr[Qn(x)D2un(x, t)

], x ∈ H, t ≥ 0,

un(x, 0) = u0(x), x ∈ H ,(1.123)

with Qn a nite rank operator strongly converging to Q. Then for each n a solution to eq.(1.123) can be achieved, using eq. (1.122) to be

un(t, x) =1√

(2tπ)nλ1 . . . λn

∫Rne− 1

2t

∑ni=1

(xi−yi)2

λi u0(y1, . . . , ym, xm+1, . . . )dy1 . . . dym ,

where we have denoted by (λ1, . . . , λn) the eigenvalues of the nite rank operator Qn. If thesequence unn of solutions were convergent, then its limit could be taken as a candidate forthe solution to eq. (1.121). However Prop. 1.7.20 tells us that Q ∈ L+(H) is not enough,since if TrQ =∞ then eq. (1.121) does not have a continuous solution. Therefore we needa stronger assumption on the operator Q, namely we have to assume Q ∈ L+

1 (H).Then, assuming Q ∈ L+

1 (H) and u0 ∈ Cb(H) we can exploit previous idea, so thatsolutions to eq. (1.123) is given by

un(x, t) =

∫H

u0(x+ y)NtQn(dy), x ∈ H, t ≥ 0 ,

with NtQn the Gaussian measure dened in Sec. 1.7.1.Thus if we now assume TrQ <∞, we have that un → u as n→∞, where u is given by

u(x, t) =

∫H

u0(x+ y)NtQ(dy), x ∈ H, t ≥ 0 .

Furthermore we can dene the operator Pt as

Ptu0(x) =

∫H

u0(x+ y)NtQ(dy), x ∈ H,uo ∈ Cb(H) ,

that forms a strongly continuous semigroup, see, e.g. [DPZ08, EN00] for details on semigrouptheory, in UCb(h). The semigroup Pt is called the heat semigroup.

1.8 Stochastic partial dierential equations 77

Remark 1.7.21. We are mainly interested in the heat eq. (1.121) for its connection with the(innite dimensional) stochastic dierential equation of the form

dX(t, x) = dW (t), t > 0, x ∈ H ,

see, e.g. [DPZ08, DPZ02, DP04] for details on innite dimensional stochastic dierentialequations.

1.8 Stochastic partial dierential equations

Let us consider the following diusion equationdX(t, ξ)−∆ξX(t, ξ)dt = f(t, ξ)dt+ dW (t, ξ) , in [0, T ]×O ,X(0, ξ) = x , in O ,X(t, ξ) = 0 , on [0, T ]× ∂O ,

(1.124)

where O ⊂ Rn is a bounded subset of Rn and we have denoted by ∂O its domain. Further-more we have

W (t, ξ) =

∞∑j=1

µjej(ξ)βj(ξ) ,

where ej is an orthonormal system in L2(O) taken as

−∆ej = λjej , in O ,ej |∂O = 0 ,

and βj is an independent system of Brownian motions in a probability space (Ω,F , (Ft)t ,P).Throughout the section we will assume

∞∑j=1

µ2j <∞ .

The process X(t, ξ) is said to be an L2(O) stochastic process on [0, T ] if for each j,t 7→

∫OX(t)ejdξ is a real valued stochastic process. Also X is said to be a solution to

(1.124) if for t ≥ 0 and for any j it holds

〈X(t), ej〉+ λj

∫OX(t)ejdξ = 〈x, ej〉+ 〈f(t), ej〉+

∫ t

0

µjdβj(s) ,

where we have denoted by 〈·, ·〉 the standard scalar product in L2(O)

Theorem 1.8.1. Let x ∈ L2(O), then there is a unique solution X to (1.124) which isL2(O)−continuous P−a.s. Moreover it holds that X ∈ L2(Ω;C([0, T ];L2(O))) and we have

E supt∈[0,T ]

|X(t)|2L2(O+E∫ T

0

|∇X(s)|2L2(Ods =1

2

∞∑j=1

µ2j t+

1

2|x|2L2(O+

∫ T

0

〈f(s), X(s)〉L2(O ds .

(1.125)


Proof. We will use the Fourier method. For each j the equationdXj(t) = −λjXj(t)dt+ µjdβj(t)〈f, ej〉dt , t ∈ [0, T ] ,

Xj(0) = 〈x, ej〉 ,(1.126)

admits a unique solution Xj . Applying Itô's formula we thus get

1

2d|Xj |2 + λj |Xj |2 =

1

2|µj |2dt+ 〈f,Xj〉 ,

which implies, that

X(t, ξ))

∞∑j=1

Xj(t)ej(ξ) ,

is a solution to eq. (1.124) and from Itô's formula we have that the estimate (1.125) holds.The uniqueness comes from Itô's formula.

Remark 1.8.2. Similarly we can show the existence and uniqueness of a solution to thediusion equation (1.124) with multiplicative noise, that is

dX(t, ξ)−∆ξX(t, ξ)dt = f(t, ξ)dt+∑∞j=1 µjXejdβj , in [0, T ]×O ,

X(0, ξ) = x , in O ,X(t, ξ) = 0 , on [0, T ]× ∂O .

(1.127)

The proof is similar to the one in Th. 1.8.1 with the same dierence that eq. (1.126) readsdXj(t) = −λjXj(t)dt+ µjXjdβj(t)〈f, ej〉dt , t ∈ [0, T ] ,

Xj(0) = 〈x, ej〉 ,

so that we have

Xj(t) = Xj(0)exp

λjt−

1

2µ2j t+

∫ t

0

µjdβj

.

Equation (1.124), resp. (1.127), models a diusion process (heating process) driven bythe Gaussian noise W , resp. XX. More precisely if q is the ux, e is the interval energyand X is the temperature, we have the energy conservation law

de(t) + div q(t) = f(t)dt+ dW (t) ,

that is the exterior energy is perturbed by the Gaussian noise W .

Example 1.8.1. Let us consider the reaction-diusion equationdX(t, ξ)−∆ξX(t, ξ)dt+ f(X)dt = dW (t) , in [0, T ]×O ,X(0) = x , in O ,X(t) = 0 , on [0, T ]× ∂O ,

(1.128)

where f : R → R is a continuous function. If f is monotonically increasing or Lipschitz, asolution to eq. (1.128) can be found by the so called Galerkin scheme, that is take

Xn =

∞∑j=1

〈X, ej〉ej =

n∑j=1

Xnj ej −∆ej = λjej ,

1.9 Kolmogorov equation for the Ornstein-Uhlenbeck process 79

with ej = 0 on ∂O, and substitute Xn in (1.128). We thus get the stochastic equationdXn

j (t, ξ) + λjXnj dt+

⟨f(∑n

j=1Xnj ej

), ej

⟩dt = µjdβj(t) , j = 1, . . . , n ,

Xnj (0) = 〈x, ej〉 ,

(1.129)

system (1.129) is of the formdYj(t, ξ) + λjYjdt+ fj

(Y 1, . . . , Y n

)dt = µjdβj(t) , j = 1, . . . , n ,

Yj(0) = 〈x, ej〉 .(1.130)

If then f is monotonically increasing, or more generally it holds(fj(Y )− fj(Y )

)· (Y − Y ) ≥ −C|Y − Y |2 ,

we have that eq. (1.130) has a solution and proves that

Xn → X , in L2(Ω;C([0, T ];L2(O))) .

1.9 Kolmogorov equation for the Ornstein-Uhlenbeck pro-

cess

1.9.1 Stochastic perturbations of linear equations

Let us consider two separable Hilbert spaces H and U with inner product 〈·, ·〉 and normgiven by ‖ · ‖. Let furthermore be ekk be a complete orthonormal basis for U and letβkk a sequence of mutually independent standard Brownian motion on a xed probabilityspace (Ω,F , (Ft)t ,P), where we have denoted by Ft is the σ−algebra generated by all βk(s)with s ≤ t.

Throughout the section we will deal with the following stochastic dierential equationdXt = AXtdt+BdWt, t ≥ 0,

X0 = x ∈ H. (1.131)

where A : D(A) ⊂ H → H and B : U → H, are linear operators and W is a cylindricalBrownian process, in the sense of denition 1.7.14, dened as

Wt =

∞∑k=1

βk(s)ek, t ≥ 0 .

Throughout the work we will assume the following hypothesis to hold.

Hypothesis 1.9.1. (i) A : D(A) ⊂ H → H is the innitesimal generator of a strongly con-tinuous semigroup etA.

(ii) B ∈ L(U,H).


(iii) for any t > 0 the linear operator Qt dened as

Qtx =

∫ t

0

esACesA∗xds, x ∈ H, t ≥ 0 , (1.132)

with C := BB∗, is of trace class.

Under assumptions 1.9.1, eq. (1.131) admits a unique (mild) solution to eq. (1.131)given by

Xxt = etAx+WA

t , t ≥ 0 ,

where the process WAt is called stochastic convolution and it is given by

WAt =

∫ t

0

e(t−s)ABdWs =

∞∑k=1

∫ t

0

e(t−s)ABekdβk(s), t ≥ 0 . (1.133)

Remark 1.9.2. A mild solution of eq. (1.131) is a mean square continuous stochastic processadapted to Wt.

We will in what follows mainly have a probabilistic approach as the one given in [DP04].Anyhow a completely analogous analytic approach can be done, as the one in [DPZ02],considering the Kolmogorov equation associated to the OU stochastic equation is

Dtu(x, t) = 12Tr[CD

2u(x, t)] + 〈Ax,Du(x, t)〉 , t > 0, x ∈ D(A),

u(x, 0) = ϕ(x), x ∈ H,ϕ ∈ Cb(H), (1.134)

with D the space derivative Dx and C := BB∗.

The stochastic convolution

The present section is devoted to show that if assumptions 1.9.1 hold, then the series denedin eq. (1.133) converges in the space L2(Ω,F ,P) and thatWA

t is a Gaussian random variableNQt . We will further give some results concerning continuity properties of the stochasticconvolution (1.133). Even if not stated we will always assume hypothesis 1.9.1 to hold.

Let us rst of notice that the generic k−th term in the series (1.133) is vector valuedBrownian integral dened as∫ t

0

e(t−s)ABekdβk(s) =

∞∑h=1

∫ t

0

⟨e(t−s)ABek, fh

⟩dβk(s) ,

where fhh is a complete orthonormal system on H.

Proposition 1.9.3 (Prop. 2.2 [DP04]). For any t ≥ 0 the series (1.133) is convergent inL2(Ω,F ,P;H) to a Gaussian random denoted WA

t ∼ N (0, Qt), where the operator Qt isdened in eq. (1.132). In particular it holds

E∣∣WA

t

∣∣2 = TrQt, t ≥ 0 .

The next result shows that WAt is continuous as a function of t. Let us rst introduce

the space of mean square continuous adapted processes on [0, T ] taking values in H,

CW ([0, T ];L2(Ω,F ,P;H)) =: CW ([0, T ];H) ,


the space of all continuous functions F : [0, T ] → L2(Ω,F ,P;H) which are adapted to W ,i.e. F (s) is Fs-measurable for any s ∈ [0, T ]; the space CW ([0, T ];H) is Banach space ifequipped with the norm

‖F‖CW ([0,T ];H) :=

(supt∈[0,T ]

E|F (t)|2) 1

2

.

Proposition 1.9.4 (Prop. 2.3 [DPZ08]). For any T > 0 we have that WA· ∈ CW ([0, T ];H).

In Prop. 1.9.4 we have proven WA to be mean square continuous, we are now to provethat it has continuous trajectories, i.e. it is continuous for P-almost all ω.

Theorem 1.9.5 (Th. 2.9 [DP04]). Let T > 0 and m ∈ N, then it exists a constant C1m,T > 0

such thatE supt∈[0,T ]

∣∣WAt

∣∣2m ≤ C1m,T .

Moreover WA· is P−almost surely continuous on [0, T ].

Let us now assume T > 0 to be xed and let us consider the stochastic convolution as arandom variable in L2(0, T ;H). Then the following tells us that the stochastic convolutiondened in eq. (1.133) is a Gaussian random variable.

Proposition 1.9.6 (Prop. 2.15 [DP04]). WA· is a Gaussian random variable on L2(0, T ;H)

with mean 0 and covariance operator Q given by

Qh(t) =

∫ T

0

g(t, s)h(s)ds, h ∈ L2(0, T ;H), t ∈ [0, T ]

where

g(s, t) =

∫ s∧t

0

e(s−r)ACe(t−r)A∗ds, t, s ∈ [0, T ] .

The Ornstein-Uhlenbck semigroup

Let us consider the Ornstein-Uhlenbck (OU) process (1.131). As already stated in Prop.1.9.3, the solution Xx

t to the equation (1.131) is normally distributed, namely Xxt ∼

N (etAx,Qt). The main object of the current section will be the corresponding transitionsemigroup Rt in Bb(H), called Ornstein-Uhlenbck (OU) semigroup, in the sense of denition1.7.17,

Rtϕ(x) = Eϕ(Xxt ) =

∫H

ϕ(y)NetAx,Qt(dy), ϕ ∈ Bb(H), t ≥ 0, x ∈ H .

Equivalently, previous expression can be rewritten as

Rtϕ(x) =

∫H

ϕ(etAx+ y)NQt(dy), ϕ ∈ Bb(H), t ≥ 0, x ∈ H . (1.135)

Remark 1.9.7. The OU semigroup dened in eq. (1.135) satises the semigroup property

Rt+sϕ = RtRsϕ, t, s ≥ 0, ϕ ∈ Cb(H) .

Furthermore setting u(x, t) := Rtϕ, we have that u(x, t) is solution to the Kolmogorovequation (1.134).


We have that the space E(H) of exponential function is stable for the semigroup Rt, i.e.RtE(H) ⊂ E(H). In fact, from eq. (1.135) we have that

Rtϕh(x) =

∫H

ei〈etAx+y,h〉NQt(dy) = ei〈e

tAx,h〉∫H

ei〈y,h〉NQt(dy) =

= ei〈etAx,h〉e− 1

2 〈Qth,h〉 = e−12 〈Qth,h〉ϕetA∗h(x), x ∈ H .

Proposition 1.9.8 (Prop. 2.17 [DP04]). (i) for all t ≥ 0, the OU semigroup Rt dened ineq. (1.135) is Feller, in the sense of Def. 1.7.17, and we have

‖Rtϕ‖0 ≤ ‖ϕ‖0, t ≥ 0, ϕ ∈ Cb(H) ;

(ii) if ϕ ∈ Cb(H) and ϕn1,n2 ⊂ Cb(H) is a two-index subsequence s.t. ‖ϕn1,n2‖0 ≤ ‖ϕ‖0for all n1, n2 ∈ N and

limn1,n2→∞

ϕn1,n2(x) = ϕ(x), x ∈ H ,

thenlim

n1,n2→∞Rtϕn1,n2

(x) = Rtϕ(x), x ∈ H, t ≥ 0 ;

(iii) the mapping[0, T ]×H → R, (t, x) 7→ Rtϕ(x) ,

is continuous for all ϕ ∈ Cb(H).

We would like to stress that, unless A = 0, Rt is not a strongly continuous semigroup onCb(H). In fact we have that

Rtϕh(x) = e−12 〈Qth,h〉ϕetA∗h(x), x, h ∈ H ,

so that taking the limit as t goes to 0

limt↓0

Rtϕh(x) = ϕh(x) = Re ei〈h,x〉 ,

hence the limit is not uniform on x ∈ H.Despite this we can introduce a dierent notion of convergence, called π−convergence,

and consequently a dierent notion of innitesimal generator for the semigroup Rt. For anyt > 0 let us set

∆tϕ :=1

t(Rtϕ− ϕ), ϕ ∈ Cb(H) ;

then we can dene the innitesimal generator L of Rt and its domain as followsD(L) =

ϕ ∈ Cb(H) : ∃f ∈ Cb(H), limt↓0 ∆tϕ(x) = f(x),∀x ∈ H, supt∈[0,1] ‖∆tϕ‖0 <∞

Lϕ(x) = limt↓0 ∆tϕ(x) = f(x), x ∈ H,ϕ ∈ D(L)

,

where L is the innitesimal generator of Rt in Cb(H). Therefore we have that Rt is notstrongly continuous but only pointwise continuous, see, e.g. [EN00, DP04, DPZ02], andreferences therein, for details on semigroup and innitesimal generator and π−convergence.

In what follows, for brevity reason, even if not stated we will use the notion of π−convergenceinstead of the standard convergence. We will avoid in general to write ϕn

π→ ϕ, but we willjust write ϕn → ϕ. The same holds for density criteria. However we shall dene thesemigroup Rt on other spaces dierent from Cb(H).


Proposition 1.9.9. Let us consider the innitesimal generator L and its domain D(L) asintroduced above. Then the following holds:

(i) D(L)d⊂ Cb(H);

(ii) if ϕ ∈ D(L) then we have that Rtϕ ∈ D(L), ∀t ≥ 0;

(iii) LRtϕ = RtLϕ;

(iv) if ϕ ∈ D(L), then Rt is dierentiable for any t ≥ 0 and

d

dtRtϕ(x) = LRtϕ(x) = RtLϕ(x), x ∈ H .

The following is a basic result on the resolvent of L. The results follows immediatelyfrom the Hille-Yosida theorem, just with the foresight of Rt being only pointwise continuous.

Proposition 1.9.10 (Prop. 2.21 [DP04]). (i) let ρ(L) be the resolvent set of L, and R(λ, L) :=(λ− L)−1 the resolvent. Then we have that ρ(L) ⊂ (0,∞) and

R(λ, L)f(x) =

∫ ∞0

e−λtRtf(x)dt, f ∈ Cb(H), λ > 0, x ∈ H .

Moreover

‖R(λ, L)f‖0 ≤1

λ‖f‖0, f ∈ Cb(H), λ > 0 .;

(ii) if f ∈ Cb(H) and fn1,n2 ⊂ Cb(H) is a two-index sequence s.t.

limn1,n2→∞

fn1,n2(x) = f(x), ∀x ∈ H ;

and ‖fn1,n2‖0 ≤ ‖f‖0 for all n1, n2 ∈ N, then we have

limn1,n2→∞

R(λ, L)fn1,n2(x) = R(λ, L)f(x), ∀x ∈ H .

Asymptotic behaviour of solutions

The present section is devoted to the study of asymptotic behaviour of the solution Xxt of

the OU process and to the statement of some result concerning existence and uniqueness ofan invariant measure µ.

As usual we assume Hyp. 1.9.1 to hold. Therefore by Hille-Yosida theorem there existM ≥ 0 and ω ∈ R such that

‖etA‖ ≤Meωt, t ≥ 0 .

We will thus assume further ω < 0 and set ω1 = −ω. Therefore we have that the linearoperator

Q∞x :=

∫ ∞0

etACetA∗xdt, x ∈ H,


is well dened and of trace class.We have already said that a solution of the OU equation (1.131) is explicitly given by

Xxt = etAx+WA

t .

Proposition 1.9.11 (Prop. 2.32 [DP04]). For any ϕ ∈ Cb(H) and any x ∈ H we have that

limt→∞

Rtϕ(x) =

∫H

ϕ(y)NQ∞(dy) .

Proof. If ϕ ∈ Cb(H) we have that

limt→∞

Rtϕh(x) = limt→∞

ei〈etAh,x〉− 1

2 〈Qth,h〉 = e12 〈Q∞h,h〉 =

∫H

ϕh(x)N∞(dx) .

Thus from previous proposition it follows that the measure NQ∞ is ergodic and invariantfor the semigroup Rt. We thus have the following.

Theorem 1.9.12 (Th. 2.34 [DPZ08]). The measure µ = NQ∞ is the unique invariantmeasure for the semigroup Rt.

Proof. Let us start proving the existence. It is enough to prove that eq. (1.120) for ϕh(x) =ei〈h,x〉, h ∈ H since it holds the approximating result in Lem. ??. In particular we havethat eq. (1.120) holds i

µ(etA∗h)e−

12 〈Qth,h〉 = µ(h), h ∈ H ⇔⟨

Q∞etA∗h, etA

∗h⟩

+ 〈Qth, h〉 = 〈Q∞h, h〉 , h ∈ H, t ≥ 0 ⇔

etAQ∞etA∗ +Qt = Q∞, t ≥ 0 ,

where we have denoted by µ the Fourier transform of µ. Last equality can be easily checkedto hold.

For the uniqueness part let us assume µ to be an invariant measure for Rt, then we get

limt→∞

µ(h) = e−12 〈Q∞h,h〉, h ∈ H ,

by the uniqueness of the Fourier transform we are done.

The transition semigroup in L2(H,µ)

Assumptions 1.9.1 are still in charge, furthermore again we assume ω > 0 and set ω1 = −ω.Let us further consider the invariant measure µ = NQ∞ of Rt. We will focus on what followson the particular case of L2, anyway many of the result can be extended to the more generalcase of Lp, p ≥ 1.

Proposition 1.9.13 (Prop. 10.1.5 [DPZ02]). The semigroup Rt has a unique extension toa strongly continuous semigroup of contractions in L2(H,µ).


Proof. By Holder inequality and from the invariance of µ we have that∫H

|Rtϕh(x)|2 µ(dx) ≤∫H

Rt|ϕh|2(x)µ(dx) =

∫H

|ϕh(x)|2 µ(dx) ,

Since Cb(H)d⊂ L2(H,µ), Rt is uniquely extendible to L2(H,µ). Therefore

‖Rtϕh‖L2(H,µ) ≤ ‖ϕh‖L2(H,µ), t ≥ 0, ϕh(x) ∈ L?2(H,µ) .

Eventually strong continuity follows from dominated convergence theorem.

The unique extension will still be denoted by Rt and it generator by L2 with domaingiven by D(L2). For any h ∈ H we have that

ϕh(x) ∈ D(L2)⇔ h ∈ D(A∗) .

Let us now dened a subspace oh the exponential function set dened as

EA(H) := spanϕh(x) = ei〈h,x〉 : h ∈ D(A∗)

⊂ E(H) .

For this space the following generalization of the approximation result Le. ?? holds.

Proposition 1.9.14 (Prop. 2.37 [DP04]). For all ϕh ∈ Cb(H) there exists a three-indexsequence ϕn1,n2,n3 ⊂ EA(H) s.t.

(i) ‖ϕn1,n2,n3‖0 ≤ ‖ϕ‖0, n1, n2, n3 ∈ N;

(ii) limn1,n2,n3→∞ ϕn1,n2,n3(x) = ϕ(x), x ∈ H.

We can thus state the following.

Theorem 1.9.15 (Th. 2.38 [DP04]). EA(H) is a core for L2(H,µ). Moreover

L2ϕ(x) =1

2Tr[CD2ϕ(x)

]+ 〈x,A∗Dϕ(x)〉 , x ∈ H,ϕ ∈ EA(H) .

We will prove eventually that, for any t > 0 and for any x ∈ H, the measures NetAx,Qtand µ = NQ∞ are equivalent. This fact allows us to obtain a representation formula for Rtϕfor all ϕ ∈ L2(H,µ). In particular we are to prove that

Rtϕ(x) =

∫H

pt(x, y)ϕ(y)µ(dy), x ∈ H ,

where pt(x, y) is the Radon-Nikodym density, namely

pt(x, y) =dNetax,QtdNQ∞

(y), x, y ∈ H . (1.136)

Let us then introduce the semigroup of linear operators on H dened as

T (t) := Q− 1

2∞ etAQ12∞, t ≥ 0 , (1.137)

We thus have the following result.

Lemma 1.9.16 (Lem. 10.3.2 [DPZ02]). Let us consider the semigroup T (t)t≥0 dened in(1.137) and by B its innitesimal generator. Then etB is a semigroup of contraction on H.

Theorem 1.9.17 (Th. 10.3.5 [DPZ02]). For any ϕ ∈ L2(H,µ) and t > 0, we have thatRtϕ, dened in eq. (1.136) is a C∞ function.


1.9.2 Stochastic dierential equations with non Lipschitz coe-cients

The general setting is as in Sec. ??. Let us consider two separable Hilbert spaces H andU and let us assume hypothesis 1.9.1 are fullled. Let us then consider a stochastic partialdierential equations of the form

dXt = (AXt + F (Xt)) dt+BdWt, t ≥ 0,

X0 = x ∈ H, (1.138)

where as before Wt is a cylindrical Wiener process denes as in eq. (??) and F : H → H isa non linear Lipschitz continuous function in the sense that it exists a constant K > 0 suchthat

|F (x)− F (y)| ≤ K|x− y|, quadx, y ∈ H .

In particular we have that a mild solution to eq. (1.138) X ∈ CW ([0, T ];H) is given by

Xt = etAx+

∫ t

0

e(s−t)AF (Xs)ds+WAt , (1.139)

where WAt is the stochastic convolution dened in Sec. ??.

We will focus on the case of H and U being innite dimensional since a wide theorydoes exist for the nite dimensional case, see, e.g. [SK91]. The main problems when dealingwith the innite dimensional setting are the unboundedness of the operator A and that theoperator C := BB∗ is not necessary of trace class. We will not state any result of existenceand uniqueness of mild solution (1.139), we refer to [DP04] §3.2 for details. However the ideais to proceed by approximating solution. In particular if C 6∈ L+

1 (H) and A is unboundedwe approximate the noise by a nite dimensional noise of the form

Wnt =

n∑k=1

βk(t)ek, t ≥ 0 ,

and then solve the problemdXn

t = (AXnt + F (Xn

t )) dt+BdWnt , t ≥ 0,

Xn0 = x ∈ H

, (1.140)

proving that limn→∞Xn = X in CW ([0, T ];H).Then we make use of the Yosida approximation of A, see, e.g. [DPZ08, DP04, EN00] for

details, given byAk = kA(k −A)−1, k ∈ N ,

and then solve the approximated problemdXn,k

t =(AkXn,k

t + F (Xn,kt )

)dt+BdWn

t , t ≥ 0,

Xn,k0 = x ∈ H

, (1.141)

and then prove that limk→∞Xn,k = Xn in CW ([0, T ];H).Throughout the work we assume hypothesis 1.9.1 are still in charge and that the operator

F is Lipschitz continuous. The present section is so structured, aggiungi.


The transition semigroup

We will devote the current section to some general results concerning the transition semi-group associated to the mild solution (1.139) using in particular the two approximatingsemigroup associated with eq. (1.140) and eq. (1.141).

Ptϕ(x) = Eϕ(Xxt ), ϕ ∈ Bb(H), t ≥ 0, x ∈ H (1.142)

Pnt ϕ(x) = Eϕ(Xn,xt ), ϕ ∈ Bb(H), t ≥ 0, x ∈ H (1.143)

Pn,kt ϕ(x) = Eϕ(Xn,k,xt ), ϕ ∈ Bb(H), t ≥ 0, x ∈ H (1.144)

Proposition 1.9.18. Let us consider the transition semigroups (1.142), (1.143) and (1.144),then we have the following

limn→∞

Pnt ϕ(x) = Ptϕ(x), ϕ ∈ Cb(H), t ≥ 0, x ∈ H ,

limk→∞

Pn,kt ϕ(x) = Pnt ϕ(x), ϕ ∈ Cb(H), t ≥ 0, x ∈ H,n ∈ N .

Furthermore the following estimates hold

‖Ptϕ‖0 ≤ ‖ϕ‖0, ‖Pnt ϕ‖0 ≤ ‖ϕ‖0, ‖Pn,kt ϕ‖0 ≤ ‖ϕ‖0, ϕ ∈ Bb(H) .

Eventually we prove that the transition semigroup is Feller.

Proposition 1.9.19 (Prop. 3.9 [DP04]). For all s, t ≥ 0 we have

Pt+sϕ = PtPsϕ, ϕ ∈ Bb(H) . (1.145)

Moreover for all ϕ ∈ Cb(H) and t ≥ 0 we have Ptϕ ∈ Cb(H).

Proof. The semigroup property (1.145) holds from the Markov property of solution to classic

SDE for Xn,k for the semigroup Pn,kt , then it holds for Pt letting n and k going to innity.

Since C1b (H)

d⊂ Cb(H), it is enough to show that Ptϕ ∈ Cb(H) for all ϕ ∈ C1

b (H), wethus have

|Ptϕ(x)− Ptϕ(y)| ≤ ‖ϕ‖1E|Xxt −X

yt | ≤MT e

TKMT |x− y|, x, y ∈ H ,

then we have proven that Ptϕ ∈ Cb(H).

Invariant measure

In order to prove the existence of an invariant measure we will consider eq. (1.138) with anegative initial time s, namely

dXt = (AXt + F (Xt)) dt+BdWt, t ≥ −s,X−s = x ∈ H

, (1.146)

where taking another cylindrical Wiener process W 1t |= Wt we have set

Wt =

Wt, t ≥ 0 ,

W 1−t, t ≤ 0 .


Then approximating as before with eq. (1.140) and eq. (1.141) problem (1.146), we retrievea solution to eq. (1.146) to be

Xx−s,t = e(t+s)Ax+

∫ t

−se(t−u)AF (Xx

−s,u)du+WA−s,t .

We can prove that there exist

lims→∞

Xx−s,0 := η, in L2(Ω,F ,P;H) , (1.147)

so that the law of Xxt is convergent to the unique invariant measure ν of Pt. Thus the

following result holds.

Theorem 1.9.20 (Th. 3.17 [DP04]). Let ν be the law of the random variable η deend in(1.147). Then the following statements hold:

(i) limt→∞ Ptϕ(x) =∫Hϕ(y)ν(dy), x ∈ H, ϕ ∈ Cb(H);

(ii) the measure ν is the unique invariant measure for Pt;

(iii) for any Borel probability measure λ on H we have

limt→∞

∫H

Ptϕ(x)λ(dx) =

∫H

ϕ(x)ν(dx), ϕ ∈ Cb(H) .

We eventually that the measure ν has nite moments.

Proposition 1.9.21 (Prop. 3.18 [DP04]). Let ν be the unique invariant measure denedabove, then for any n ∈ N we have ∫

H

|x|2nν(dx) <∞ .

The transition semigroup in L2(H, ν)

We will denote by ν the unique invariant measure for the semigroup Pt dened is Sec. 1.9.2.We will prove that the semigroup Pt can be uniquely extended to a strongly continuoussemigroup of contractions in L2(H, ν) with innitesimal generator K2. Again Pt can moregenerally be extended to a semigroup on Lp, we will avoid the general case, treating onlythe simpler case of p = 2.

In particular Prop. ?? still holds so that we can uniquely extend the semigroup Pt to asemigroup of contractions in L2(H, ν) which it will still be denoted by Pt.

Then from Hille-Yosida theorem we have that the innitesimal generator K2 of Pt onL2(H, ν) is closed densely dened and it satises the estimate

|λR(λ,K2)| ≤ 1 .

Then dening by K0 the Kolmogorov operator associated with eq. (1.138) as

K0ϕ =1

2Tr[CD2ϕ] + 〈x,A∗Dϕ〉+ 〈F (x), Dϕ〉 = Lϕ+ 〈F (x), Dϕ〉 , ϕ ∈ EA(H) , (1.148)


where L is the OU operator dened in Sec. ??.From the estimate on the moments of ν we eventually have that F (x) ∈ L2(H, ν) so that

K0 is a well-dened operator taking values in L2(H, ν).We have rst of all the following result.

Proposition 1.9.22 (Prop. 3.19 [DP04]). For any ϕ ∈ EA(H) we have that

Ptϕ = ϕ(x) + E∫ t

0

K0ϕ (Xxs ) ds, t ≥ 0, x ∈ H ,

where K0 is the Kolmogorov operator (1.148).

Similarly to what dome in Sec. ?? we will prove that EA(H) is a core for K2 implyingthat K2 is the closure of the Kolmogorov operator K0 dened in (1.148). In order to achievethis goal we need rst to introduce the denition of dissipative operator and then to statethe so-called Lumer-Philips theorem.

Denition 1.9.23 (Dissipative operator). Let be A a linear operator A : D(A) ⊂ H → H.Then A is said to be dissipative if

‖(λI −A)ϕ‖H ≥ λ‖ϕ‖H ≤1

λ, ∀ϕ ∈ D(A), λ > 0 .

Remark 1.9.24. Equivalently an operator A is said to be dissipative if

Re 〈Aϕ,ϕ〉 ≤ 0, ∀ϕ ∈ D(A) .

Any dissipative operator is closable. Furthermore a dissipative operator for which (λI −A) is surjective is called m-dissipative, an operator with dense domain is m-dissipative i itis the innitesimal generator of a strongly continuous semigroup of contractions in H. Wethen have the following.

Theorem 1.9.25 (Lumer-Philips). [Th. 3.15 [EN00]] Let be A : D(A) ⊂ H → H bea densely dened dissipative operator in H, then the closure A of A generates a stronglycontinuous semigroup of contractions i the range of (λI −A) is dense in H for some (andhence all) λ > 0.

We can now prove the following fundamental result.

Theorem 1.9.26 (Th. 3.21 [DP04]). The operator K2 is the closure of the operator K0 inL2(H, ν).

Comparison of ν with a Gaussian measure

In this section we are assuming hypothesis 1.9.1 with M = 1 and growth bound ω. Further-more we set

κ := inf

〈F (x)− F (y), x− y〉

|x− y|2: x, y ∈ H

and assume that ω + κ < 0. In addition we assume C = I and F ∈ Cb(H). Eventually weset ω1 := −(ω + κ).

Let us further denote ν the invariant measure of the semigroup Pt dened in Sec. 1.9.2and by µ := NQ∞ the unique invariant measure of the OU semigroup dened in Sec. ??


whose innitesimal generator L ∈ Cb(H). It can be proven, see, e.g. [DP04] §2.4 that theOU semigroup under these assumptions is strong Feller. We eventually recall that

Q∞ =

∫ ∞0

etAetA∗xdt, x ∈ H .

The aim of the current section is to show that ν is absolutely continuous w.r.t. µ. Wewill follow [DP04] Sec. §3.7.1, an analogous procedure can be found in [DP04] Sec. §3.7.2.

Let us consider the Kolmogorov operator in Cb(H) dened as

Kϕ = Lϕ+ 〈F (x), Dϕ〉 , ϕ ∈ D(L,Cb(H)) , (1.149)

where we have denoted by D(L,Cb(H)) the domain of the OU operator dened on Cb(H).Let us now introduce the approximating operators as follows

Kn,kϕ(x) = Ln,kϕ(x) + 〈F (x), Dϕ(x)〉 , ϕ ∈ D(Ln,k, Cb(H)), (1.150)

Ln,kϕ(x) =1

2Tr[PnD

2ϕ]

+⟨Akx,Dϕ(x)

⟩, ϕ ∈ D(Ln,k, Cb(H)), (1.151)

where Pn is the orthogonal projection of dimension n and Ak are the Yosida approximationsof A.

We rst of all show that the operator K dened in eq. (1.149) is m-dissipative and thattherefore generates a strongly continuous semigroup of contractions. We further show thatthe resolvent is given by the Laplace transform of the semigroup.

Lemma 1.9.27 (Lem. 3.31 [DP04]). The operator K dened in eq. (1.149) is m-dissipativein Cb(H). Moreover for any λ > π‖F‖20 we have that

(λ−K)−1 = (λ− L)−1(1− Tλ)−1 ,

where

Tλϕ =⟨F,D(λ− L)−1ϕ

⟩, ϕ ∈ Cb(H) .

Furthermore we have that

R(λ,K)f(x) =

∫ ∞0

e−λtPtf(x)dt, x ∈ H, f ∈ Cb(H) .

Remark 1.9.28. Everything done since now in the present section holds also for f ∈ Bb(H)

We can then state the following result.

Theorem 1.9.29 (Th. 3.33 [DP04]). For any f ∈ Bb(H) we have∫H

fdµ =

∫H

fdν +

∫H

⟨F,DL−1f

⟩dν , (1.152)

where DL−1f :=∫ t

0DRtf(x)dt = limλ→0D(λ−L)−1f(x). Moreover ν is absolutely contin-

uous w.r.t. µ.


Proof. Let us just prove that ν << µ. Let Γ ⊂ H be a Borel set s.t. µ(Γ) = 0. Then fromthe fact that Rt is strong Feller we have that NetAx,Qt << µ and eq. (1.136) holds

Rt1Γ(x) = NetAx,Qt(Γ) = 0 ,

for all t > 0 and x ∈ H. Thus D(λ − L)−11Γ = 0 for all x ∈ H. Then by eq. (1.152)

follows thatν(Γ) = µ(Γ) = 0 .

Part II

Mathematical nance

93

Chapter 2

The binomial asset model

2.1 Introduction

Here follows a brief introduction about the main techniques and the results will be treatedthroughout the course.

2.1.1 History: Milestones

• Harry Markovitz, in his PhD thesis Portfolio Selection (1952) introduced the conceptsof mean return, variance/covariance for a given portfolio and investment diversica-tion. However he didn't use stochastic calculus techniques.

• Robert Merton (1969) developed a new approach to assets' price, based on an equilib-rium argument.

• Four years later in 1973, Fisher Black and Myron Scholes in their work The pricingoptions and liabilities stetted the famous formula, which takes their names, for pricingcontingent claim. In 1997 Merton and Scholes won the Nobel Prize for their work(Black was ineligible since already dead).

• Harrison, Kreps and Pliska (1979-1983) provided a rigorous time-continuous stochas-tic formulation for the BS-option price formula, also allowing to price other (non-European) derivatives. Indeed they found a way of pricing exotic options (ex. con-tracts whose payos depend on the assets' history or contracts that may be close beforethe expiration time)

2.1.2 Sketch of the course

The road map of the course is built on two blocks: the discrete-time models and thecontinuos-time models.

The Discrete-time models are essentially based on coin tosses and 1-dimensional randomwalk. Despite their simplicity, we can here dene the concepts of non-arbitrage and fair price,strictly related to the Fundamental theorems of asset pricing. The main techniques usedare the martingales theory and Markov processes, also related to change of measure and the

95

96 2. The binomial asset model

Radon-Nykodyn derivative. We introduce the Black&Scholes formula in order to discoverthe fair price of European-type option. Eventually, the Utility maximization problem isstudied.

The part concerning continuous-time model will deal with the Brownian motion, withregards to its formal derivation and its mathematical properties. Then the Itô integral andItô's formula will be dened. Then two fundamental mathematical results, such as Girsanovtheorem and Feynman-Kac formula, will be introduced both under a mathematical point ofview along with their nancial application. Eventually, a brief introduction to jump processwill be treated.

2.2 The one-period binomial model

Let (Ω,F ,P) be a nite probability space where Ω = ω1, ωn, . . . , ωN is a nite set with

|Ω| = N ∈ N and let P a probability measure over Ω s.t. P(ωi) = pi > 0 with∑Ni=1 pi = 1.

Moreover let us introduce a ltration Ftt=0,1,...,T , with T ∈ N+ be the nal horizon ofinvestment, with F0 = F and Ft−1 ⊆ Ft for all t = 1, . . . , T − 1. From an heuristic point ofview Ft represents the information available to the investor up to time t. We will assumethat buy/sell actions occur at time t ∈ 0, 1, . . . , T which can be performed with respectto a specic basket of assets. In particular we will assume it exists a unique risk-free asset(such as a bond or a bank account) and d ∈ N+ risky assets. For the sake of simplicity wewill assume further that the market is friction less, i.e. there are no transactions costs.

Let us then consider a collection of real valued random variables Z = Ztt=0,...,T suchthat Z is adapted to the ltration F , namely for any t = 1, . . . , T we have that Zt is Ft-measurable, and that Z is predictable w.r.t. F , i.e. the values of Z at time t depends onlyon Ft−1.

Denition 2.2.1 (Discrete-time nite market model). An adapted stochastic process

S = Stt=0,1,...,T = (S0t , S

1t , . . . , S

dt )t=0,1,...,T , (2.1)

taking values in R1+d is called a discrete-time nite market model.

In Def. 2.2.1 we have denoted with S0 the risk-free asset. In particular denoting byr > 0 its interest rate and assuming, with no loss of generality, that it has value 1 at timet = 0, i.e. S0

0 = 1, it immediately follows that its value at a generic time t it is given by

S0t = Πt

k=1(1 + r) = (1 + r)t .

Eventually we have denoted by Sit , i = 1, . . . , d, the value of the i−th asset at time t ≤ T .

Denition 2.2.2 (Portfolio strategy). A predictable stochastic process H = Htt=1,...,T

with values in R1+d is called a portfolio strategy (equivalently trading strategy).

As before we will use the notation 0 for the risk free asset and i = 1, . . . , d for the riskyassets.

The portfolio value process associated to the strategy H is the stochastic process Vtt,t = 1, . . . , T dened by

Vt =⟨Ht, St

⟩=

d∑k=0

Hkt S

kt ∀t = 1, . . . , T .

2.2 The one-period binomial model 97

Denition 2.2.3. A strategy H such that

d∑k=0

Hkt S

kt =

d∑k=0

Hkt+1S

kt . (2.2)

is called a self-nancing strategy.

The main feature of a self-nancing strategy is that, according to eq. (2.2), from time tto time t+ 1 we can only rearrange money so that no infusions or withdrawals are allowedand the purchase of a new asset must be nanced by the sale of an old one.

Thus, if we are dealing with a self-nancing trading strategy, from the fact that S00 = 1,

we have that

V0 =⟨H1, S0

⟩=

d∑k=0

Hk0 S

k0 .

We can also simplify equation (2.1), in fact we are mainly interested in the d risky assetsdue to the deterministic nature of the risk-less bond. Let us thus dene the discountedprice processes by rescaling all the asset price process w.r.t. the bank account, so that wewill deal with the following process

St = (S1t , . . . , S

dt ) =

(S1t

S0t

, . . . ,Sdt

S0t

).

Remark 2.2.4. For every predictable process Htt=1,...,T ∈ Rd there exists a unique self-

nancing trading strategy Htt=1,...,T ∈ R1+d such thatHkt = Hk

t ∀(k, t) ∈ 1, . . . , d1, . . . , T,H0

1 = 0 .

We shall always use H to indicate the self nancing (1 +d)-dimensional trading strategyH. The discounted value process is then dened as

Vt =VtSt0

∀t = 0, . . . , T .

Denoting by

∆Vt+1 := Vt+1 − Vt, t = 0, . . . , T − 1 ,

∆St+1 := St+1 − St =(S1t+1 − S1

t , . . . , Sdt+1 − Sdt

), t = 0, . . . , T − 1 ,

it can be proved that

1. ∆Vt+1 = 〈Ht+1,∆St+1〉

2. VT = V0 + (H · S)T = V0 +∑Tl=1 〈Hl,∆Sl〉 ,

where (H · S)t =∑tl=1 〈Hl,∆Sl〉.

Remark 2.2.5. Exploiting (1) and (2), we have that the discounted value process does notdepend on H0

t t=1,...,T .


2.2.1 Completeness and arbitrage

Denition 2.2.6. A contingent claim (equivalently a nancial contract) is a ow ofpayments, represented by X = Xtt=1,...,T , which may depend on the assets values up totime t. In particular at every time t = 1, . . . , T we have that

Xt = Φ(S0, . . . , St) ,

for a suitable function Φ.

Usually the contingent depends only on the terminal value of the asset, namely Xt =Φ(St), but it sometimes may happen that the whole history of values of the asset aect thecontingent claim, such type of contingent claims are called path-dependent options.

Example 2.2.1. Let us give few examples in order to gain some condence with the conceptof contingent claim. Let us consider for simplicity a market with one risk less asset B withinterest rate r > 0 and one risky asset S.

European call option The most standard contingent claim is the so called European calloption. In particular a given investor at time t = 0 can stipulate this type of contract,for instance with a bank, that gives him the opportunity (but not the obligation) tobuy the asset S at a maturity time T at a xed price K. It is clear that if happensthat at time T the asset S worth more than K it is convenient for the investor to buythe asset at price K in order to sell it immediately (or to keep the asset) at its marketprice ST , so that he gains ST −K > 0;

Asian call option Let us now give an example of path-dependent option. The setting is asbefore, just now the nal payo of the contingent claim is determined by the averageof all the values that the asset S takes from time t = 0 up to time t = T and now juston the values that S takes at maturity time T .

It happens then that a contingent claim is nothing but a random variable X over(Ω,F ,P). Furthermore it is clear that the privilege of buying an asset at a given pricedoes not come for free, but an investor has to pay something in order to stipulate such typeof contract. How much has an investor to pay to buy this contract? Is there exist a fairprice? Is there a procedure that lead everyone to assign the same price for this contract?Previous questions are the main problems one deal with in mathematical nance.

Denition 2.2.7. A contingent claim X is called attainable if the following holds

∃V0 ∈ R+, ∃ a self-nancing strategy H s.t. X = V0 +

T∑l=1

〈Hl,∆Sl〉 .

The amount of money V0 is called fair price of the contingent claim X. Roughlyspeaking V0 is the price that makes an investor indierent between choosing X or H.

Denition 2.2.8 (Complete market). The market is called complete if every contingentclaim is attainable, that is there exits a self-nancing predictable strategy that replicatesthe value of the contingent claim X.


Denition 2.2.9 (Arbitrage). We will say that we have an arbitrage possibility/opportunityif there exists a predictable process H with values in Rd such that

(H · S)T (ω) ≥ 0 ∀ω ∈ Ω,

(H · S)T (ω) > 0 for at least one ω ∈ Ω .

Conversely, we will say that the market is free of arbitrage (FOA).

Let us pay some more attention on the arbitrage condition. Such condition is also calledin nance, the no-free meal condition. In fact what denition 2.2.9 says is that in a FOAmarket no gain can be made without taking any risk. Completeness of the market andabsence of arbitrage are the two main assumptions that are (almost) always assumed tohold in mathematical nance (even if in the real world they does not hold!)

The following is a fundamental result.

Proposition 2.2.10 (Law of one price). In a FOA and complete market model, there existsan unique fair price for every attainable contingent claim.

2.2.2 The Binomial Model

Exercise 2.2.1. One period binomial model. Let us consider a one period binomialmodel, i.e. T = 1 and let us consider a market with two possible states, namely Ω = ωd, ωuwith P(wd) = pd ∈ (0, 1). For simplicity let us consider one risk-less bond S0 with interest

rate r > 0 and one risky asset S1 with return u if ω = ωu and d if ω = ωd, with u > d > 0.We thus have the following situation

S00 = 1 S0

1 = 1 + r S10 = s0 > 0

S11(ωu) = s0u

S11(ωd) = s0d

(1− pd)

pd

Find the non-arbitrage condition for the one period binomial model.

Exploiting the result in Ex. 2.2.1 we have the following fundamental result.

Proposition 2.2.11. The one-period Binomial Model is complete i 0 < d < 1 + r < u.

Proposition 2.2.12. Let 0 < d < 1 + r < u and consider the 1-period binomial model, thensuch model is complete, namely, given X discounted payo of a contingent claim

X : Ω→ R, X is F-measurable, xd := X(ωd) and xu := X(ωu)

then we have to prove that there exists an initial capital V0 and a self-nancing portfolio Hsuch that

X = V0 + (H · S)1 . (2.3)

This means that for every contingent claim X there exists an initial endowment V0 and areplicating strategy H such that the following holds X = V0 + (H · S)1.

Proof. Exercise.


By the Law of one price, see Prop. 2.2.10 we have that V0 determined as in Prop. 2.2.12is the unique price of the Contingent Claim X.

Proposition 2.2.13 (Law of one price). There exists a unique value V0 such that eq. (2.3)holds.

2.2.3 Towards Risk Neutral Measure (RNM) and Hedging

In the real world it happens that we do not know the probability pu that a given asset goesup neither the probability pd that the asset price goes down. Therefore saying that thetoday's price of a contingent claim written on a given asset S1 with maturity time tomorrowis its expected value makes no sense. How do we compute then the fair price of a givencontingent claim? In order to go beyond this problem to possible approaches are usuallyexploited in nance, the rst is known as the backward approach or hedging portfolio whereasthe second takes into account the so called risk neutral measure.

The hedging portfolio

The aim of a hedging strategy, is to construct a portofolio of d risky assets and one risk-lessbond whose nal value is the same of the terminal value of a given contingent claim. Fromthe the FOA principle the initial capital that has be invested in the portfolio and the fairprice of the claim should coincide.

The portfolio is built via backward iteration and it is known is standard literature ashedging portfolio, see, e.g. [Shr04].

Let us consider for the moment the 1-period binomial market model introduced in Ex-ample 2.2.1. According to what said above we aim to nd π0 and π1, the number of bondsthe investor should buy/sell, resp. share of the risky asset to buy/sell.

At initial time t = 0 the portfolio worth

V0 = π0S00 + π1S

10 , (2.4)

whereas at maturity T = 1, taking into account that the bond is deterministic and at timeT = 1 it worth S0

1 = S00(1 + r), we have

V1(ω) =

π0S

00(1 + r) + π1S

11(ωu), if ω = ωu ,

π0S00(1 + r) + π1S

11(ωd), if ω = ωd ,

. (2.5)

Computing system (2.5) for π0S00(1 + r) we have

V1(ωu)− π1S11(ωu) = V1(ωd)− π1S

11(ωd) ,

so that eventually we get

π1 =V1(ωu)− V1(ωd)

S11(ωu)− S1

1(ωd), π0 =

V1(ωd)− π1S11(ωd)

S00(1 + r)

. (2.6)

Example 2.2.2. Let us consider the following market, composed for the sake of simplicityby one risky asset and one risk-less bond,


S00 = 1 S0

1 = 1 S10 = 1

S11(ωu) = 2

S11(ωd) = 1

2

Let us further consider the case of a standard European call option written on theunderlying S1 with strike price K = 1, so that it nal payo is given by

Φ(S11) = max

0, S1

1 −K

=:(S1

1 −K)

+,

and we have the following

Φ(S11)(ω) =

1, if ω = ωu ,

0, if ω = ωd ,. (2.7)

Since our goal is to construct the replicating portfolio we have that the nal payo ofthe claim given in eq. (2.7) must be equal to the nal value of the hedging portfolio. Thus,from eq. (2.6) we have that

π1 =1

2− 12

, π0 = −1

3.

Computing thus explicitly the value of the portfolio we have

V0 = π0S00 + π1S

10 =

1

3,

V1(ωu) = π0S00 + π1S

11(ωu) = 1 = Φ(S1

1)(ωu) ,

V1(ωd) = π0S00 + π1S

11(ωd) = 0 = Φ(S1

1)(ωd) ,

(2.8)

The risk neutral measure

The second approach to option pricing is the so called risk-neutral pricing, namely we haveto nd an equivalent probability measure Q, equivalent w.r.t. the unknown real worldprobability P, such under the measure Q, the investor is indierent between investing in themarket and buying the option.

Throughout the section we will assume to be in a complete and FOA market, namely weassume that 0 < d < 1 + r < u holds.

Let us then compute the initial value of the portfolio, given the quantity π0 and π1 ineq. (2.4) and (2.5). Then we have

V0 = π0S00 + π1S

10 =

V1(ωd)− π1S11(ωd)

S00(1 + r)

S00 +

V1(ωu)− V1(ωd)

S11(ωu)− S1

1(ωd)S1

0 =

=1

(1 + r)

[S1

0(1 + r)V1(ωu)− S10(1 + r)V1(ωd) + S1

1(ωu)V1(ωd)− S11(ωd)V1(ωu)

S11(ωu)− S1

1(ωd)

]=

=1

(1 + r)

[S1

0(1 + r)− S11(ωd)

S11(ωu)− S1

1(ωd)V1(ωu) +

S11(ωu)− S0

0(1 + r)

S11(ωu)− S1

1(ωd)V1(ωd)

].

(2.9)

Let us then dene

qd =S1

0(1 + r)− S11(ωd)

S11(ωu)− S1

1(ωd), qu =

S11(ωu)− S0

0(1 + r)

S11(ωu)− S1

1(ωd),


or according to the notation introduced above

qu =u

u− d=u− (1 + r)

u− dand qd = 1− qd = − d

u− d=

1 + r − du− d

, (2.10)

then qu and qd are the so called risk neutral probabilities. It can be seen thatqd > 0 , qu > 0 ,

qd + qu = 1 ,;

so that qd and qu dene a new probability measure over Ω, usually denoted as Q. We thushave

Q(ωd) = qd, Q(ωu) = qu .

This new probability measure Q is called in nance risk-neutral measure and under themeasure Q it holds V0 = EQ[X], see Prop. 2.2.12.

In particular we have that the fair price of a contingent claim written on S1 is given by

V0 =1

(1 + r)[quV1(ωu) + qdV1(ωd)] ,

and by the law of the unique price V0 should coincide with V0 found via the hedging portfolio.

Exercise 2.2.2. Show that in the market of Example 2.8, the fair price of the optioncomputed with the risk neutral probabilities is the same of the fair price found in 2.8.

Denition 2.2.14. A probability measure over Ω is a risk-neutral measure (RNM) i

S10 = EQ[S1

1 ] .

Therefore it has been proved that the problem of not knowing the real world probabili-ties can be overcome introducing a new probability measure such that they incorporate allinvestors' risk premia and the investor, under this new probability measure, is indierentbetween choosing the risky investment and the non risky one, from here comes the namerisk-neutral probability.

Therefore by Ex 2.2.1 we have that for the 1-period Binomial model the non-arbitragecondition implies the existence of a Risk Neutral Measure. What can we say about theinverse implication?

Proposition 2.2.15. Let us consider a 1-period binomial-model, then the market is free ofarbitrage (FOA) if and only if there exists a risk neutral measure.

Previous proposition can be generalized to state an elementary, but interesting versionof the rst fundamental theorem of asset pricing.

Theorem 2.2.16 (1st Fundamental Theorem of Asset Pricing). Let (Ω,F ,P) be a niteprobability space on which is dened a market model composed of 1 risk-less asset and drisky assets. The following are equivalent:

1. the market is FOA;

2. ∃ Q ∼ P such that EQ[(H · S)t] = 0 ∀t = 1, . . . , T and ∀H replacing strategy.

2.3 The multi-period binomial Model 103

2.3 The multi-period binomial Model

Let us now consider the case of a multi-period binomial model. In particular let us considerT ∈ N+, T > 1 and a sequence of i.i.d. Bernoulli-type random variables Zt t=1,...,T suchthat

P(Zt = u) = pu ∈ (0, 1) ,

P(Zt = d) = 1− pu .

S10

S11(ωu)

S11(ωd)

S12(ωd, ωd)

S12(ωu, ωu)

S12(ωu, ωd)

pu

pd

pu

pu , pd

pd

Thus the price evolution process S is given by

S1t = ZtS

1t−1 .

Let us further assume that the market is FOA, i.e. we assume u > 1 + r > d.

Remark 2.3.1. In the present setting, previous condition is necessary in order to have amarket FOA but it is not sucient. Anyway we will assume the market to be FOA andu > i+ r > d to hold.

2.3.1 Pricing under the risk-neutral measure and the hedging porto-folio

The theory developed in Sec. 2.2.3 holds with few adjustment in the present setting. Inparticular we have that the hedging and the risk-neutral approach lead to the same results.

Let us consider a FOA multi-period binomial model and a contingent claimX with payofunction g = g(ST ). By Prop. 2.3.2, the contingent claim X can be replicated, i.e. thereexist a strategy H and an initial endowment V0 such that X = V0 + (H · S)T . Moreoverby the FFTAP, see Thm. 2.2.16, there exists a measure Q ∼ P such that EQ[(H · S)T ] = 0.Then

EQ[X] = EQ[V0 + (H · S)T ] = V0

This means that in order to compute the fair price V0 we have to evaluate the expectedpayo. Note that

S1T = S1

0 uydT−y ,

where y stands for the number of up-jumps in 1, · · · , T. Since every movement in theprice evolution S is a Bernoulli trial with probability qu = Q(Zu = u), it turns out thaty ∼ Bin(T, qu). That said, it follows

V0 = EQ[X] = EQ[g(S1T )] = EQ[g((1 + r)−TS1

T )]

=

T∑y=0

(T

y

)qyu(1− qu)T−yg

((1 + r)−TS1

0 uydT−y

).


Proposition 2.3.2 (Backward approach to replicating strategy). Let us consider a FOAmulti-period binomial model, then the model is complete, namely every contingent claim Xhas a representation of the form

X = V0 + (H · S)T ,

with V0 ∈ R and H a predictable process.

Proof. Consider

V0 + (H · S)T−1 +HT∆ST = VT−1 +HT∆ST = X

Applying the 1-period Binomial Model technique to nd the one period hedging strategy,we can compute the value for HT and VT−1. Then as considering VT−1 as a new contingentclaim and applying the same scheme we nd the value for HT−1 and VT−2 such that thefollowing holds VT−2 +HT−1∆ST−1 = VT−1. Iterating, we construct H and V0 is uniquelydetermined.

In particular the backward approach in the multi-period binomial model is just an iter-ation of multiple one-period binomial model.

Exercise 2.3.1. Let us consider a market with one risky asset S1 and one risk-less bondS0 with interest rate r = 0 and initial value S0

0 = 4. Let us consider a three-period binomialmodel, namely T = 3. The asset evolves as

S10 = 100

S11(ωu) = 120

S11(ωd) = 80

S12(ωd, ωd) = 60

S12(ωu, ωu) = 140

S12(ωu, ωd) = 100

S13(ωd, ωd, ωd) = 40

S13(ωu, ωu, ωu) = 160

S13(ωu, ωu, ωd) = 120

S13(ωu, ωd, ωd) = 80

Let us further consider a claim with nal payo given by

Φ(S13) =

(S1

3 − 100)

+.

Compute the fair price of the claim Φ with both the hedging portfolio and the risk neutralprobabilities.

Chapter 3

Continuous time models

3.1 Towards the Black-Scholes formula

Let us x a maturity time R 3 T < ∞ and let us consider a time interval [0, T ]. Let usthen divide the time interval in N ∈ N+ subintervals of equal length δN = T

N such that⋃Nk=1

[(k − 1)δN , kδN

]= [0, T ].

Fixed N , let us consider a multi-period Binomial model with

Sk(ω) =

uNSk−1 if ω = ωu

dNSk−1 if ω = ωd(3.1)

where

uN = eσ√δN+αδN and dN = e−σ

√δN+βδN .

Moreover, let rN = rδN , then exploiting eq. (2.10) the risk neutral probability Q is givenby

qN =erN − dNuN − uN

. (3.2)

We further have that

qN →1

2, as N →∞

We have that that under the risk neutral measure QN the risk neutral price of a contin-gent claim C with payo function F = F (ST ) is the discounted expected value of the futurepayo, i.e.

H0 =1

(1 + rN )NEQN [F (SN )]

=1

(1 + rN )N

N∑k=0

(N

k

)qk(1− q)N−kF (ukdN−kS0) , (3.3)

see, e.g. [Pas11, Pas07].

105

106 3. Continuous time models

Let us now consider an European Put option, characterized by strike price K and ma-turity time T and payo F = (K − SN )

+, exploiting eq. (3.3) it's price is given by

P(N)0 =

1

(1 + rN )NEQN

[(K − SN )+

]=

1

(1 + rN )NEQN

[(K − S0e

XN )+],

where

XN =

N∑k=0

Y(N)k

is an auxiliary random variable dened in order to take into account the number of up/downmoves.

Indeed Y(N)k for k = 0, . . . , N are independent, identically distributed random variables

of Bernoulli type s.t.

Y(N)k =

σ√δN + αδN with probability qN ,

−σ√δN + βδN with probability 1− qN .

For each N , XN are i.i.d. random variable with nite mean and variance and applyingthe central limit theorem we have that

XNL−→ X ∼ N

((r − σ2

2

)T, σ2T

),

since

EQN [XN ]→(r − σ2

2

)T and VarQN [XN ]→ σ2T .

Exploiting previous result, it makes sense to dene P0 = limN→∞ PN0 as the (Black-Scholes)price of an European Put (with strike price K and maturity T ). In particular, the followingholds.

Theorem 3.1.1. Let be PN0 the price of a European put option written on the underlying Swith strike price K and maturity time T in the N−period model described in eq. (3.1) withparameters uN and qN in eq. (3.2), then it exists P0 dened as

P0 = limN→∞

PN0

such that it holdsP0 = e−rTEQ

[(K − S0e

X)+]

, (3.4)

with

X ∼ N((

r − σ2

2

)T, σ2T

).

Proof. The proof follows by the denition of PN0 and the fact that XNL−→ X.

Theorem 3.1.2. The price P0 in eq. (3.4) can be explicitly computed as

P0 = Ke−rTΦ(−d2)− S0Φ(−d1) (3.5)

3.2 Martingale Pricing Theory 107

where

d1 =ln[S0

K

]+(r + σ2

2

)T

σ√T

d2 =ln[S0

K

]+(r − σ2

2

)T

σ√T

.

Denition 3.1.3 (Black-Scholes price). The price P0 in eq. (3.5) is called the Black-Scholesprice for an European put option.

Using previous results for European options, see, e.g. [Shr04, Pas11] we have that theprice for an European call option is given by

C0 = S0Φ(d1)−Ke−rTΦ(d2) (3.6)

with d1 and d2 as before.Furthermore we have that for an european call with maturity time T and strike price K,

and for an European Put option with the same strike price and maturity we have that

C(S0, 0) = P (S0, 0) + S0 −Ke−rT . (3.7)

The previous relation is called Put-Call Parity.So far, we computed the European options' prices at the initial time t = 0. These results

can be generalized at any given time t ∈ [0, T ]. The Black-Scholes formulas to price a Calland a Put option with strike price K and maturity time T at a given time t are

C(St, t) = StΦ(d1)−Ke−r(T−t)Φ(d2) (3.8)

P (St, t) = Ke−r(T−t)Φ(−d2)− StΦ(−d1) (3.9)

where

d1 =ln[StK

]+(r + σ2

2

)(T − t)

σ√T − t

d2 = d1 − σ√T − t =

ln[StK

]+(r − σ2

2

)(T − t)

σ√T − t

.

Exercise 3.1.1. Compute the price of an European Call Option with payo function givenby

g(x) = (x− K)+

3.2 Martingale Pricing Theory

As for the discrete market model, also in continuous time we can price a contingent claimwith the so called risk-neutral measure. Girsanov Th. 1.4.4 represents in this sense the maintool when one is to price a contingent claim Φ under the risk-neutral measure.

Denition 3.2.1. A probability measure Q is said to be risk-neutral if Q is equivalent tothe real world probability measure P and the discounted price e−rtSt is a Q−martingale.


Then, assuming the asset price S to evolve according to a geometric Brownian motion ofmean µ, we want to nd a dierent measure Q, equivalent to the original one P, such thatthe new drift of S is equal to the interest rate r. Then this happen, it can be easily shownthat the discounted stock price is a Q−martingale. Let us see this in details.

Let us x a maturity time T < ∞ and let us consider a bank account whose value attime t is denoted by Bt and a risky asset whose price at time t is given by St, t ∈ [0, T ].We want to construct a portfolio πt = (πSt , π

Bt ) where πSt , resp. π

Bt , is the unit of money

invested in the bank account, resp. risky asset.Thus the value of the so constructed portfolio at time t is given by

Vt = πSt St + πBt Bt . (3.10)

In nancial mathematics, and in particular for application to hedging problems, a crucialrole is played by the notation of self-nancing portfolio.

Denition 3.2.2 (Self-nancing portfolio). A portfolio V = Vtt∈[0,T ] is self-nancingif for any time t ∈ [0, T ] it satises

dV πt = πSt dSt + πBt dBt .

In other words a portfolio is self-nancing if there is no exogenous infusion or withdrawalof money so that the purchase of a new asset must be nanced by the sale of an old one.

By applying the Itô-Doeblin Lemma we have the so called continuous-time self-nancingcondition

St dπSt +Bt dπ

Bt + dSt dπ

St + dBt dπ

Bt = 0 .

Theorem 3.2.3 (First fundamental theorem of asset pricing). If a market model has a riskneutral probability measure, then the market is free of arbitrage.

Proof. See, e.g. [Shr04, Th.5.4.7].

Notice indeed that the 1st FTAP provides a condition one can apply in order to checkwhether or not a model admits arbitrage opportunities.

In this scenario we have an equivalent formulation for the arbitrage condition stated fora discrete market model in Def. 2.2.9. In particular, an arbitrage opportunity is a self-nancing strategy which has zero initial value and probability strictly greater than zero tohave a non null value at a later time. In other words there is the opportunity of beating themarket.

More formally we have that an arbitrage possibility on a nancial market is a self-nancing portfolio π such that it holds simultaneously

V π(0) = 0

P(V π(T ) ≥ 0) = 1

P(V π(T ) > 0) > 0 .

A market is said to be arbitrage free if there are no arbitrage possibilities.Recall that a market model is complete when every contingent claim can be replicated.

The 2nd FTAP gives the requirements for a market to be complete.


Theorem 3.2.4 (Second fundamental theorem of asset pricing). Let us consider a mar-ket model that has a risk-neutral probability measure. The model is complete, i.e. everycontingent claim can be hedged, if and only if the risk-neutral probability is unique.

Proof. See, e.g. [Shr04, Th.5.4.9].

In particular, from the fact that all discounted price processes Sit are martingale w.r.t.the EMM Q, thus also the discounted value process V is a Q-martingale.

Exploiting the completeness, i.e. the existence of the unique EMM Q, and the lack ofarbitrage opportunities of a market, the price of every contingent claim can be computed asthe Q-expectation of its deated value at the nal time T . If a contingent claim C can bereplicated by a strategy πC , then the price of the claim has to be equal to the initial value of

the replicating strategy πC(0) otherwise an arbitrage opportunity occur, i.e. V πC

(0) = C(0).Indeed, by denition of replicating strategy it holds

V πC

(T ) = C(T )

but since V πC

Ntis a martingale under the EMM

V πC

(0)

N0= EQ[

V πC

(T )

NT]

that leads to

C(0) = EQ[C(T )

NT] ,

where Nt denotes the chosen numeraire.In particular let us focus on the case in which we choose as numeraire the bank account

Bt evolving according todBt = rBtdt , (3.11)

with r a positive constant representing the deterministic interest rate. This means that inthe following the cash account is used as deation factor.

Let us further consider a stock S with dynamic

dSt = µSt dt+ σSt dWt , (3.12)

with µ and σ some deterministic constants and Wt is a P-Brownian motion where asusual we denote by P the real-world measure.

Let us now dene Zt := StBt

be the discounted price process, where St solves eq. (3.12)and Bt solves eq. (3.11).

Under previous assumptions and exploiting the Itô Lemma we have that

dZt = (µt − r)Zt dt+ σZt dWt

and moreover, by using the 1st FTAP 3.2.3 we have that Zt has to be a Q-martingale.Applying Girsanov Th. 1.4.4 we have that Zt is a Q-martingale if and only if it exists aQ−Brownian motion W s.c. it can be written as

Wt = Wt +

∫ t

0

ηs ds


with ηs = µs−rσs

. The process ηt is known in literature as market price of risk.Since, in the present setting, the risk-neutral measure Q is uniquely determined, then

the 2nd FTAP (3.2.4) guarantees that the market is complete.Let us notice that under the new measure Q the asset price S in eq. (3.12) evolves

accordingdSt = rSt dt+ σtStdWt .

Remark 3.2.5. If we add a further noise source, e.g. if we consider a second, independentfrom the rst Brownian motion then ηt is not uniquely determined and we can use one ofits (innitely many) realization to dene Zt as a martingale. Furthermore the market is nolonger complete.

The Black-Scholes price

Girsanov theorem 1.4.4 nds perhaps its best application in nance. The main idea is thatin a FOA market an asset price S should have deterministic evolution equal to the risk-lessbond with interest rate r > 0 or equivalently the discounted stock price e−rtSt has to be amartingale. Therefore when we are to price a contingent claim F written on the underlyingS we have to compute the expectation w.r.t. a suitable measure Q s.t. the discounted priceis a martingale. In order to do this we need the stock price to evolve according this newmeasure. Here comes the need for a theorem that allows us to change the measure of thedriving noise.

Let us then assume we are dealing with a stock price S evolving, in a standard probabilityspace (Ω,F , (Ft)t ,P), according to

dSt = µStdt+ σStdWPt , (3.13)

with W a standard P−Brownian motion and µ and σ some real constants. Our aim is tond an equivalent measure Q s.t. e−rtSt is a martingale.

Formally, since σ 6= 0, we can divide everything in eq. (3.13) by σ and denote ξ := µ−rσ .

Applying then Girsanov theorem 1.4.4 it exists Q ∼ P s.t.

WQt = ξt+W P

t .

Then substituting the Q−Brownian motion into eq. (3.13) we have that under Q theasset evolves according to

dSt = µStdt+ σSt

(dWQ

t − ξdt)

= rStdt+ σStdWQt , (3.14)

then computing, via Itô's lemma, it is easy to show that e−rtSt is a Q−martingale.Let us suppose we want now to price an European call option with nal payo given by

Φ(S) = (ST −K)+ written on the underlying S evolving asdSt = Stµdt+ StσdWt ,

S0 = s0 ,,

for some constants µ and σ and Wt a standard Brownian motion.Then the fair price P at time t = 0 is given by

C := Q[e−rTΦ(ST )

].


We have seen previously how we can nd an equivalent measure Q, which exists and isunique from Th. 3.2.3 and Th. 3.2.4, under which the discounted stock price is a martingale.Then, under Q the asset S evolves as

dSt = Strdt+ StσdWQt , S0 = s0 , ,

with r > 0 the interest rate. Applying Itô's lemma to the function X = f(S) = logS wehave that the return X evolves as

dXt = (r − σ2

2 )dt+ σdWQt , X0 = x0 , .

We can easily see that Xt ∼ N(

(r − σ2

2 ), σ2t).

Then we can compute the fair price for the option Φ as

C := EQ[e−rT (XT − logK)+

],

whose solution is exactly the BS price in eq. (3.6)

Exercise 3.2.1. Shows that

C := EQ

[e−rT

(eXT −K

)+

]= N(d1)−Ke−rTN(d2) ,

with N the cumulative distribution of the standard Gaussian law, and

d1 =ln[S0

K

]+(r + σ2

2

)T

σ√T

d2 =ln[S0

K

]+(r − σ2

2

)T

σ√T

.

HintWrite the expectation as

C := EQ

[e−rT

(eXT −K

)+

]= e−rTEQ

[(eXT −K

)1XT>logK

],

and write the Brownian motion as

WT =√TW1 W1 ∼ N (0, 1) .

What we have done above is to take the risk-less asset as numéraire so that we considerthe asset price divided by the value of the risk-free asset. In particular we focus on

Sit :=SitBt

,

with Si the i−th risky asset and B the risk-less bond.Anyway taking into account the cash account process as a numéraire leads to consider

an equivalent martingale measure (EMM or MM for short) which is not always the bestpossible choice. Hence we might want to use dierent processes as numéraire. Let us in fact


consider a (dierent) process Nt as numéraire. Then a dierent EMM arises. Is there anylink between the two martingale measures and how can we switch between them?

Let us consider a given probability measure Q on a probability space (Ω,F) and a strictlypositive Q-martingale such that

EQ[Mt] = 1 ∀t ∈ [0, T ] .

Then it is possible to dene a new probability measure P on the same space (Ω,F) suchthat for any A ∈ F

PM (A) := EQ[MT1A] . (3.15)

To see that P is indeed a probability measure let us observe that

(i) PM (Ω) = 1;

(ii) PM (A) ≥ 0 for any A ∈ F follows by the positiveness of M ;

(iii) PM(ti≥1 Ai

)= EQ[MT1ti≥1Ai

]= EQ[∑

i≥1MT1Ai]

=∑i≥1 PM (Ai).

Moreover Q(A) = 0 ⇐⇒ PM (A) = 0 and hence the measures Q and PM are equivalent.

Exercise 3.2.2. Show that, for any random variable X, it holds

EPM [X] = EQ [MTX] .

(Hint)Use (3.15) to show that

EQ[MT1A] = EPM[1A

],

then the claim follows approximating any r.v. by indicator function and applying densitycriterion.

It is thus natural to take into consideration the Radon-Nikodym density dPMdQ so that we

will usually write

EPM [X] = EQ[dPMdQ

X

].

Therefore we now can switch between the measure as follows

EPMt [X] =

EQt

[dPMdQ X

]EQt

[dPMdQ

] =EQT

[dPMdQ X

]Mt

.

3.2.1 Hedging and replicating

Let us consider a bank account with a dynamic of the following type

dBt = rtBt dt

and a risky asset which follows the dynamic

dSt = µtSt dt+ σtSt dWt


where Wt is a standard Brownian Motion. The value process associated to a portfolioπ = (πSt , π

Bt ) is

Vt = πSt St + πBt Bt ,

and if it is self-nancing its dierential, according to Itô's formula is

dVt = πSt dSt + πBt dBt =(πSt µtSt + πBt rtBt

)dt+ πSt σtSt dWt

=

(πSt StVt

µt +πBt BtVt

rt

)Vt dt+

πSt StVt

σtVt dWt

= (rt + (µt − rt)θt)Vt dt+ θtσtVt dWt (3.16)

where

θt =πSt StVt

represents the fraction of the portfolio value invested in the risky assent and consequently,(1− θt) the fraction deposited in the bank account.

If Vt is the value of a given security at time t and we want to replicate it by a self-nancing strategy π composed of security S and cash B, i.e. π = (πS , πB) then by (3.16) θtis the fraction of wealth invested in the security S at time t. Note that this quantity doesnot change if a dierent measure is chosen since the measure aects only the drift term.

Delta Hedging

One of the main issue in mathematical nance is how to hedge the risk we take sellingor underwriting a contract. We will just give here few introductory notions on hedging, adetailed expository of the topic can be found in [DNØP+09, Shr04, FS11b].

The simplest dynamic hedging strategy is the so called ∆-hedging.In a Black-Scholes model, consider an investor that wants to protect himself from the

risk taken selling an option written on an underlying S with payo function F = F (St, t)by investing in the underlying asset. Recall that by assumption of Black-Scholes model, theunderlying follows an evolution of the following type

dSt = µSt dt+ σSt dWt .

The investor earns an initial wealth equal to F (0, S0) from the sale of the option, and hisaim is to allocate this amount into the stock and the bank account, so as to equal the valueof the option and the value of his portfolio, i.e. so as to construct a replicating portfolio. Letπ represent the amount invested in the underlying, then the value process satises

dV = (rV + (µ− r)π) dt+ σπdWt . (3.17)

In order to replicate the value of the option, it must hold

V (T ) = F (t, ST ) ,

and more generally, we look for the strategy π such that

X(t) = F (t, St) .


The previous condition is veried if the processes X and F solve the same SDE. By applyingItô formula, the process F (t, St) satises the following SDE

dF (t, S(t)) =

(∂Ft(t, St) +µSt∂SF (t, S(t)) +

1

2σ2S2

t ∂SSF (t, St)St

)dt+σSt∂SF (t, St) dWt

(3.18)then by equating the two SDEs (3.17) and (3.18) we have that π must satisfy

π = ∂SF (t, S(t))St ,

or equivalently,π

St=∂F

∂S(t, S(t)) = ∆F .

This means that in order to replicate the option, the number of shares to be held, i.e. πSt,

equals the derivative of the option price with respect to the underlying, called the ∆ of theoption.

In the (Black-Scholes) Call case ∆Call = Φ(d1) > 0 whereas in the Put case ∆Put =Φ(d1)− 1 < 0.

Now, being the portfolio deterministic and from the non-arbitrage condition, we musthave that the evolution of the portfolio equals the risk-less bond with deterministic interestrate r > 0, namely

dVt =

(µ∂SFSt − ∂tF − µSt∂SF −

1

2σ2S2

t ∂SSF

)dt = rVtdt = r (∆St − F ) dt = r∂SFSt−rFdt .

In conclusion we have the following fundamental result.

Theorem 3.2.6 (Black-Scholes PDE). Under the setting introduced above we have that thefair price of a European call option C written on the underlying S is the solution to

∂tF + 12σ

2S2t ∂SSF + rSt∂SF = rF ,

F (T, ST ) = (ST −K)+,

. (3.19)

It can be shown, see, e.g. [Shr04, Pas11], that the solution to eq. (3.19) coincide withthe fair price given in eq. (3.8).

A (slightly) dierent method of ∆-hedging an options position is the following. Assumethat we short an option with payo function F (t, S(t)) and we want to hedge the risk byinvesting in the underlying. The portfolio value process is given by

V (t, St) = ϕtSt − F (t, St) ,

where ϕt represent the amount invested in the stock and it's the process we want to determinein order to vanish the portfolio's risk. Note that

dV (t, St) = ϕt dSt − dF (t, dt)

=

(− ∂tF + (ϕt − ∂SF )µSt −

σ2S2t

2∂SSF

)dt+ (ϕt − ∂SF )σSt dWt .

Therefore ϕ = ∂F∂x (t, St) is the choice that reduces the risk to zero by vanishing the random-

ness of V . Therefore, at each time t, the investor has to build a portfolio composed by −1unit of the derivative and ∆ unit of the underlying.


Notice that in order to reach a risk-balance at every t ∈ [0, T ] we should be able to tradecontinuously in time. This is unrealistic since in real markets trading occur at xed times.Moreover, in the Black-Scholes model, the market is frictionless, while in the real marketstransactions incur fees.

Delta-Gamma Hedging

If the value of the underlying changes too quickly the ∆−hedging is not enough to cover therisk. Thus some other countermeasures has to be taken. Here comes the Γ−hedging.

As before, let F = F (St, t) denote the payo of a derivative on the underlying S. The Γis thus dened as the second derivative of the value function w.r.t. the underlying, i.e.

ΓF =∂∆F

∂F=∂2F

∂S2.

In the (Black-Scholes) Call case we have

ΓCall = ΓPut =Φ′(d1)

σSt√T − t

.

The Greeks

The derivatives of an option value with respect to the parameters involved are known asGreeks. They reect the sensitivity of the option price and then are used to (try to) measurethe risk of a nancial position w.r.t. the variation of the relevant parameters. In additionto ∆ and Γ introduced above, the standard Greeks for an option F are

Rho : ρ =∂F

∂r

Theta : Θ =∂F

∂t

Vega : ν =∂F

∂σ

For an European Call option, we have

ρCall = K(T − t)e−r(T−t))Φ(d2) ,

ΘCall = −σStφ(d1)

2√T − t

− rKe−r(T−t)Φ(d2) ,

νCall = S(t)φ(d1)√T − t ,

.

Dividend-Paying Stocks

Up to now we have always assumed that shareholders don't receive dividends. Clearly thisassumptions often fail to be true in reality. Therefore what does happen when the underlyingpay dividends? How do the BS pricing formulas eq. (3.8)-(3.9) change?

Denition 3.2.7. A stock S is said to pays continuous dividends at a constant dividendyield D0 if in the innitesimal time interval dt a value equal to D0 S dt is given to the holdersof a share of the stock S.


In the present case, the price of a Call option written on the underlying S with strikeprice K and maturity time T becomes

C(St, t) = Ste−D0(T−t)Φ(d′1)−Ke−r(T−t)Φ(d′2)

where

d′1 =ln[StK

]+(r +D0 + σ2

2

)(T − t)

σ√T − t

and d′2 =ln[StK

]+(r +D0 − σ2

2

)(T − t)

σ√T − t

.

Analogous formula holds for the Put options. It is further trivial to see that if D0 = 0 weare in the standard BS setting, we refer to [Shr04] for a detailed treatment of the topic.

A dierent situation arise when the dividends are payed at xed times. Typical case ofdiscrete dividends are bonds or interest rate derivative that usually pays dividend every sixmonths.

Therefore the BS formula (3.8) reads as

C(S0, 0) = (S0 −D)Φ(d1)−Ke−r(T−t)Φ(d2)

where D is the actualized value of these dividends which have been payed within time Tand

d1 =ln[S0−DK

]+(r + σ2

2

)T

σ√T

and d2 = d1 − σ√T =

ln[StK

]+(r +D0 − σ2

2

)T

σ√T

.

Also in this case, an analogous formula holds for the Put option.Moreover, it can be shown a Put-Call Parity relation is still in charge for both the

continuous and the discrete dividends. For the particular case of the discrete dividend eq.(3.7) reads

C(S0, 0)− P (S0, 0) = S0 −D −Ke−rT .

Exercise 3.2.3. Write the explicit formula for the case of an asset paying continuous divi-dends.

3.3 Packages and Exotic options

3.3.1 Packages

More complicate derivatives can be constructed by combination of Call and Put options.They can be priced by superposition principle. Here follows some examples.

Bull Spread Let 0 < K1 < K2 <∞. A Bull Spread's payo at expiration T is given by

f(T ) =

0 ST ≤ K1

ST −K1 K1 < ST ≤ K2

K2 −K1 K2 < ST

,

see, Fig. 3.1a. An investor buys a bull spread option if he expects the stock price toincrease (here the name bull).

3.3 Packages and Exotic options 117

ST

fT

K1 K2

(a) Bull spread

ST

fT

K1K2K3K4

(b) Bear spread

Let us notice that this payo function can be regarded as the dierence of the payo oftwo call options on the same underlying and maturity, with respectively strike pricesK1 and K2. Indeed, let

C1(T ) :=

0 ST ≤ K1

ST −K1 K1 < STand C2(T ) :=

0 ST ≤ K2

ST −K2 K2 < ST

then we havef(T ) = C1(T )− C2(T ) .

Thus a Bull Spread can be achieved by buying (long position) the call option withstrike price K1 and selling (short position) the call with the higher strike price K2.

It follows that the price of such a Bull Spread at time t < T can be computed exploitingeq. (3.6),

f(t) = C1(t)− C2(t)

= St[Φ(d1(K1))− Φ(d2(K2))]− e−r(T−t)[K1Φ(d2(K1))−K2Φ(d2(K2))]

where

d2(Ki) =ln[St]− ln[Ki] +

(r − σ2

2

)(T − t)

σ√T − t

and d1(Ki) = d2(Ki)+σ√T − t for i = 1, 2.

Bear Spread Let 0 < K1 < K2 <∞. The derivative's payo at expiration T is given by

f(T ) =

K2 −K1 ST ≤ K1

K2 − ST K1 < ST ≤ K2

0 K2 < ST

.

Conversely to a bull spread, a bear spread leads to prot if the stock price decreases,see, Fig. 3.1b.

Buttery Spread Let 0 < K1 < K3 < ∞ and K2 = K1+K3

2 . The derivative's payo atexpiration T is given by

f(T ) =

0 ST ≤ K1

ST −K1 K1 < ST ≤ K2

K − 3− ST K2 < ST ≤ K3

0 K3 ≤ ST


Straddles In this case the derivative's payo function is given by f(T ) = |ST −K|.

Strangles Let 0 < K1 < K2 < ∞ and dene K ′2 = K1+K2

2 and K ′1 = K2−K1

2 . In this casepayo at expiration T is given by

f(T ) =

0 ST ≤ K1

ST −K1 K1 < ST ≤ K2

K − 3− ST K2 < ST ≤ K3

0 K3 ≤ ST

Condor Option Let 0 < K1 < K2 < K3 < K4 < ∞ such that K4 −K3 = K2 −K1. Itspayo is given by the function

f(T ) =

K2 −K1 ST ≤ K1

K2 − ST K1 < ST ≤ K2

0 K2 < ST ≤ K3

ST −K3 K3 < ST ≤ K4

K4 −K3 K4 ≤ ST

Exercise 3.3.1. Plot the payo functions for the buttery spread, straddles, strangles andthe condor option and from the plot guess what are the investor's expectations for the stockprice.

3.3.2 Exotic options

3.4 Main nancial models

We are now to introduce the main models used in mathematical nance.

3.4.1 Equity market models

Generalized Geometric Brownian Motion

Let Wt, with t ≥ 0 be a Brownian motion with associated natural ltration F(t) t ≥ 0.Moreover let αt and σt be two adapted (w.r.t. F(t)) stochastic processes.

Let us consider then the Itô process X(t), t ≥ 0 dened as

Xt =

∫ t

0

σs dWs +

∫ t

0

(αs −

σ2s

2

)ds

or, analogously, as

dXt = σ(t) dWt +

(αt −

σ2t

2

)dt .

Applying then Itô's lemma we have that

(dXt)2 = σ2

t dt .

3.4 Main nancial models 119

Let us consider an asset price S = St such that its behaviour follows

St = S0eXt = S0 exp

[ ∫ t

0

σs dWs +

∫ t

0

(αs −

σ2s

2

)ds

](3.20)

for a given positive constant S0 ∈ R+. The stochastic process S is given as a function ofthe process X. In particular S = f(t,Xt) = eXt for f(t, x) = f(x) = S(0)ex. Applying theItô-Doeblin formula

dSt = f ′(Xt) dXt +1

2f ′′(Xt)(dXt)

2

αtSt dt+ σtSt dWt .

The process α represents the instantaneous mean rate of return while the process σ(t) theinstantaneous volatility. Note that both processes can be random.

Previous dynamic includes all possible models of a positive, continuous asset price drivenby a single Brownian motion.

In particular if α and σ are constant functions, we have that St is as in Example 1.4, i.e.

S(t) = S0 exp

[(α− σ2

2

)t+ σW (t)

].

Note that this process St is not a martingale, whereas the martingale property of the Brow-nian motion could be misleading.

Let us now, instead of S0 exp[σWt], consider the following slight modication S0 exp[σW (t)−

12σ

2t]. Then we have the following result.

Proposition 3.4.1. Let Wt, with t ≥ 0 be a Brownian motion with associated naturalltration F(t) t ≥ 0 and σ a positive constant. Let Zt be the Ft-adapted process

Zt = exp[σWt −

1

2σ2t].

Then Zt is a martingale.

Proof. For any s, t such that s < t,

E[Zt| Fs] = E[

exp

[σWs −

1

2σ2t

]exp[σ(Wt −Ws)]

∣∣∣∣Fs]= exp

[σWs −

1

2σ2t

]E[

exp[σ(Wt −Ws)]| Ft]

= exp

[σWs −

1

2σ2t

]E[

exp[σ(Wt −Ws)]]

but since Wt −Ws ∼ N(0, t− s) then E[

exp[σ(Wt −Ws)]]

= exp[σ2

2 (t− s)]. Therefore

E[Zt|Fs] = exp

[σWs −

1

2σ2t

]exp

[σ2

2(t− s)

]= exp

[σWs −

1

2σ2s

]= Zs ,

then the claim follows.


2 4 6 8 10

0.5

1.0

1.5

2.0

2.5

3.0

(a) Geometric Brownian motions, µ = 0 σ = .1

2 4 6 8 10

10

20

30

40

50

60

70

(b) Geometric Brownian motions, µ = .3 σ = .4

3.4.2 Siegel's paradox

Consider two countries, Germany and America, and the spot rate of their currencies, theDeutsche Mark and the Dollar, denoted respectively by rDM(t) and r$(t).

We consider the Deutsche Mark as benchmark and we study the dollar/Deutsche markforeign-exchange rate Xt of $/DM, which behaves as a Geometric Brownian motion. ThenXt solves the following SDE

dXt

Xt= (rDM(t)− r$(t)) dt+ σXdB

∗t .

where B∗ is a Brownian motion under the measure P∗ ∼ P, risk neutral measure from theGerman point of view.

Now, we consider the process Yt = 100 × X−1t . Yt represents a security which pays

continuous dividends, and in particular σY = −σX and ln[Yt] = − ln[Xt] + c0 where c0 is asuitable constant (c0 = ln[100]).

The question is whether the following relation

dYtYt

= (r$(t)− rDM(t)) dt+ σY (t) dB∗t

holds from the point of view of an American investor. By applying Itô's Lemma, we havethat

dYtYt

=d(100×X−1

t )

100×X−1t

=dX−1

t

X−1t

.

Since

d

(1

Xt

)= −

(1

Xt

)2

dXt +d 〈X〉tX3t

= −(

1

Xt

)2

dXt +σ2X(t)

Xtdt

we obtain

dYtYt

= −dXt

Xt+ σ2

X(t) dt

= (r$(t)− rDM(t) + σ2Y (t)) dt+ σY (t) dB∗t .

3.4 Main nancial models 121

The previous equality implies that the US investor asks for a risk premium equals to σ2Y (t).

Viceversa if the dollar is used as benchmark, then the German investor asks for a riskpremium equals to σ2

X(t).Consider a more general case. Xt = (X0

t , X1t , . . . , X

nt )t∈[0,T ], for a given, positive value

T on the probability space (Ω,F ,P). The dynamic of X is the following

dXt

Xt= µ(t) dt+ σ(t) dBt .

A standard choice for the numéraire is to choose a component of the vector, let say the rstone X0, and normalize the vector with respect to it, i.e.

Xt =

(1,X1t

X0t

, . . . ,Xnt

X0t

)We consider a self-nancing trading strategy φ = (φ)t = (φ0

t , φ1t , . . . , φ

nt ) and its related

value process Vt(φ) = φt · Xt.

Proposition 3.4.2. The trading strategy φt is self-nancing with respect to the process Xif and only if it's self-nancing with respect to the normalized process Xt .

Then, by the self-nancing assumption, it follows that dVt(φ) = φt · dXt or analogously,

Vt(φ) = φ0 · X0 +∫ t

0φz · Xz.

Let CT be an attainable contingent claim, or in other words a contingent claim for whichthere exists a self-nancing trading strategy φ such that CT = VT (φ). Considering its naldiscounted value it follows that

CT =CTX0T

=VT (φ)

X0T

= VT (φ).

Now since Vt is a local martingale under P?, we have that

Ct = EP? [VT (φ)|Ft] = EP? [CT |Ft]

andCt = XT

0 Ct = EP? [CT |Ft] .The question is how and if the choice of the numéraire aects the previously analysis.Consider

Xt =

(X0t

Xit

, . . . ,Xi−1t

Xit

, 1,Xi+1t

Xit

. . . ,Xnt

Xit

).

the related value process V =V

Xiand the contingent claim price's process C =

C

Xiand

recall the notation E(Y ) = exp[Y − 1

2 〈Y 〉]. Then the following proposition holds.

Proposition 3.4.3. Xt is a martingale process under the equivalent measure Q∗i = E(Ni)P∗where

Ni(t) =

∫ t

0

σi(s)− σ0(s) dB∗s .

To conclude, note that by the previous proposition it follows that the probability measureP∗ is indeed a martingale measure with respect to the numéraire X0 if and only of Q∗i is amartingale measure w.r.t. the numéraire Xi. Therefore the previous paradox comes from apoor decision in choosing the martingale measure.


3.5 The Merton model

Chapter 4

Interest Rates

This chapter mainly follows [CC12], other exhaustive treatment of the topic can be foundin [BM06, Shr04].

4.1 Allowing for stochastic interest rates in the Black-

Scholes Model

4.1.1 The hedging portfolio

When we derived the Black-Scholes price in eq. (3.6) one of the assumption we made was toconsider a deterministic interest rate r. It appears that in the real world this assumptionsnever holds. This fact has lead to a generalization of the BS model where the interest rateis now assumed to be a stochastic process. Of course to consider a stochastic interest rater leads on one side to deal with more realistic models but on the other side we end up withmodels that need more sophisticated tools in order to be studied.

The rst problem we encounter is that we can no longer discount the expected futureoption value by the deterministic quantity e−rT . This led Merton in [Mer73] to considerthe price of a bond B having the same maturity T as the option as discount factor. Moreformally, let us consider an option C with a maturity time T > 0 and let us denote byB(t, T ) the prices at time t < T of a "risk-less" bond which pays 1 at maturity time T . Wefurther assume the price B to evolve according the stochastic dierential equation

dB(t;T )B(t;T ) = α(t, B(t;T ))dt+ ν(t, B(t;T ))dWB(t) ,

B(T ;T ) = 1 ,, (4.1)

with α and ν two suitable function according to what introduced in Sec. 1.4.1. It canbe easily seen that, if we choose ν ≡ 0 and α = r we immediately retrieve the standardassumption of constant interest rate B(t) = e−r(T−t).

As usual we will further assume to deal with a stock S whose price evolves according todS(t)S(t) = µ(t, S(t))dt+ σ(t, S(t))dWS(t) ,

S(0) = s0 ,, (4.2)

123

124 4. Interest Rates

again for µ and σ suitable function. In total generality we also assume the two Brownianmotion to have correlation ρ ∈ [−1, 1],

EdWB(t)dWS(t) = ρdt .

Under these assumptions we have that the value of an option written on the underlyingS will be a function f of the asset S the time t and also of the risk-less bond B, i.e.f = f (t, S,B). Applying Itô's lemma 1.1.16 we have that the option price evolves accordingto

df

f=

1

f

(ft + µSfS + αBfB +

1

2σ2S2fS,S +

1

2ν2B2fB,B + ρσνSBfS,B

)︸︷︷︸

µf

dt+

+σS

ffS︸︷︷︸

σf

dWS(t) +νB

ffS︸︷︷︸

νf

dWB(t) ,

, (4.3)

where we have denoted for short fx the partial derivative of f w.r.t. the variable x, i.e.

fx :=∂f

∂x.

In order to retrieve the fair price at time t = 0 of the option f with maturity T , asintroduced in Sec. 3.2.1, we will construct a replicating portfolio, we will thus denote byπS , resp. πB , resp. πf , the numbers of euro on the portfolio invested in the stock S, resp.the bond B, resp. the option f .

Thus the instantaneous return on the portfolio is given by

πS(t)dS(t)

S(t)+ πB(t)

dB(t)

B(t)+ πf (t)

df(t)

f(t)=(πS(t)(µ− α) + πB(t)(µf − α)

)dt+

+(πS(t)σ + πB(t)σf

)dWS(t) +

(πB(t)νf − (πS(t) + πB(t))ν

)dWB(t) ,

(4.4)

where we have exploited the condition of zero aggregate investment

πS(t) + πB(t) + πf (t) = 0 .

As in the standard hedging portfolio, see, e.g. Sec. 3.2.1, we choose πS and πB s.t. thestochastic terms vanish, in particular from eq. (4.4) we have

πS(t)σ + πB(t)σf = 0 , (πB(t)νf − (πS(t) + πB(t))ν = 0 ,

that, exploiting the particular form of σf and νf introduced in eq. (4.3), leads to thecondition

πB = −πS σσf

, ⇒ f = SfS +BfB . (4.5)

Thus we have constructed a risk-less portfolio whose instantaneous return is now givenby (

πS(t)(µ− α) +−πS σσf

(µf − α)

)dt = πS(t)σ

(µ− ασ− µf − α

σf

)dt .

4.1 Allowing for stochastic interest rates in the Black-Scholes Model 125

Eventually, from the fact that the instantaneous return should be zero, as involves zero netinvestment, we retrieve the condition

µ− ασ

=µf − ασf

, (4.6)

substituting now the denition of µf and σf in eq. (4.3) into eq. (4.6) and exploitingcondition (4.5), we have the following partial dierential equation

ft + α(SfS + PfP ) +1

2σ2S2fS,S +

1

2ν2B2fB,B + ρσνSBfS,B − αf = 0 , (4.7)

and from eq. (4.5) we get the following

ft +1

2σ2S2fS,S +

1

2ν2B2fB,B + ρσνSBfS,B = 0 , (4.8)

endowed with suitable boundary condition according to the option we are dealing with. Inparticular, if we are to consider an European call option with strike price K we will have toconsider the terminal condition f(T ) = maxS(T )−K, 0.

4.1.2 Solving for the option price

Let us now introduce the new state variable X dened by

X =S

KB,

here B is used to discount between maturity and current time, so that X may be interpretedas the price per share of stock in units of the present value of the exercise price. In the caseof a constant interest rate we have

B(t, T ) = e−r(T−T ) .

Exploiting then Itô's lemma we have that

dX

X= (µ− α+ ν2 − ρσν)dt+ σdWS

t − νdWBt ,

so that the instantaneous rate of return of X is given by

Var [X] = E[(sigmadWS

t − νdWBt

)2]= (σ2 + ν2 − 2ρσν)dt = V (τ)2dτ .

Dene now τ = T − t the time to maturity and the price

h(X, τ) =f(S,B, τ)

KB(τ),

so that, applying the chain rule, we have that

1

2σ2S2fS,S +

1

2ν2B2fB,B + ρσνSBfS,B =

1

2V 2(τ)X2 ∂

2h

∂X2KB ,


and together with the relation∂

∂tf =

∂

∂τh ,

yields that eq. (4.48) now becomes

∂

∂τh =

1

2V 2(τ)X2 ∂

2h

∂X2KB . (4.9)

In the particular case of an European call option we have that the initial and boundaryconditions are

h(0, τ) = 0 , h(X, 0) = maxX − 1, 0 , (4.10)

so that the solution to the PDE (4.9) with boundary conditions (4.10) is given by

h(X, τ) = XΦ(d1)− 1Φ(d2) , (4.11)

with Φ the cumulative distribution function of the standard Gaussian random variable and

d1 =log(X1 + V 2

2 τ

V 2√τ

, d2 = d1 − σ2√τ ,

with

V 2 =1

τ

∫ τ

0

V 2(s+ t)ds .

Coming then back to the original variables we retrieve that

f(S,B, τ) = SΦ(d1)−KBΦ(d2) , (4.12)

with

d1 =1

V 2√τ

(log

S(t)

K− logB(τ) +

V 2

2τ

), d2 = d1 − V 2

√τ . (4.13)

The pricing formula (4.12)-(4.13) generalises the Black-Scholes formula (3.6) in the caseof a non deterministic interest rate. In fact the bond with same maturity as the option isused to do stochastic discounting and the average volatility V replaces the σ of the classicalcase. If we adopt the common practice of calculating an implied V from market data thenthere is no need to estimate separately ρ, ν and σ. This observation also helps to explain therobustness of the Black-Scholes model when used with implied volatility, as the volatility socalculated is observationally compatible with a wide class of deterministic time functions ofρ, ν and σ and not just with a constant σ.

4.2 Change of Numeraire

An approach equivalent to the one introduced in Sec. 4.1.2 is the so called change ofnumeraire, already introduced in Sec. 3.3.2.

Let us consider as usual an asset S evolving according todS(t)S(t) = µ(t, S(t))dt+ σ(t, S(t))dW P(t) ,

S(0) = s0 ,, (4.14)

4.2 Change of Numeraire 127

under suitable condition on the coecients µ and σ and the Brownian motion W evolvesaccording the real world probability measure P. Let us further consider two derivativeswritten on the underlying S whose price will be denoted by two functions U and V .

Proceeding thus as in Sec. 3.2 and exploiting Itô's formula (1.1.16), we have the followingprocesses

dS(t) = (r + ηtσ)S(t)dt+ σS(t)dW P(t) ,

dU(t) = (r + ηtσU )U(t)dt+ σUU(t)dW P(t) ,

dV S(t) = (r + ηtσV )V (t)dt+ σV V (t)dW P(t) ,

,

where the process ηt := µ−rσ is the market price of risk introduced in Sec. 3.2 and from Itô's

lemma we have σU = σSSUUS and σV = σS

SV VS . An application now of Girsanov theorem

1.4.4 leads to

dS(t) = rS(t)dt+ σS(t)dWQ(t) , (4.15)

dU(t) = rU(t)dt+ σUU(t)dWQ(t) , (4.16)

dV (t) = rV (t)dt+ σV V (t)dWQ(t) , (4.17)

(4.18)

where

WQ(t) := W P(t) +

∫ t

0

η(s)ds ,

is a Brownian motion evolving according to the risk-neutral probability measure Q equivalentto P.

Let us now consider a (possibly stochastic) risk-free interest rate r and the money marketaccount

A(t) = exp

(∫ t

0

r(s)ds

),

so that it satises the dierential equation

dA(t) = r(t)A(t)dt .

Applying Itô's lemma we thus have that the evolution of U and V discounted by theprocess A satisfy

dU(t)

A(t)= σU

U(t)

A(t)dWQ(t) , (4.19)

dV (t)

A(t)= σV

V (t)

A(t)dWQ(t) , (4.20)

(4.21)


which shows that the processes U(t)A(t) and V (t)

A(t) are Q−martingales. We thus have

U(t)

A(t)= EQ

t

[U(T )

A(T )

], (4.22)

V (t)

A(t)= EQ

t

[V (T )

A(T )

], (4.23)

(4.24)

where we have denoted for short by EQt the expectation under Q conditioned w.r.t. the

ltration at time t, namely

EQt := EQ [ ·| Ft] .

From eq. (4.25) and (4.26) we eventually have

U(t) = EQt

[exp

(∫ T

t

r(s)ds

)U(T )

], (4.25)

V (t) = EQt

[exp

(∫ T

t

r(s)ds

)V (T )

], (4.26)

(4.27)

We have here used the money market accountA as numeraire so that the term exp(−∫ Ttr(s)ds

)acts as a stochastic discount factor. In particular we have that the value at time t of U isthe discounted value of U(T ) from T back to t averaged over all possible realisation.

However it may happen that it is more convenient to use other processes as discountfactors. For instance in the previous case we could have used the process V as numeraire,then we would consider the process Y := U

V . Proceeding exactly as before we retrieve nowthat the following

U(t) = V (t)EQt

[U(T )

V (T )

], (4.28)

where again Q is a probability measure equivalent to P. In particular eq. (4.28) may providea more convenient pricing relationship, as it happens if V is the price of a bond that maturesat the same time as the instrument U .

We now give two dierent application of the change of numeraire introduce above.

4.2.1 Option pricing under stochastic interest rate

The rst application of the result stated above is the BS price under stochastic interest ratetreated in Sec. 4.1.2.

Let us consider the market model introduced in Sec. 4.1.2 composed by a stock price Sevolving according to

dS(t)S(t) = µ(t, S(t))dt+ σ(t, S(t))dWS(t) ,

S(0) = s0 ,, (4.29)


a stochastic interest rate r evolving according todB(t;T )B(t;T ) = α(t, B(t;T ))dt+ ν(t, B(t;T ))dWB(t) ,

B(T ;T ) = 1 ,, (4.30)

and an option price f evolving according to

df

f=

1

f

(ft + µSfS + αBfB +

1

2σ2S2fS,S +

1

2ν2B2fB,B + ρσνSBfS,B

)︸︷︷︸

µf

dt+

+σS

ffS︸︷︷︸

σf

dWS(t) +νB

ffS︸︷︷︸

νf

dWB(t) ,

, (4.31)

Let us further consider the condition (4.6)

µ− ασ

=µf − ασf

= η , (4.32)

where as usual η denotes the market price of risk associated with the stock price noiseWS .

Substituting eq. (4.32) into eq. (4.29) and (4.31) we have

dS(t)

S(t)= (µ+ ησ)dt+ σdWS(t) , (4.33)

df(t)

f(t)= (µ+ ησf )dt+ σfdW

S(t) + νdWSB(t) , (4.34)

(4.35)

where both WS and WB evolve according to the real world probability measure P. Ex-ploiting Girsanov theorem we can now dene two new Brownian motions WS and WB thatevolve according to the equivalent risk-neutral probability measure Q so that eq. (4.36) and(4.37) become

dS(t)

S(t)= µdt+ σdWS(t) , (4.36)

df(t)

f(t)= µdt+ σfdW

S(t) + νdWSB(t) . (4.37)

(4.38)

Introducing now the stochastic money market account

A(t) = exp

(∫ t

0

µ(s)ds

),

we have that the two processes

dS(t)

A(t)=S(t)

A(t)σdWS(t) , (4.39)

df(t)

A(t)=f(t)

A(t)σfdW

S(t) +f(t)

A(t)νdWSB(t) , (4.40)

(4.41)


are Q−martingales, so that we have

f(t) = EQt

[A(t)

A(T )f(T )

]. (4.42)

On the other side we could also choose the bond price B as numeraire, so that, denoting byQ the equivalent risk measure when we choose B as numeraire, we get

f(t) = EQt

[A(t)

A(T )f(T )

]= EQ

t

[AB(t)

B(T )f(T )

]. (4.43)

From the fact that the value at maturity time T is given by B(T ) = 1 we eventually have

f(t)

B(t)= EQ

t [f(T )] . (4.44)

In order to calculate eq. (4.44) we need to have the explicit form of stochastic pricedynamics, in particular from Itô's lemma we have

dS(t)/B(t)

S(t)/B(t)=(ν2 − ρσν

)dt+ σdWS(t) + νdWB(t) , (4.45)

with instantaneous variance given by

Var[σdWS(t) + νdWB(t)

]= (σ2 + ν2 − 2ρσν)dt = V 2dt .

We can thus rewrite eq. (4.45) as

dS(t)/B(t)

S(t)/B(t)=(ν2 − ρσν

)dt+ V dWS/B(t) . (4.46)

Applying now Girsanov theorem and introducing the new Brownian motion

dWS/B(t) = dWS/B(t) +

∫ t

0

ν2 − ρσνV

ds

we havedS(t)/B(t)

S(t)/B(t)= V dWS/B(t) . (4.47)

Applying eventually Feynman-Kac theorem 1.3.23 we have that f(t)B(t) is given by the

solution of 12V

2∂X,Xf(t)B(t) + ∂t

f(t)B(t) = 0 ,

f(X,T ) = Φ ,(4.48)

where Φ is the nal payo of the option f . The solution to eq. (4.48) is given by eq. (4.11).


4.2.2 Change of numeraire under multiple source of risk

Let us now consider a market where d+1 dividend-paying stocks are traded and let us denotefor short the d−dimensional vector as S =

(S0, . . . , Sd

). In particular we will assume that

they evolves under the risk-neutral probability measure Q asdSi(t)Si(t) =

(r − qi

)dt+

∑nj=0 σ

i,j(t,S(t))dWQj (t) ,

Si(0) = si0 , i = 0, . . . , d ,, (4.49)

where qi is the dividend yield on the i−th asset Si, we refer to Sec. [?] for details ondividends-paying stocks.

Let us further consider an option written on S whose price is given by a function fk(t,S),k = 1, . . . ,m. By Itô's formula we thus have that fk satises

dfk(t)fk(t)

= rdt+∑nj=0 ν

k,j(t,S(t))dWQj (t) ,

fk(0) = Φ , k = 1, . . . ,m ,, (4.50)

where as before from Itô's formula we have that

fkνk,j =

n∑i=0

σi,jSi∂Sifk .

We can now dene as done before the money market account

A(t) = exp

(∫ t

0

r(s)ds

),

with a (possibly stochastic) risk-free interest rate r. Proceeding as before we obtain

fk(t,S) = EQt

[e−

∫ Ttr(s)dsfk(T,S(T))

].

Let us now suppose we use the 0−th asset S0 as numeraire, so that we focus on the

processes Zi = Si

S0 , i = 1, . . . , d and on Y k = fk

S0 , k = 1, . . . ,m. Applying as usual Itô'sformula we have that Zi and Y k evolves according to

dZi(t)

Zi(t)=

q0 − qi −n∑j=0

σ0,j(σi,j − σ0,j)

dt+

n∑j=0

(σi,j − σ0,j)dWQj (t) ,

dY k(t)

Y k(t)=

q0 −m∑j=1

σ0,j(νk,j − σ0,j)

dt+

m∑j=1

(νk,j − σ0,j)dWQj (t) ,

.

Since the stocks are paying dividends they are not Q−martingales, in order to overcome thisproblem we dene the new processes

Zi(t) = Zi(t)e−(q0−qi)t , Y k(t) = Y k(t)e−q0t ,


so that we have that they evolves according to

dZi(t)

Zi(t)=

− n∑j=0

σ0,j(σi,j − σ0,j)

dt+

n∑j=0

(σi,j − σ0,j)dWQj (t) ,

dY k(t)

Y k(t)=

− m∑j=1

σ0,j(νk,j − σ0,j)

dt+

m∑j=1

(νk,j − σ0,j)dWQj (t) ,

.

Applying thus Girsanov theorem 1.4.4 and dening the Brownian motion under Q

Wj(t) = WQj (t)−

∫ t

0

σ0,j(s)ds ,

we have that

dZi(t)

Zi(t)=

n∑j=0

(σi,j − σ0,j)dWj(t) ,

dY k(t)

Y k(t)=

m∑j=1

(νk,j − σ0,j)dWj(t) ,

,

so that Zi and Y k are Q−martingales and thus we have

Y k(t) = EQt

[Y k(T )

]= EQ

t

[e−q

0(T−t)Y k(T )],

if we now consider the original function fk we eventually have that the price at time t ofthe option fk is given by

fk(t) = S0(t)EQt

[e−q

0(T−t) fk(T )

S0(T )

]. (4.51)

4.3 Interest rate option problem

4.3.1 Interest rate caps, oors and collars

As it is possible to stipulate contract on traded securities, it is possible to make contractsover interest rates. These types of contracts are known as caps, oors and collars.

An interest rate cap is an agreement written on some reference interest rate R, forinstance 6−months LIBOR, that sets the borrowing rate at the market rate R if R < Rcapand limits the rate to Rcap if the market rate R > Rcap, see, gure 4.1a.

It can be immediately seen that an interest rate cap is a call option on the interest rate,and in particular it is used for the same reasons, that is it is an insurance against the interestrate on the underlying oating rate asset rising above a certain level. If the interest rate risesabove the cap rate, the buyer of the cap eectively receives a payo which is the dierencebetween the current market rate and the cap rate. There also can be a series of rate resetsover the life of a cap, hence the cap is nothing but a portfolio of call options on R. Each

4.3 Interest rate option problem 133

1 2 3 4 5

0,2

0,4

0,6

0,8

1

Rcap

R

Payo

(a) Payo on a long position in an in-terest rate cap

1 2 3 4 5

0,2

0,4

0,6

0,8

1

Rfloor

R

Payo

(b) Payo on a long position in an in-terest rate oor

1 2 3 4 5

−1

−0,5

0,5

1

Rfloor Rcap

R

Payo

(c) Payo on a long position in a capand short position in a oor

Figure 4.1

component of the cap is know as caplet. For the caplet over the period between ti and ti+1,the cap rate Rcap is compared with the reference rate at ti, that is Ri. However, the payofor this period is settled at time ti+1, and therefore it pays in arrears.

An interest rate oor is the reverse of an interest rate cap. Such a long position would beof interest to a lender who wants to guarantee that a lending rate will not fall below a certainpre-specied rate. That is if R > Rfloor the lenders receives the market rate, conversely ifR < Rfloor then the lender receives the oor rate Rfloor, see, e.g. 4.1b.

The last type of derivative is the so called interest rate collar. It correspond to a longposition in an interest rate cap and in a short position in an interest rate oor. The cor-responding price for a collar is thus strictly less than the price of a cap or a oor, see, .4.1c.

Concerning the payo structure of interest rate caps and oors we have the followingsituation. The present value at timeti of a caplet payo received at time ti+1 for e 1 principalamount is

PV ( caplet payoi) =(ti+1 − ti) maxRi −Rcap, 0

1 +Ri(ti+1 − ti)=

= (ti+1 − ti) maxRi −Rcap, 0P (ti, ti+1) ,

(4.52)


Rcap

value of cap at T = 0

Observe Rk

Payment = τLmaxRk −Rcap, 0

τ 2τ kτ (k + 1)τ nτ

k−th caplet

R

Figure 4.2: Payo structure of an interest rate cap

Figure 4.3

where P (ti, ti+1) is the price at time ti of a pure discount bond maturing at time ti+1. Theunderlying reference interest rate R is quoted so that

(1 +Ri(ti+1 − ti))−1 = P (ti, ti+1) ,

which allows us to use the prices of a pure discount bonds as discount factors in eq. (4.52).Conversely for an interest rate oor, the present value at time ti of the oorlet payo forperiod (ti, ti+1) is

PV ( oorlet payoi) =(ti+1 − ti) maxRfloor −Ri, 0

1 +Ri(ti+1 − ti)=

= (ti+1 − ti) maxRfloor −Ri, 0P (ti, ti+1) ,

(4.53)

see g. 4.2 for the payo situation for an n−period interest rate cap.

We can further relate the above problem of pricing interest rate caps and oors to theproblem of pricing put and call options on bonds, hence interest rates cap and oors canbe reduced to a portfolio of options on bonds. The main dierence is that bonds are tradedinstruments so that they can be used in order to construct the hedging portfolio, that isthe basic instrument if one is to price a derivative security. This observation is fundamentalsince we cannot construct hedging portfolio directly with the reference interest rate as it isnot a traded instrument.

4.3 Interest rate option problem 135

0 t T

Figure 4.4: The time line for interest rate processes

Thus, exploiting eq. (4.52), we can rewrite the caplet payo, setting τ = ti+1 − ti, as

PV ( caplet payoi) =τ

1 +RiτmaxRi −Rcapτ

τ, 0 =

=τ

1 +Riτmax1 +Riτ − (1 +Rcapτ)

τ, 0 =

max1− 1 +Rcapτ)

1 +Riτ, 0 =

= (1 +Rcapτ) max 1

1 +Rcapτ− 1

1 +Riτ, 0 .

(4.54)

Let us now set Xc = 11+Rcapτ

, so that eq. (4.54) becomes

PV ( caplet payoi) = (1 +Rcapτ) maxXc −1

1 +Riτ, 0 =

= (1 +Rcapτ) maxXc − P (ti, ti+1, 0 .

Hence the caplet payo is equivalent, up to the factor (1+Rcapτ) to the payo of a bondput option with maturity time ti written on an underlying bond with maturity time ti+1

and exercise price Xc.Eventually we conclude the present section briey highlighting why it is that the interest

rate option problem is so much more dicult problem than the option pricing problem ina world of deterministic interest rate. There are mainly two reasons, the rst concerns thenumber of underlying asset that can be considered innite. Second, it is not clear whatunderlying asset to use.

Let us for the moment focus on the rst problem, that is the innite dimensional natureof the problem. The main dynamic we try to model is the one of the yield curve out of arange of maturities. Interest rate derivatives derive their values from the evolution of thissurface, that is in principle an innite dimensional object, even though it can be seen thatits evolution can be well captured by a nite number of xed maturity forward rates.

Concerning the second issue, we note that one may use at least three dierent quantitiesto describe the par rates in a yield curve, that are the instantaneous interest rate r(t) agreedat time t for borrowing starting at time t, the price at time t P (t, T ) of pure discount bondmaturing at time T and the instantaneous interest rate f(t, T )agreed at time t for borrowingstarting at time T , see gure 4.4.

The relationship between these quantities is illustrated in gure 4.5, the details of whichwill treated extensively in the following sections. Other quantities such as yields and thediscretely compounded rates that are quoted in markets can also be determined in terms ofr(t), P (t, T ) and f(t, T ).


r(t)

f(t;T ) P (t;T )

r(t)=f(t;T ) P (t;T )=EQ[e−

∫Tt r(s) ds

]

P (t;T )=e−∫Tt f(t;s) ds

Figure 4.5: Relationship between spot rate, forward rate and bond price

Therefore we have seen how to reduce the problem of pricing an interest rate cap (oroor) to the problem of pricing an option on a bond. The main dicult in this process isthat it is in fact a two step process; let us denote by TB the maturity time of the bond andby TC the maturity time of the option, then we have rst to solve for the bond price overthe interval [TC , TB ] to determine the possible option payos at time TC . Then we need tosolve the bond option pricing problem over the interval [0, TC ], see g. 4.6.

X

TC TB time

Solve option pricing PDE Solve bond pricing PDE

Boundary condition at TC

Figure 4.6: The two pass nature of bond option pricing

4.4 Modelling interest rate dynamics

4.4.1 The relationship between interest rates, bond prices and for-ward rates

The relationship between interest rates, bond prices, yield to maturity and forward ratesis key topic to understand. The present section is devoted to the link among these threenancila objects, we refer to [BM06, CC12] for a detailed tretament of the topic.

Let us x a positive maturity time T < ∞, then for any t ∈ [0, T ] let P (t, T ) be theprice at time t of a pure discount bond paying 1 at maturity time T . We will further denoteby ρ(t;T ) the yield to maturity, that is the continuously compounded rate of return causing

4.4 Modelling interest rate dynamics 137

the bond price to satisfy the maturity condition P (T ;T ) = 1, namely we have

P (t;T )eρ(t;T )(T − t) = 1 , (4.55)

or equvalently eq. (4.55) can be espressed as

ρ(t;T ) = − logP (t;T )

T − t. (4.56)

We can then dene the instantaneous spot rate of interest, r(t), is the yield on thecurrently maturing bond, namely

r(t) = ρ(t; t) . (4.57)

If we now let t→ T in eq. (4.56), we get

limt→T− logP (t;T )

T − t= ∂tρ(T ;T ) = r(T ) ,

where we have denoted by ∂tρ(T ;T ) the partial derivative of ρ(t;T ) w.r.t. the rst variablet, often called running time, evaluated at time t = T .

Let us now instead consider an investor who is holding a bond with maturity time T1.Furthermore he would like to know what return he would earn between the maturity timeT1 and a second xed time T2 (T2 > T1), if he contracted now at time t. The a new object,that is the forward rate, has to be taken into account. In particular the investor has toconsider the forward rate f(t;T1, T2) dened as P (t;T1) = P (t;T2)ef(t;T1,T2)(T2−T1), i.e.

f(t;T1, T2) =1

T2 − T1log

P (t;T1)

P (t;T2). (4.58)

The following diagram better explain the relationships that exits between f(t;T1, T2),P (t;T1) and P (t;T2).

t T1 T2

f(t;T1, T2)

P (t;T1)

P (t;T2)

To see how the forward rate f(t;T1, T2) represents the implicit rate of interest currentlyavailable at time t on riskless loans from T1 to T2, let us consider the following set oftransactions. Let us buy one bond with maturity time T2 and cashow −P (t;T2), let us

further short sell P (t,T2)P (t,T1) bonds with maturity T1 so that the cash ow is P (t;T2).

In particular a set of transactions at time t involving zero net cashow has the net eect

of investing P (t,T2)P (t,T1) at time T1 to yield one for sure at time T2. Furthermore the forward

rate dened above is simply the continuously compounded rate of interest earned on thisinvestment.


Remark 4.4.1. Note that calculation of f(t, T1, T2) involves only bond prices observable att.

To treat directly with the instantaneous forward rate dened in eq. (4.55) as

f(t;T ) := f(t;T, T ) , (4.59)

instead of the instataneous spot rate r leads to a series of advantages, both under a nancialand mathematical point of view. In particular this is known as the Heath-Jarrow-Mortonmodel and we address the detailed treatment of this particular model in a later chapter.

Remark 4.4.2. Eq. (4.59) is the instantaneous rate of return the bond holder can earn byextending the investment an instant beyond T .

Let us now set in eq. (4.59) T1 = T and T2 = T + x and let us evaluate

f(t;T ) = limx→0

1

x(logP (t;T )− logP (t;T + x)) = − lim

x→0

∂x logP (t;T + x)

∂xx= −∂T

P (t;T )

P (t;T ),

(4.60)where according to the notation introduced above we have denoted by ∂T the partial deriva-tive w.r.t. the second variable T of P (t;T ).

From eq. (4.60) it follows immediately that the price of the pure discount bond may bewritten in terms of the forward rate as

P (t;T ) = e−∫ Ttf(t,s)ds , (4.61)

In a world of certainty all securities, in equilibrium, must earn the same instantaneousrate of return so as to exclude the possibility of riskless arbitrage opportunities, this equi-librium applyed to discounted bonds leads to

r(t) =∂tP (t;T )

P (t;T ); (4.62)

then from eq. (4.62) we get∂s logP (s;T ) = r(s) ,

and integrating w.r.t. time t and exploiting the terminal condition P (T ;T ) = 1,∫ T

t

∂s logP (s;T )ds = logP (s;T )|Tt = logP (T ;T )−logP (t;T ) = − logP (t;T ) = −∫ T

t

r(s)ds ,

and eventually we obtain the relation

P (t;T ) = e−∫ Ttr(s)ds . (4.63)

Comparing then eq. (4.63) with eq. (4.55) we have that the relationship between theyield to maturity ρ(t;T ) and the spot rate r(t), in a world of certainty, reads

ρ(t;T ) =1

T − t

∫ T

t

r(s) ds , (4.64)

substituting then eq. (4.63) into eq. (4.60) reveals the relationship between the spot rater(t) and the forward rate f(t;T ), againg in a world of certainty, that is

f(t;T )r(t) , ∀ T ≥ t . (4.65)


This last eq. (4.65) is a degenerate version of the fact that the expected instantaneousspot rate for time T is equal to the instantaneous forward rate for time T , evaluated at timet.

We will later see how the relationships (4.63)-(4.64)-(4.65) generalise quite naturallyin a world of uncertainty to be the corresponding relationships under suitable probabilitymeasures.

Remark 4.4.3. Another important set of interest rates are the so-called LIBOR rates. Wedo no treat this topic here, we refer to [CC12, Ch.26] or to [BM06] for details.

4.4.2 Modelling the spot rate of interest

In nancial modelling the key feature in modelling nancial instruments is the volatility, itis less clear what is the best form the volatility function should take. Forms usually used inempirical studies and in development of term structure and interest rate derivative modelsto be discussed later are of the general form

σ(t, r) = σrγ , γ ≥ 0 , (4.66)

so that the interest rate process becomes

dr(t) = κ(θ − r(t))dt+ σr(t)γdW (t) , (4.67)

with W a standard Brownian motion and κ, θ and σ some suitable parameters. The partic-ular form of the drift in eq. (4.67) is known as mean reverting property and it is a typicalfeature of interest rates. In particular can be empirically seen that an interest rate r willtend to move to the average price over time; θ is called long term mean level in the sense thatall future trajectories of the spot rate r will evolve around a mean level θ in the long run, κ isthe speed of reversion and characterizes the velocity at which such trajectories will regrouparound θ, eventually σ is the instantaneous volatility and it measures the uncertainty. Adetailed treatment of dierent spot rate models with suitables parameters to choose can befound in [BM06].

An important criticism of modelling the volatility by use of functional forms such as(4.66) is that volatility of interest rates depend on the level of interest rates. This seemsto run against what is often observed as it is possible to identify historical periods whenrates were high but relatively stable or rates were low but fairly volatile. It is also oftenargued that volatility of interest rates should also be a function of the news arrival process,i.e. dW . One way to capture this eect would be to allow the volatility σ to itself follow astochastic process, this is particular feauture is called stochastic volatility model, this topicis addressed to later chapter.

The Vasicek model

The Vasicek model assume the interest rate to evolves according to the Ornstein-Ouhlenbeck(OU) process introduced in eq. (1.5). Among its main characteristic it is analyticallytractable and it allows for explicit solutions when one is to price claim written on it, on theother hand it allows the price to go below zero. This last feature has lead many practitionersto prefer the CIR model, to be introduce later on. We refer to [BM06] for a detailed treatmentof this model.


Let us consider a stochastic interest rate R evolving according to the followingdRt = (α+ βRt)dt+ σWt ,

R0 = r0 ,, (4.68)

with α, β and σ ∈ R and r0 > 0. Eventually Wt is a standard Brownian motion adapted tothe natural ltration Ft.

Proposition 4.4.4. A solution to eq. (4.68) is given by

Rt = r0e−βt +

α

β(1− e−βt) + σe−βt

∫ t

0

eβsWsds (4.69)

Proof. Apply Itô's lemma to

f(t,Xt) = r0e−βt +

α

β(1− e−βt) + σe−βtXt ,

with Xt =∫ t

0eβsWsds. Then applying Itô's lemma to f we have the desired result.

In order to compute the marginal distribution of eq. (4.69) we need the following result.

Theorem 4.4.5. Let Wt, t ≥ 0, be a standard Brownian motion and f a deterministicfunction. Then we have that

I(t) :=

∫ t

0

f(s)dWs ∼ N(

0,

∫ t

0

f2(s)ds

).

Proof. From the fact that I0 = 0 and the martingale property of the stochastic integral itimmediately follows

EIt = I0 = 0 .

Applying thus the Itô isometry we have that

Var [It] = E(∫ t

0

f(s)dWs

)2

= E∫ t

0

f2(s)ds =

∫ t

0

f2(s)ds .

The fact that I is Gaussian distributed follows from the moment generating function, inparticular we have that

EeθIt = eθ2

2

∫ t0f2(s)ds , ∀ θ ∈ R .

Thus we can write previous expression as

EeθIt−θ2

2

∫ t0f2(s)ds = 1 . (4.70)

Noticing now that eq. (4.70) is a generalized Brownian motion, it follows that it is amartingale. Thus it has always mean equal to 1 and the claim follows.

We can now compute the marginal distribution for the Vasicek model.

Proposition 4.4.6. We have that eq. (4.69)

Rt ∼ N(e−βtr0 +

α

β(1− e−βt), σ

2

2β(1− e−2βt

).


2 4 6 8 10

-2

-1

1

2

3

(a) Vasicek model, κ = .1 σ = 1 θ = .3 r0 = 1

2 4 6 8 10

-1.0

-0.5

0.5

1.0

1.5

2.0

(b) Vasicek model, κ = 2 σ = .3 θ = .3 r0 = −1

In particular from Prop. 4.4.6 we have that R can assume negative values with non nullprobability. This feature led to the development of alternative interest rate models.

Exercise 4.4.1. Exploiting Th. 4.4.5 prove Prop. 4.4.6.

Remark 4.4.7. Eq. (4.68), and in general any interest rate model, is usually given in thefollowing form

dRt = κ(θ −Rt)dt+ σdWt . (4.71)

This is due to the nancial meaning of the drift components. In particular a drift as in eq.(4.71) we have that κ is called long term mean and θ is the mean-reversion parameter.

The Cox-Ingersoll-Ross model

A very widely used stochastic process in nance is the the Cox-Ingersoll-Ross (CIR) process,frequently reered as Feller or square root process. Its interest for nance applications liesin the fact that (for certain parameter constellations) the process is guaranteed to generatepositive values, and its density function is known. This is clearly a desirable feature for amodel of interest rate processes.

In this section we seek to give some insight into why this process generates positive (or atleast non-negative outcomes), and also why its conditional transitional probability densityfunction is a chi-squared distribution.

Let us then consider an interest rate R evolving according todRt = (α− βRt) dt+ σ

√RtdWt ,

R0 = r0 > 0 ,, (4.72)

with α, β and σ ∈ R+ and Wt a standard Brownian motion.

As rst feature it can immediately be seen that, for any t ≥ 0, we have that Rt ≥ 0w.p.1. Conversely, the most important drawback of using eq. (4.104) is that, in spite ofwhat happens for the Vasicek model (4.68), we do not have a closed form solution. Despitethis fact, exploiting It ô's formula, we can compute the marginal distribution for R.

Let us then consider f(t, Rt) = eβtRt, then by Itô's lemma 1.1.16 we have that

df(t, Rt) = αeβtdt+ σeβt√RtdWt ,


integrating thus both side we get

eβtRt = r0 +α

β(eβt − 1) + σ

∫ t

0

eβs√RsdWs .

Taking now the expectation we eventually have

ERt = e−βtr0 + e−βtα

β(eβt − 1) . (4.73)

It can be immediately be seen that the expectation in eq. (4.73) is the same as the expec-tation for the Vasicek model (4.68).

Let us then compute the variance. Let us then denote by Xt = eβtRt, by Itô's lemmacomputed above we have that Xt solves

dXt = r0 +α

β(eβt − 1) + σ

∫ t

0

eβs√RsdWs .

Exploiting then Itô's lemma applied to the function f(Xt) = X2t we have

dX2t =

(2αeβtXt + σ2eβtXt

)dt+ 2σe

βt2 X

32t dWt , X

20 = r2

0 , .

Computing the expectation we get

Ee2βtR2t = e−2βtr2

0 +2α+ σ2

β

(r0 −

α

β

)(e−βt − e−2βt

)+α(2α+ σ2

2β2

(1− e−2βt

),

eventually we have

Var [Rt] = ER2t − ERt

2 =σ2

βr0

(e−βt − e−2βt

)+α(ασ2

2β2

(1− 2e−βt + e−2βt

).

We can further compute the long time mean and variance, that is

limt→∞ERt =α

β,

limt→∞Var [Rt] =ασ2

2β2,.

4.4.3 Modelling forward rates

Let us recall that f(t, T ) denotes the forward rate at time t for instantaneous borrowing attime T . Heath, Jarrow, and Morton in [HJM92] propose to model the forward rate as thedriving stochastic process. In particular they write the process for f(t, T ) in the form of astochastic integral equation, namely for a xed maturity time T <∞

f(t;T ) = f(0;T ) +

∫ t

0

α(s, T ) ds+

∫ t

0

σ(s, T ) dW (s) , t ∈ [0, T ] , (4.74)

where W is a standard Brownina motion, α(t, f), resp. σ(t, f), is the drift, resp. thevolatility, of the forward process, assumed to be regular enough, f(0, T ) is the initial forwardrate curve that can be obtained from currently observed yields.


0.2 0.4 0.6 0.8 1.0

1

2

3

4

(a) CIR model, κ = .1 σ = 1 θ = .3 r0 = 1

0.2 0.4 0.6 0.8 1.0

0.7

0.8

0.9

1.0

1.1

(b) CIR model, κ = 2 σ = .3 θ = .3 r0 = 1

Remark 4.4.8. In eq. (4.74) we assume α(t, T, ω) and σ(t, T, ω), where by ω we indicatesthat the coecients α and σ may have some randomness. In what follows for the sake ofsimplicity we omit the dependance on ω.

Remark 4.4.9. We can also allow for possible dependence of α and σ on r(t) to capture theeect of the level of interest rates on volatility.

In what followsthe discussion of this section we only allow one noisy term W in order toalleviate the mathematical notation. In dierential form we may write eq. (4.74) as the

f(t;T ) = α(t, T ) dt+ σ(t, T ) dW (t) , t ∈ [0, T ] ,

f(0;T ) = x ,. (4.75)

We shall defer for the moment discussion of appropriate functional forms for α and σ.The aim of the calculations in the rest of this section is to determine the stochastic processesfor the spot rate r(t) and bond price P (t;T ) implied by eq. (4.74).

The reader needs to be wary that the correct path from the forward rate dynamics tothe spot rate dynamics has to occur at the level of the stochastic integral equations. Thuswe must set T = t in eq. (4.74) for f(t;T ) to obtain the stochastic integral equation forr(t). From this latter we can obtain the stochastic dierential equation for r(t). Let usstress that it is not correct to obtain the stochastic dierential equation for r(t) simply bysetting T = t in eq. (4.75) for f(t;T ). This has to do with the fact that the path historymatters here and we need to work with the stochastic integral equations, which are properlydened mathematical objects. In fact the stochastic dierential equation is merely a (veryconvenient) short-hand notation.

Recalling that the spot rate of interest and forward rate are related by r(t) = f(t, t) wesee from eq. (4.74) that r(t) satises the stochastic integral equation

r(t) = f(0; t) +

∫ t

0

α(s, t) ds+

∫ t

0

σ(s, t) dW (s) , t ∈ [0, t] , (4.76)

eq. (4.76) can be equivalently written as

dr(t) =

(f2(0; t) + α(t, t) dt+

∫ t

0

αt(s, t) ds+ +

∫ t

0

σt(s, t) dW (s)

)dt+ σ(t, t) dW (t) ,

(4.77)where we have denoted for short by αt and σt the partial derivative w.r.t. t.


Close inspection of eq. (4.77) reveals that this process for r(t) is more general than thetypes of processes considered before. In fact the dierence stems from the third componentof the drift term, i.e. ∫ t

0

σt(s, t) dW (s) ;

this term is a weighted sum of the noisy terms dW (s) from initial time 0 to current time t,thus, it is a path-dependent term that makes eq. (4.77) non Markovian.

We are now to determine the bond price dynamics implied by the forward rate dynamics(4.74). Before proceeding we compute a quantity that will be useful in simplifying theexpression for the bond price that we shall calculate below. We note that integrating eq.(4.76) w.r.t. time we get∫ t

0

r(s) ds =

∫ t

0

f(0; s) ds+

∫ t

0

∫ s

0

α(p, s)) dp ds+

∫ t

0

∫ p

0

σ(p, s) dW (p) ds ,

exploiting then stochastic Fubini's theorem, see, e.g. [Fil01], we get∫ t

0

r(s) ds =

∫ t

0

f(0; s) ds+

∫ t

0

∫ t

p

α(p, s) ds dp+

∫ t

0

∫ t

p

σ(p, s) ds dW (p) . (4.78)

In order we determine the equation for the bond price that is implied by the forward rateprocess (4.74), we recall rst the denitional relationship between bond prices and forwardrates, i.e.

P (t;T ) = e∫ Ttf(t;s)ds , logP (t;T ) =

∫ T

t

f(t; s)ds . (4.79)

Substituiting now eq. (4.74) into eq. (4.79) we obtain

logP (t;T ) = −∫ T

t

f(0; s) ds−∫ T

t

∫ t

0

α(p, s) dp ds−∫ T

t

∫ t

0

σ(p, s) dW (p) ds ,

and after applying again stochastic Fubini's theorem we get

logP (t;T ) = −∫ t

0

f(0; s) ds−∫ t

0

∫ T

t

α(p, s) ds dp−∫ t

0

∫ T

t

σ(p, s) ds dW (p) .

Exploiting now the fact that∫ T

t

f(0; s)ds =

∫ T

0

f(0; s)ds−∫ t

0

f(0; s)ds ,

and that for any function g we have∫ t

0

∫ T

t

g(p, s) ds dp =

∫ t

0

(∫ p

t

g(p, s)ds−∫ T

p

g(p, s)ds

)dp =

∫ t

0

(−∫ t

p

g(p, s)ds−∫ T

p

g(p, s)ds

)dp ,

we then obtain

logP (t;T ) = −∫ T

0

f(0; s) ds−∫ t

0

∫ t

p

α(p, s) ds dp+

∫ t

0

∫ T

p

σ(p, s) ds dW (p)+

+

∫ t

0

f(0; s) ds+

∫ t

0

∫ T

p

α(p, s) ds dp+

∫ t

0

∫ t

p


(4.80)


Then from eq. (4.78) we can represent the last three terms in eq. (4.80) as∫ t

0

f(0; s) ds+

∫ t

0

∫ T

p

α(p, s) ds dp+

∫ t

0

∫ t

p

σ(p, s) ds dW (p) =

∫ t

0

r(s) ds ,

whereas from eq. (4.79) we obtain

−∫ T

0

f(0; s)ds = logP (0;T ) .

Hence eq. (4.80) becomes

logP (t;T ) = logP (0;T ) +

∫ t

0

r(s) ds−∫ t

0

∫ t

p

α(p, s) ds dp+

∫ t

0

∫ T

p


We then have that the log bond price B(t;T ) = logP (t;T ) solves

dB(t;T ) = (r(t)− αB(t;T )) dt+ σB(t;T )dW (t) , (4.81)

where we have dened

αB(t;T ) :=

∫ T

t

α(t, s) ds , σB(t;T ) := −∫ T

t

σ(t, s) ds . (4.82)

An application of Itô's lemma to P (t;T ) = eB(t;T ) we obtain

dP (t;T ) =

(r(t)− αB(t;T ) +

1

2σ2B(t;T )

)P (t;T )dt+ σB(t;T )P (t;T )dW (t) . (4.83)

Equation (4.77)(4.83) form a linked system for the instantaneous spot rate and the bondprice. This system will typically be non-Markovian and many applications in the HJMframwork seek simplications which reduce this system to Markovian form. In particularthe steps from the forward price dynamics to the bond price dynamics are summarised asfollows

f(t;T ) = f(0;T ) +∫ t

0α(s, T ) ds+

∫ t0σ(s, T ) dW (s)

r(t) = f(0; t) +∫ t

0α(s, t) ds+

∫ t0σ(s, t) dW (s)

dB(t;T ) = (r(t)− αB(t;T )) dt+ σB(t;T )dW (t)

dP (t;T ) =(r(t)− αB(t;T ) + 1

2σ2B(t;T )

)P (t;T )dt+ σB(t;T )P (t;T )dW (t)

T = t

B(t;T ) = −∫ Ttf(t; s) ds

P (t;T ) = eB(t;T )


Eventually we are back from the question we started the section with. The relationshipbetween interest rates, bond prices, yield to maturity and forward rates can be summarizedwith the following diagram.

r(t)

f(t;T ) P (t;T )

r(t)=f(t;T ) P (t;T )=EQ[e−

∫Tt r(s) ds

]

P (t;T )=e−∫Tt f(t;s) ds

4.5 Interest rate derivatives: one factor spot rate

In the present section we are to treat the problem of pricing options on interest rate derivativesecurities. The essential feature of this problem is that we need to take account of thestochastic nature of interest rates. We have seen in section 4.1.2 one approach to thisproblem, namely modelling the price of pure discount bonds as a stochastic process andmaking this one of the stochastic factors upon which the value of the option depends. Thegeneral approach is due to Merton, see, e.g. [Mer73].

There are however a number of practical diculties in attempting to implement thisapproach. In particular it requires specication of the average expected return varianceover the time interval to maturity, together with the covariance between return and theinstantaneous short term rate. Also it is not clear in practice how best to estimate thesevariances and covariances.

Nevertheless, Merton's approach has guided the development of many of the subsequentinterest rate option models. A characteristic of the stock option model is that there is onebasic approach.

For interest rate contingent claims however there does not seem to be one basic approachbut rather a range of alternative approaches. These dier according to what is taken as theunderlying factor, which is usually one of the instantaneous spot interest rate, the bondprice or the forward rate. Further, some models are presented in a discrete time frameworkand some in a continuous time framework. An important distinction between alternativeapproaches is whether the initial term structure, that is the currently observed yield curve,is itself to be modelled or to be taken as given. This modelling choice will determine whetherthe resulting models involve the market price of interest rate risk.

In this chapter we survey models of interest rate derivatives which take the instantaneousspot rate of interest as the underlying factor. The by-now familiar continuous hedgingargument is extended so as to model the term structure of interest rates and from this otherinterest rate derivative securities. This basic approach is due to Vasicek [Vas77] and hencewe shall often refer to it as the Vasicek approach.

4.5.1 Arbitrage models of the term structure

We consider the perspective of an investor standing at time 0 and observing various marketrates that enable him to compute the initial forward f(0, T ), and consequently the initial

4.5 Interest rate derivatives: one factor spot rate 147

0 t T

e 1

P (0, T ) observed P (t, T ) =?

Figure 4.9: The time line for bond pricing problem

bond price curve P (0, T ), for any maturity T . This investor wishes to price at any time t ≤ Ta pure default-free discount bond that pays e1 at time T . The investor seeks the arbitrage-free bond price that does not allow the possibility of arbitrage opportunities between bondsof diering maturities. Furthermore the investor wishes the bond price so obtained to beconsistent with the currently observed initial bond price curve, see, e.g. 4.9.

Initially, we assume that the price of a default free bond is a function of only the currentshort term rate of interest and time. Thus we write P (r(t), t, T ) to denote the price at timet of a discount bond maturing at time T , having maturity value of e 1, when the currentinstantaneous spot rate of interest is r(t), which is assumed to be riskless in the sense thatmoney invested at this rate will always be paid back, i.e.,

P (r(T ), T, T ) = 1 .

We also assume the short term rate follows the diusion process

dr(t) = µr(r(t), t)dt+ σr(r(t), t)dWt , (4.84)

Applying Itô's lemma we get

dP

P=

1

P

(∂

∂tP + µr

∂

∂rP +

1

2σ2r

∂2

∂r2P

)dt+

σrP

∂

∂rPdWt = µP (r, t, T )dt+ σP (r, t, T )dWt .

(4.85)

Let us now consider an investor who at time t invests e 1 in a hedge portfolio containingtwo default free bonds maturing at times T1 and T2 respectively, held in the dollar amountsQ1 and Q2. Using then the notation Pi, i = 1, 2, to denote the price of the bond maturingat time Ti, i = 1, 2, we can write

return hedge portfolio = Q1dP1

P1+Q2

dP2

P2= (Q1µP1

+Q2µP2) dt+ (Q1σP1

+Q2σP2) dWt ,

(4.86)

where µPi , resp. σPi , denotes the expected return, resp. the standard deviation, of thebond of maturity Ti, i = 1, 2. This return can be made certain by choosing the amounts Q1

and Q2 such thatQ1

Q2= −σP2

σP1

. (4.87)

Thus from eq. (4.86) the dollar return on the now riskless hedge portfolio is

(Q1µP1+Q2µP2

) dt .


Absence of arbitrage implies that this return must be the instantaneous spot rate of interestr, so that given that the original investment is e 1, that is Q1 + Q2 = 1, then this lastcondition states that

(Q1µP1 +Q2µP2) dt = rdt ,

after rearrangement gives

Q1(µP1 − r) +Q2(µP2 − r) = 0 ,

which eventually combined with eq. (4.87) yields the condition for no arbitrage betweenbonds of any two maturities, namely

µP1 − rσP1

=µP2 − rσP2

. (4.88)

Since the maturity dates T1 and T2 were arbitrary, the ratio

µP − rσP

,

must be independent of maturity time T . Let λ(r, t) denote the common value of this ratiofor bonds of an arbitrary maturity T , thus we have

µP (r, t, T )− r(t)σP (r, t, T )

= λ(r, t) . (4.89)

The quantity λ can be interpreted as the market price of interest rate risk per unit of bondreturn volatility.

Thus eq. (4.89) asserts that in equilibrium bonds are priced so that instantaneous bondreturns equal the instantaneous risk free rate of interest plus a risk premium equal to themarket price of interest rate risk times instantaneous bond return volatility. Substitutionfrom eq. (4.85) of the expressions for µP and σP results in the PDE for the bond price

∂P∂t + (µr − λσr)∂P∂r + 1

2σ2r∂2P∂r2 − rP = 0 ,

P (r(T ), T, T ) = 1 .(4.90)

To solve eq. (4.90), either analytically or numerically, we need to specify the drift µrand diusion σr as well as form of the market price of risk term λ(r, t). One commonassumption is that this latter term is constant. To formally derive this result, involves somevery particular assumptions about how the capital market operates. These conditions arediscussed briey in the next subsection. To give a proper theoretical basis to the choice ofλ(r, t) it would be necessary to construct a dynamic general equilibrium model and relateλ(r, t) to investor preferences. This is the approach adopted in [CIJR85].

4.5.2 The martingale representation

Just as in the case of the stock option model we are able to obtain a martingale representationof the pricing relationship; we note from the no arbitrage condition (4.89) substituted intoeq. (4.85) yields the stochastic bond price dynamics under the condition of no arbitragethat is

dP

P= (r + λσP (r, t, T ))dt+ σP (r, t, T )dWt . (4.91)


Exploiting then Girsanov theorem 1.4.4, we can introduce a new probability measure Pand thus dene a new Wiener process

dWt = dWt + λ(t)dt , (4.92)

under which dPP evolves according to

dP

P= rdt+ σP (r, t, T )dWt (4.93)

so that the discounted process is a Q−martingale.However unlike in the stock option situation, the spot rate r is here stochastic, so we

need to dene the money market account as

A(t) = exp

∫ t

0

r(s)ds

, (4.94)

which, applying Itô's lemma can be shown to satisfy the following

dA(t) = rA(t)dt .

We can then dene the bond price in units of the money market account

Z(r, t, T ) =P (r, t, T )

A(t),

and again by Itô's lemma we have that it satises

dZ

Z= σP (r, t, T )dWt . (4.95)

We immediately have from eq. (4.95) that Z is a P−martingale, that is

Et [Z(r(T ), T, T )] = Z(r(t), t, t) ,

which in terms of the original bond price can be expressed as

P (r, t, T ) = Et[A(t)

A(T )P (r(T ), T, T )

],

or exploiting the fact that P (r(T ), T, T ) = 1 together with eq. (4.94) as

P (r, t, T ) = Et

[exp

∫ T

t

r(s)ds

]. (4.96)

To derive the interest rate dynamics under the measure P let us use eq. (4.92) to rewriteeq. (4.84) as

dr(t) = (µr(r(t), t)− λ(t)σr(r(t), t)) dt+ σr(r(t), t)dWt , (4.97)

Applying then Feynman-Kac theorem 1.3.23 gives us again the PDE (4.90)

∂P

∂t+ (µr − λσr)

∂P

∂r+

1

2σ2r

∂2P

∂r2− rP = 0 , (4.98)


equipped with the suitable boundary condition.Thus, just as in the stock option situation, we have two representations of the bond price,

that is the PDE (4.98) and the expectation (4.96) under the interest rate dynamics (4.97).Let us eventually emphasize the discounted cash ow interpretation of the representation

(4.96). The factor

exp

∫ T

t

r(s)ds

,

discounts back to t the euro received at T , for one particular path followed by the interestrate r(t). Since r(t) is stochastic this quantity is in fact a stochastic discount factor. Toobtain then the discounted value at t of the e 1 received at maturity time T we need toaverage over the range of possible paths followed by r(t) under the measure P.

It is also of interest to contrast the bond price expression (4.96) with the correspondingexpression in eq. (4.63) for a world of certainty, and we see how this is generalised in anatural way to the world of uncertainty. We thus have a complete analogy with the stockoption price derivation of section ?? with the exception that the pricing relationships hereinvolve the market price of interest rate risk λ.

4.5.3 On some specic model

A variety of term structure models are obtained by specifying dierent forms for µ and σ inthe interest rate process, eq. (4.97) and/or dierent forms for the market price of risk term.

Let us now focus on two fundamental models we have already introduced in section 4.4.2,that is the Vasicek model and the Cox-Ingersoll-Ross model (CIR).

The Vasicek model

The Vasicek model, see, e.g. [Vas77] holds a special place in the interest rate term structureliterature as it was the earliest model. Its basic assumptions are to take µr = κ(θ − r) andσr = σ, with κ, θ and σ some positive constants, also assume a constant market price ofinterest rate risk λ. Setting γ = κθ − λσ the bond pricing PDE (4.90) becomes

∂P

∂t+ (γ − κr)∂P

∂r+

1

2σ2 ∂

2P

∂r2− rP = 0 . (4.99)

In order to solve the above PDE let us ansatz a solution of the form

P (t, T ) = ea(t,T )−b(t,T )r(t) , (4.100)

together with the condition

a(T, T ) = 0 , b(T, T ) = 0 , (4.101)

so that the boundary condition P (r(T ), T, T ) = 1 is attained. From the ansatz (4.100) wehave that

∂P

∂r= −bP , ∂2P

∂r2= b2P ,

∂P

∂t= −(

∂

∂ta+

∂

∂tb)P .

Substituting then into eq. (4.99) we have(1

2σ2b2 − ∂

∂ta− ∂

∂tb

)+

(κb− ∂

∂tb− 1

)r = 0


which is true for any t and any r only if it holds(12σ

2b2 − ∂∂ta−

∂∂tb)

= 0 ,(κb− ∂

∂tb− 1)r = 0 .

We then get that a and b have to satisfy∂∂ta = 1

2σ2b2 − bγ ,

∂∂tb = κb− 1 ,

subject to the initial condition (4.107). We then have that solution of the above system isgiven by

a(t, T ) =(γκ −

σ2

2κ2

)(T − t) +

(γκ2 − σ2

2κ3

) (e−κ(T−t) − 1

)+ σ2

4κ3

(e−κ(T−t) − 1

)2,

b(t, T ) = 1−e−κ(T−t)κ .

The corresponding expression for the yield to maturity is

ρ(t, T ) = − logP (t, T )

(T − t)=a(t, T )− b(t, T )r(t)

(T − t)=

=

(γ

κ− σ2

2κ2

)(1 +

e−κ(T−t) − 1

κ(T − t)

)+

σ2

4κ3

(e−κ(T−t) − 1

)2(T − t)

+1− e−κ(T−t)

(T − t)r(t) .

By letting (T − t)→∞ we nd that the yield at innite maturity is given by

ρ∞ =γ

κ− σ2

2κ2,

so that the bond price can be expressed as

P (r, t, T ) = exp

(e−κ(T−t) − 1

)κ

(r − ρ∞)− ρ∞(T − t)− σ2

4κ3

(e−κ(T−t) − 1

)2. (4.102)

The Vasicek model is now of historical interest, but it contains all the basic ingredientsneeded to deal with the more sophisticated models, that is a technique to solve the pricingPDE and the idea of relating the parameters of the model to information that can beobtained from the currently observed yield curve. The last observation also makes evidentone of the shortcomings of the Vasicek model. By setting t = 0 in eq. (4.103) we obtain

P (r0, 0, T ) = exp

(e−κT − 1

)κ

(r0 − ρ∞)− ρ∞T −σ2

4κ3

(e−κT − 1

)2. (4.103)

If γ is chosen so as to match the long term yield ρ∞ then we only have two parameters,κ and σ, left to make expression (4.103) consistent with the entire currently observed yieldcurve P (r0, 0, T ). Clearly this is impossible as at the most we could choose κ and σ to t twopoints exactly, or alternatively choose them to obtain some sort of least squares t. Theseobservations suggest that one possibility to develop a model that ts the currently observedyield curve is to make at least one, if not more, of the quantities κ, θ and σ time varying.


We would then have at our disposition a whole set of values of say κ (if it were allowed tobe time varying) with which to match the theoretical model to the currently observed yieldcurve.

We know that under P the bond price dynamics are given by eq. (4.93). However atthat point in our development we did not have an explicit expression for ∂P

∂r . The solution(4.100) now enables us to calculate this expression, in fact substituting this expression into(4.93) we get

dP

P= (r − λσb(t, T )) dt− σb(t, T9dWt .

The last equation indicates that the standard deviation of bond return is −σb, wherethe minus sign simply indicates that a positive shock to the interest rate dynamics resultsin a negative shock to the bond price dynamics. This is merely a reection of the fact thatinterest rates and bond prices are inversely related. Furthermore the interest rate dynamicsunder P becomes

dr(t) = (γ − κr(t))dt+ σdWt .

The Cox-Ingersoll-Ross model

The Cox-Ingersoll-Ross model (CIR) introduced in [CIR85], assumes the interest rate r(t)to evolve according to

dr(t) = κ(θ − r(t))dt+ σ√r(t)dWt , (4.104)

being as in the Vasicek model κ, θ and σ some positive constants. As discussed in section4.4.2 this process guarantees non-negative spot interest rate sample paths. The CIR modelemploy a dynamic general equilibrium framework to derive the bond pricing equation andunder specic assumptions about investor preferences, in order to obtain a tractable bondpricing equation we assume that the market price of interest rate risk is a function of r givenby

λ(r) = λ√r ,

being λ a constant. In the present model the bond pricing PDE (4.90) becomes

∂P

∂t+ (κθ − (κ+ λσ)r)

∂P

∂r+

1

2σ2r

∂2P

∂r2− rP = 0 . (4.105)

Given the very similar structure to the PDE (4.99) encountered in the Vasicek model,being the only dierence the r in front of the second derivative, we again try a solution ofthe form

P (t, T ) = ea(t,T )−b(t,T )r(t) . (4.106)

As for the Vasicek model, in order the terminal condition to holds we get

a(T, T ) = 0 , b(T, T ) = 0 . (4.107)

From the ansatz (4.106) we have that

∂P

∂r= −bP , ∂2P

∂r2= b2P ,

∂P

∂t= −(

∂

∂ta+ r

∂

∂tb)P .

which substituted then into eq. (4.105) yield(κθb− ∂

∂ta

)+

(1

2σ2b2 + (κ+ λσ)b− ∂

∂tb− 1

)r = 0 ,


which is true for any t and any r only if it holds(κθb− ∂

∂ta)

= 0 ,(12σ

2b2 + (κ+ λσ)b− ∂∂tb− 1

)r = 0 .

(4.108)

The dierence compared to the solution of the Vasicek model is the term b2 in the ordi-nary dierential equation (4.108). This is in fact the well-known Ricatti ordinary dierentialequation whose solution is known. In particular its solution is given by

a(t, T ) = 2κθβσ2

(−β (T−t)

φ1− φ1−φ2

φ1φ2log φ1−φ2e

β(T−t)

φ1−φ2

),

b(t, T ) = 2σ2

1−e−β(T−t)φ1e−β(T−t)−φ2

,

φ1 = −κ+λσ2 + β

σ2 , φ1 = −κ+λσ2 − β

σ2 ,

β =√

(κ+ λ)2 + 2σ2 ,

we refer to [] to details on computation of the above solutions.In order to be able to calibrate the model to market data we needed the additional

exibility required by allowing the coecients in (4.104) to be time varying. This can beeasily done assuming that all of the coecients κ, θ, σ and λ in the PDE (4.105) becometime varying.

4.5.4 Pricing bond options

Let us assume that the short-term rate follows the process (4.84). We also assume that thereare no arbitrage opportunities in the bond market. Thus the price of the discount bond ofany maturity is still given by the solution to the PDE (4.90).Let C(r, t) denote the price attime t of a call option of maturity time TC written on a bond having maturity T > TC .

By Itô's lemma we have that

dC

C=

1

C

(∂

∂tC + µC

∂

∂rC +

1

2σ2C

∂2

∂r2C

)dt+

σCC

∂

∂rCdWt = µC(r, t, T )dt+σC(r, t, T )dWt .

(4.109)Consider an investor who at time t invests e 1 in a hedge portfolio containing the bond

of maturity T held in the euro amount QP and the option of maturity TC held in the euroamount QC . The euro return on this hedge portfolio over time interval dt is given by

return hedge portfolio = QPdP

P+QC

dC

C= (QPµP +QCµC) dt+ (QPσP +QCσC) dWt ,

(4.110)and as before the hedge portfolio is rendered riskless by choosing QP and QC such that

QPQC

= −σCσP

. (4.111)

The absence of arbitrage means that the hedge portfolio can only earn the same returnas the original e 1 invested at the risk-free rate, that is

(QPµP +QCµC) dt = rdt . (4.112)


t TC T

solve (4.90)solve (4.114)

P (r(T ), T, T ) = 1P (r(TC), TC , TC) = 1C(r(t), t)

Figure 4.10: The two step process for the bond option pricing problem.

Recalling that QP +QC = 1, so that conditions (4.111)-(4.112) imply

µC − rσC

=µP − rσP

,

But by eq. (4.89) we know that in an arbitrage-free bond market µP−rσP

is equal to themarket price of interest rate risk. Thus we arrive at the no arbitrage condition between theoption and bond markets, that is

µC(t, s)− r(t)σC(t, s)

=µP (t, s)− r(t)

σP (t, s)= λ(r, t) , (4.113)

This has the now familiar interpretation that in the absence of riskless arbitrage theexcess return risk adjusted on both the bond and the option are equal. The common factorto which they are equal is the market price of risk of the spot interest rate, the underlyingfactor. Equation (4.113) yields the PDE (4.90) for the bond price P whilst for the optionprice C, the PDE

∂C

∂t+ (µr − λσr)

∂C

∂r+

1

2σ2r

∂2C

∂r2− rC = 0 , (4.114)

which in the case of a European call option, with strike price K, on the bond must be solvedon the time interval 0 ≤ t ≤ TC subject to the boundary conditions

C(r(TC), TC)) = maxP (r(TC), TC , T )−K, 0 ,C(∞, t) = 0 ,

(4.115)

where the last condition is a consequence of the result that P (∞, t, T ) = 0, i.e. the bondvalue declines to zero as the interest rate becomes larger.

As above it is a two-pass structure of the solution process. We must rst solve the partialdierential equation (4.90) with boundary condition P (r(T ), T, T ) = 1 for the bond priceP (r(s), s, T ) on the time interval TC ≤ s ≤ T . Then the value P (r(TC), TC , T ) is used inthe solution of the partial dierential equation (4.114) with the boundary condition (4.115),see, e.g. 4.10.

In order to obtain the martingale representation for the option price we follow almostidentical steps to those we followed above to obtain the martingale representation for thebond price. First we observe from the no-arbitrage condition (4.113) that

µC = r + λσC ,


which substituted into eq. (4.109) gives the arbitrage free option price dynamics, that is

dC

C= (r + λσC)dt+ σCdWt ,

or written in terms of W , that is the Wiener process under the probability measure P,

dC

C= rdt+ σCdWt . (4.116)

If we then set

Y (r, t) =C(r, t)

A(t),

that is the option price measured in units of money market account, we get via Itô's formula

dY

Y=σrC

∂C

∂rdWt ,

which immediately implies that Y is a P−martingale, that is

Et [Y (r(TC), TC)] = Y (r, t) ,

or in terms of the option price can be expressed as

C(r, t) = Et[e−

∫ TCt r(s)dsC(r(TC), TC)

]. (4.117)

If for example we wish to price a European call option on a bond then the maturitycondition is

C(r(TC), TC) = maxP (r(TC), TC , T )−K, 0

The interest rate dynamics under the measure P are given by

dr(t) = (µr − λσr)dt+ σrdWt ,

and applying the Feynman-Kac formula to (4.117) we will retrieve the option pricing PDE(4.114).

One of the main problem if one is to evaluate the expectation (4.117) is that one needsthe joint distribution of

e−∫ TCt r(s)ds ,

and

C(r(TC), TC) .

This calculation may turn out to be dicult. A simpler calculation may be obtained byusing the so-called forward measure, that is the measure P∗ under which the instantaneousforward rate equals the expected future forward rate, see subsection 4.6.3 for details, whichconsists in choosing as numeraire a bond of maturity TC . That is we consider

Y (r, t, TC , T ==C(r, t, TC , T )

P (r, t, TC), (4.118)


so that its dynamic under P is given by

dY

Y= −σP (t, TC)(σC − σP (t, TC))dt+ (σC − σP (t, TC))dWt . (4.119)

Following what done in section 4.2.2, exploiting Girsanov theorem, we can then dene anew process

dW ∗t = dWt − σP (t, Tc)dt , (4.120)

and a new probability measure P∗ under which the new Wiener process W ∗ is taken. Underthis new measure P∗ eq. (4.119) becomes

dY

Y= (σC − σP (t, TC))dW ∗t ,

so that we clearly have that Y is a P∗−martingale. If we now use the denition of Y (4.118),together with the fact that it is a P∗−martingale, we get

C(r(t), t, TC , T ) = P (r(t), t, TC)E∗t [C(r(TC), TC , TC , T )] . (4.121)

The main dierence now between eq. (4.117) and eq. (4.121) is the way the stochasticdiscount is done. In fact in eq. (4.117) the stochastic discounting is done along eachstochastic interest rate path from t and TC , and since these paths are stochastic this termmust appear under the expectation operator. Conversely in eq. (4.121) the discounting fromt to TC is done using the bond of maturity TC , which is known to the investor at time t andhence this term does not need to appear under the expectation operator.

It is also of interest to obtain the dynamics under P∗ for the relative bond price

X(r, t, TC , T ) =P (r, t, T )

P (r, t, TC).

This follows by noting that under the measure P∗ we have

dP (t, T )

P (t, T )= (r + σP (t, T )σP (t, TC))dt+ σP (t, T )dW ∗t ,

so that we eventually get from Itô's lemma

dX

X= (σP (t, T )− σP (t, TC))dW ∗t .

4.6 The Heath-Jarrow-Morton framework

The framework to model interest rate developed in Sec. 4.4.3 was focused on the dynamicsof the instantaneous spot rate of interest. That is the market price of interest rate riskappears in the pricing relationships.

An alternative, and more general approach, to model interest rate is the so called Heath-Jarrow-Morton (HJM) approach. This approach starts from the dynamics of the forwardrate and it has as input factors the initial term structure and the volatility of the associated

4.6 The Heath-Jarrow-Morton framework 157

forward rate. Then the dynamics of the spot interest rate can be developed starting fromthe dynamic of the forward rate. Furthermore it appears that the spot interest rate is animportant economic variable whose evaluation completely determines the evolution of thebond prices.

Nevertheless there are some mathematical drawback in the HJM approach, that is thespot rate dynamics does no longer posses the Markov property being its evolution pathdependent.

Also, as mentioned above, in order to compute the dynamic of the spot interest rate,the volatility of the forward rates is needed. To observe the aforementioned volatility is farfrom being an easy task, and it often happens that the forms of the volatility functions ischosen for analytical convenience rather than on the basis of empirical evidence.

On top of that the non-Markovian feature makes dicult the connection to the deter-ministic counterpart of the ricing problem.

We will assume in the present chapter that the path dependence of the forward ratevolatility functions arises from dependence on the instantaneous spot interest rate. Doingso we are able to write the instantaneous spot rate process as a nite dimensional Markoviansystem, whose dimension depends on the exact form of the volatility function.

4.6.1 The basic structure

The starting point of the HJM model of the term structure of interest rates is the stochasticintegral equation for the forward rate.

Let us then consider a complete probability space (Ω,F ,P), and let t ∈ [0, T ], withT <∞ a nite positive terminal time. Let us thus consider the forward rate

f(t, T ) = f(0, T ) +

∫ t

0

α(s, T, ω(s))ds+

∫ t

0

σ(s, T, ω(s))dW (s) , (4.122)

with α(s, T, ω(s)), resp. σ(s, T, ω(s)), the instantaneous drift, resp. the volatility function, attime s of the forward rate f(s, T ), and the notation ω(s) means that the instantaneous driftα and the volatility function σ could depend, as said above, on path dependent quantities,such as the instantaneous spot rate. Eventually W (t) is a standard Brownian motion.

Remark 4.6.1. Considering T = t, from eq. (4.122), we can retrieve the spot rate dynamicr(t) = f(t, t) as

r(t) = f(0, t) +

∫ t

0

α(s, T, ω(s))ds+

∫ t

0

σ(s, T, ω(s))dW (s) . (4.123)

Recall that the connection between bond prices and forward rate is given in eq. (4.61)to be

P (t, T ) = e−∫ Ttf(t,s)ds ,

and that an application of Itô's lemma together with the stochastic Fubini's theorem toP (t, T ) = eB(t;T ) leads to eq. (4.83), that is

dP (t, T ) = (r(t)− b(t, T ))P (t, T )dt+ σB(t, T )P (t, T )dW (t) , (4.124)

with b(t, T ) = αB(t, T ) + 12σ

2B(t, T ) and σB(t, T ) = −

∫ Ttσ(t, s, ω(s))ds, see subsection ??.


A quantity of interest is the money market account

A(t) = exp

∫ t

0

r(s)ds

,

such a quantity may be used in order to dene the relative bond price

Z(t, T ) =P (t, T )

A(t), 0 ≤ t ≤ T ,

Exploiting thus the fact that A solves the dierential equation

d

dtA(t) = r(t)A(t) ,

and applying Itô's formula we have that the relative bond price satises

dZ(t, T ) = b(t, T )Z(t, T )dt+ σB(t, T )Z(t, T )dWt . (4.125)

4.6.2 The arbitrage pricing of bonds and bond options

Bonds can be priced using the same hedging portfolio that has been used in previous section4.5.4. The instantaneous excess bond return, risk adjusted by its volatility must equal themarket price of interest rate risk.

The relevant bond dynamics in the current context are given by eq. (4.124), so that wehave

b(t, T )

σB(t, T )= market price of interest rate risk = −φ(t) ,

which implies the followingb(t, T ) + φ(t)σB(t, T ) = 0 . (4.126)

Using then the explicit form for b and σB we have that eq. (4.126) can be rewritten as

∫ T

t

α(t, s, ω(t))ds− 1

2

(∫ T

t

σ(t, s, ω(t))ds

)2

+ φ(t)

∫ T

t

σ(t, s, ω(t))ds = 0 .

Keeping t xed and taking the derivative w.r.t. T we have that

α(t, T, ω(t))−

(∫ T

t

σ(t, s, ω(t))ds

)σ(t, T, ω(t)) + φ(t)σ(t, T, ω(t)) = 0 ,

which eventually can be rearranged to

α(t, T, ω(t)) = −σ(t, T, ω(t))

[φ(t)−

∫ T

t

σ(t, s, ω(t))ds

]. (4.127)

Equation (4.127) is the forward rate drift restriction that was rst reported by Heath,Jarrow and Morton in [HJM90]. If the bond market is free of arbitrage opportunities thenthe forward rate drift, the forward rate volatility and the market price of interest rate riskmust be tied together as shown by eq. (4.127). Heath, Jarrow and Morton showed also


in [HJM90] that in fact this condition is both necessary and sucient for the absence ofarbitrage, and it is know in literature as HJM drift condition.

Up to this point Heath, Jarrow and Morton have not done anything conceptually dierentfrom the standard arbitrage approach of section 4.5.4, the only dierence is that in the HJMframework, eq. (4.126) is used in a dierent way. In the standard arbitrage approachused in section 4.5.4, eq. (4.126) becomes a partial dierential equation for the bond priceas a function of the assumed driving state variable (usually the instantaneous spot rate).Conversely, in the HJM approach, the condition (4.126) becomes the forward rate driftrestriction that is used to conveniently express the bond price dynamics under an equivalentprobability measure.

Exploiting condition (4.126), eq. (4.124) and (4.125) become

dP (t, T ) = (r(t)− φ(t)σB(t, T ))P (t, T )dt+ σB(t, T )P (t, T )dWt ,

dZ(t, T ) = −φ(t)σB(t, T )Z(t, T )dt+ σB(t, T )Z(t, T )dWt .(4.128)

Substituting then eq. (4.127) into eq. (4.123) we get

r(t) = f(0, t)+

∫ t

0

σ(τ, t, ω(τ))

∫ t

τ

σ(τ, s, ω(τ))dsdτ−∫ t

0

σ(τ, t, ω(τ))dτ+

∫ t

0

σ(τ, t, ω(τ))dWτ .

(4.129)The main advantage in the HJM framework is that equations (4.128)-(4.129) can be

written in terms of a dierent Brownian motion. In particular let us dene Wt a Wienerprocess under a dierent probability measure P as

dWt = dWt − φ(t)dt ,

so that equations (4.128)-(4.129) become

dP (t, T ) = r(t)P (t, T )dt+ σB(t, T )P (t, T )dWt ,

dZ(t, T ) = σB(t, T )Z(t, T )dWt ,

r(t) = f(0, t) +

∫ t

0

σ(τ, t, ω(τ))

∫ t

τ

σ(τ, s, ω(τ))dsdτ +

∫ t

0

σ(τ, t, ω(τ))dWτ .

(4.130)

Alternatively the spot rate in eq. (4.130) can be written as the solution to

dr(t) =

(∂

∂tf(0, t) +

∂

∂t

(∫ t

0

σ(τ, t, ω(τ))

∫ t

τ

σ(τ, s, ω(τ))dsdτ

)+

∫ t

0

∂

∂tσ(τ, t, ω(τ))dWτ

)dt+σ(t, t, ω(t))dWt .

(4.131)At the same time it might be convenient to deal with the logarithm of the bond price,

that is B(t, T ) = logP (t, T ). Exploiting Itô's lemma, we have that B satises

dB(t, T ) =

(r(t)− 1

2σ2B(t, T )

)dt+ σB(t, T )dWt . (4.132)

The essential characteristic of the reformulated stochastic dierential and integral equa-tions is that the market price of risk term φ(t), which is empirically dicult to evaluate, iseliminated from explicit consideration. Exploiting then Girsanov theorem, see section 1.4.1,we can derive the expression for the Radon-Nikodym derivative

dPdP

= exp

−1

2

∫ t

0

φ2(s)ds+

∫ t

0

φ(s)dWs

.


We can then take the conditional expectation at time t under the probability measureP, Et in eq. (4.130) to have

Et [dZ(t, T )] = 0 ,

so that we have that Z(t, T ) is a martingale under the measure P, that is

Et [Z(T, T )] = Z(t, T ) ,

or equivalently in terms of the bond price

P (t, T ) = Et[A(t)

A(T )

]= Et

[exp

−∫ T

t

r(s)ds

]. (4.133)

Equation (4.133) is the fundamental bond pricing equation of the HJM framework, theargument of the exponential function in eq. (4.133) should be interpreted as the stochasticdiscount factor under P. The actual implementation of eq. (4.133) will depend on the formchosen from the forward rate volatility function, also closed form analytical expressions forthe bond price may be obtained with appropriate assumptions on the volatility function.

Let us suppose we wish to price at time t an option on the bond, for example a Europeancall option on the bond with terminal payo given by

Φ(P ) = maxP −K, 0 ,

with the option maturing at time TC . Under the risk-neutral measure P we can discountthe payo at time TC back to t using the stochastic discount factor

exp

−∫ TC

t

r(s)ds

. (4.134)

Multiplying then the payo by the discount factor (4.134) we nd that under one realisationof the spot-rate process under P the option value at t is given by

exp

−∫ TC

t

r(s)ds

maxP (TC , T )−K, 0 .

The value of the option, C(t, TC), is then obtained by taking the expectation of thisquantity under the risk-neutral measure P, that is

C(t, TC) = Et[

exp

−∫ TC

t

r(s)ds

maxP (TC , T )−K, 0

].

In general, if we have some spot interest rate contingent claim with payo at t = TCgiven by a general function Φ(r(TC), TC), then its value V at t is given by

V (t, TC) = Et[

exp

−∫ TC

t

r(s)ds

Φ(r(TC), TC))

]. (4.135)

In order to obtain a pricing partial dierential equation for V (t, TC) we need to obtainthe Kolmogorov backward equation associated with eq. (4.131). In general this is not aneasy task due to the non-Markovian term∫ t

0

∂

∂tσ(τ, t, ω(τ))dWτ .


that appears in the drift of the equation.We could then try to follow the argument used in section 4.5.4 and use the hedging

argument approach to derive the bond option pricing formula. The main problems are thatin section 4.5.4 there was one underlying factor, r(t), driving the uncertainty of the market,and an application of Itô's lemma gave us the dynamic. In both cases the dynamics of thehedging portfolio could be obtained. Here we have not so far been so precise about thefactor upon which the volatility function σ(t, T ) depends.

In order then to emulate the hedging argument approach we need to be more specicabout the dynamics of these underlying rates so that we could then obtain the option pricedynamics by applying Itô's lemma. Equation (4.133) for the bond price and eq. (4.135) forthe interest rate derivative hold for quite general specications of the volatility function σ.If we are to implement these expressions then we would need to specify the dynamics underP of all stochastic factors entering into the specication of the volatility function σ.

4.6.3 Forward risk adjusted measure

We have just seen in eq. (4.135) the explicit form for the value of a spot interest rate con-tingent claim at time t and concerning the price P (t, TC) of a pure discount bond maturingat time TC we have derived eq. (4.133). Using then the results of section 4.2.2 we canexpress the value of the interest rate contingent claim, using P (t, TC) as the numeraire, sothat dening the quantity Y = V

P we would obtain

V (t, TC) = P (t, TC)E∗t[V (TC , TC)

P (TC , TC)

],

since in the current notation we have that V (TC , TC) = Φ(r(TC), TC) and P (TC , TC) = 1,so that we get

V (t, TC) = P (t, TC)E∗t [Φ(r(TC), TC)] , (4.136)

where the expectation is taken under the measure P∗ is known as T-forward measure, andin particular under this measure it holds

E∗t [f(T, T )] = f(t, T ) . (4.137)

In order to see this, let us dierentiate eq. (4.133) w.r.t. T to get

∂

∂TP (t, T ) = Et

[∂

∂Texp

−∫ T

t

r(s)ds

]= Et

[− exp

−∫ T

t

r(s)ds

∂

∂T

∫ T

t

r(s)ds

]=

= Et[

exp

−∫ T

t

r(s)ds

r(T )

]= −P (t, T )E∗t [r(T )] ,

(4.138)

where the last equality follows applying eq. (4.136) with Φ = r(T ). Rearranging then thelast result we get

− ∂

∂TlogP (t, T ) = E∗t [r(T )] .

Eventually from the fact that

f(t, T ) = − ∂

∂TlogP (t, T ) ,


we have proved eq. (4.137).The main advantage of eq. (4.136) over eq. (4.135) is that the stochastic discount term

exp

−∫ TC

t

r(s)ds

,

is replaced by the non-stochastic term P (t, TC). The eective value of the present methoddepends on how easy (or dicult) it is to evaluate the expectation E∗t .

4.6.4 Reduction to Markovian form

The diculty in implementing and estimating HJM models arises from the non-Markoviannoise term in the stochastic integral equation (4.130) for r(t). In particular this componentdepends on the history of the noise process from time 0 to current time t. So that dependingupon the specication of the volatility function the second component of the drift term couldalso depend on the path history up to time t.

In the present section we are to consider a class of functional forms of σ that allowthe non-Markovian representation of r(t) and P (t, T ) to be reduced to a nite dimensionalMarkovian system of stochastic dierential equations. In particular we investigate volatilityfunctions of the forward rate which have the general form

σ((t, T, ω(t)) = Q(t, T )G(ω(t)) , 0 ≤ t ≤ T , (4.139)

where G is an appropriately well-behaved function. A general form for the function Q wouldbe of the form

Q(t, T ) = Pn(T − t)e−λ(T−t) , (4.140)

where Pn is the polynomial

Pn(x) = a0 + a1x+ · · ·+ anxn .

This form is particularly useful since it allows the term structure of the volatility to exhibithumps as observed in implied forward rate volatilities from cap prices.

Remark 4.6.2. This structure includes forward rate volatilities for a number of importantcases in the literature, such as

• λ > 0, G(ω(t)) = 1 leads to the Hull-White model ;

• λ > 0, G(ω(t)) = g′(r(t)) leads to the Ritchken-Sankarasubramanian model, see, e.g.[RS95];

• λ > 0, G(ω(t)) =√r(t) leads to the CIR model, see, e.g. [CIR85];

• λ > 0, G(ω(t)) = g(r(t), f(t, τ)) leads to the Chiarella-Kwon model, see, e.g. [CK01].


The main goal of the present section is to express eq. (4.131) and the equation for theforward rate f as a Markovian system of stochastic dierential equations. In particular from(4.130) we have that the forward rate evolves according to

df(t, T ) = σ(t, T, ω(t))

∫ T

t

σ(t, s, ω(t))dsdt+ σ(t, T, ω(t))dWt , (4.141)

considering then the drift term in eq. (4.141) and under the volatility specication made insection 4.5.4 that is

σ(t, T, ω(t)) = σe−λ(T−t)G(ω(t)) , (4.142)

we then get

σ(t, T, ω(t))

∫ T

t

σ(t, s, ω(t)) = σ2G2(ω(t))e−λ(T−t)∫ T

t

e−λ(s−t)ds =

= −σ2G2(ω(t))e−λ(T−t) e−λ(T−t) − 1

λ= σ2(t, T, ω(t))e−λ(T−t) e

−λ(T−t) − 1

λ.

Therefore we have that eq. (4.141) can be written as

df(t, T ) = σ2(t, T, ω(t))e−λ(T−t) e−λ(T−t) − 1

λdt+ σ(t, T, ω(t))dWt . (4.143)

Concerning eq. (4.131) for r(t) we have that

∂

∂t

(∫ t

0

σ(τ, t, ω(τ))

∫ t

τ

σ(τ, s, ω(τ))dsdτ

)=

∫ t

0

(∂σ(τ, t, ω(τ))

∂t

∫ t

τ

σ(τ, s, ω(τ))ds+ σ2(τ, t, ω(τ))

)dτ .

(4.144)From eq. (4.142) we also have that

∂σ(τ, t, ω(τ))

∂t= −λσ(τ, t, ω(τ)) ,

so that the r.h.s. of eq. (4.144) reduces to∫ t

0

(−λσ(τ, t, ω(τ))

∫ t

τ

σ(τ, s, ω(τ))ds+ σ2(τ, t, ω(τ))

)dτ ,

and eq. (4.131) becomes

dr(t) =

(∂

∂tf(0, t)− λσ(τ, t, ω(τ))

∫ t

τ

σ(τ, s, ω(τ))dsdτ + σ2(τ, t, ω(τ))dτ

)dt+

− λ∫ t

0

σ(τ, t, ω(τ))dWτdt+ σ(t, t, ω(t))dWt ,

(4.145)

and from eq. (4.130) we have that

r(t)− f(0, t) =

∫ t

0

σ(τ, t, ω(τ))

∫ t

τ

σ(τ, s, ω(τ))dsdτ + σ(t, t, ω(t))dWt . (4.146)


Using then eq. (4.146) into eq. (4.145) the spot rate is simplied as

dr(t) =

(∂

∂tf(0, t) + λf(0, t) + ψ(t)− λr(t)

)dt+ σ(t, t, ω(t))dWt , (4.147)

where we have dened the variable

ψ(t) =

∫ t

0

σ2(τ, t, ω(τ))dτ . (4.148)

The function ψ(t) plays a central role in allowing us to transform the original non-Markovian dynamics to Markovian form., it can be interpreted as a variable summarisingthe path history of the forward rate volatility.

4.6.5 On some specic models

At this point eq. (4.147) is still non-Markovian since the integral in the drift term involvesthe history of the path dependent forward rate volatility. To proceed any further we needto consider specic functional forms for G(ω(t)) in the volatility specications (4.142).

The Hull-White extended Vasicek model

Let us then consider G(ω(t)) = 1 so that the subsidiary variable ψ(t) in (4.148) becomes

ψ(t) =

∫ t

0

σ2(τ, t, ω(τ))dτ =

∫ t

0

σ2e−2λ(t−τ)dτ =σ2

2λ(1− e−2λt) . (4.149)

Setting then

θ(t) =∂

∂tf(0, t) + λf(0, t) +

σ2

2λ(1− e−2λt) , (4.150)

so that eq. (4.147) becomes

dr(t) = (θ(t)− λr(t)) dt+ σdWt , (4.151)

that is eventually the Markovian form we were looking for. Clearly this is the extendedVasicek model with the long run mean allowed to be time varying. Furthermore, note thatthe expression we have obtained for θ(t) in eq. (4.150) is the same as the one we obtainedin section 4.5.4 when we worked directly from the expression for the bond price obtainedfrom the continuous arbitrage approach, which takes the spot rate process as the drivingdynamics. Also, if we set λ = 0 we obtain the continuous time specication of the Ho-Leemodel.

In order to price an option in this framework we have to solve the following pricingequation

1

2σ2 ∂

2

∂r2C + (θ(t)− λr(t)) ∂

∂rC +

∂

∂tC − rC = 0 , (4.152)

endowed with the appropriate boundary condition for the specic option.


The general spot rate model

Let us now assume that the function G(ω(t)) depends on the spot interest rate r(t), thatis G(ω(t)) = g(r(t)), in the present case we need to separately handle the non-Markovianterm appearing in the drift of eq. (4.147). The subsidiary variable becomes then

ψ(t) =

∫ t

0

σ2e−2λ(t−τ)g2(r(τ))dτ , (4.153)

which leads to, after dierentiating in eq. (4.153)

dψ(t) =(σ2g2(r(t))− 2λψ(t)

)dt .

We are thus dealing with the two-dimensional Markovian system

dr(t) =

(∂

∂tf(0, t) + λf(0, t) + ψ(t)− λr(t)

)dt+ σg(r(t))dWt ,

dψ(t) =(σ2g2(r(t))− 2λψ(t)

)dt ,

(4.154)

see, e.g. [RS95] for details.Let us now dene the innitesimal generator L of eq. (4.154) as

L =1

2σ2g2(r(t))

∂2

∂r2+

(∂

∂tf(0, t) + λf(0, t) + ψ(t)− λr(t)

)∂

∂r+(σ2g2(r(t))− 2λψ(t)

) ∂∂r

,

(4.155)so that the Kolmogorov for the transition probability density µ is

Lµ+∂

∂tµ = 0 .

In particular, denoting by V the price of a given derivative instrument, we have thepricing partial dierential equation

LV +∂

∂tV − rV = 0 ,

equipped with appropriate boundary conditions, e.g. V (r, T, T ) = 1 for bonds or V (r, Tc, T ) =maxP (r, TC , T )−K, 0 for call options.Remark 4.6.3. In the special case of g(r(t)) =

√r(t) we are dealing with the extended CIR

model of the form

dr(t) =

(∂

∂tf(0, t) + λf(0, t) + ψ(t)− λr(t)

)dt+ σ

√r(t)dWt ,

dψ(t) =(σ2r(t)− 2λψ(t)

)dt ,

Chapter 5

Appendix

5.1 Recall on functional analysis

Let us rst state some classical from functional analysis and operator theory. In the presentsection we will just recall some denition that will be used later on in order to avoid confusionwith the used notation. We refer to [Bre11, EN00] for a deeper treatment of the topic.

Denition 5.1.1. Let (X, ‖ · ‖X) and (Y, ‖ · ‖Y ) be two Banach spaces. Let T : X → Y bea continuous linear operator. Let us further dene

‖T‖ := sup‖T (x)‖Y : x ∈ BX = supx∈X\0

‖T (x)‖Y‖x‖X

<∞

with BX the unit ball in X. Then ‖ · ‖ is a norm and denoting by L(X,Y ) the space of alllinear continuous operator form X to Y we have that (L(X,Y ), ‖ · ‖) is a Banach space.

We will denote for short L(X,X) = L(X).

Proposition 5.1.2. Let (X, ‖ · ‖X) and (Y, ‖ · ‖Y ) be two Banach spaces. A linear operatorT : X → Y is bounded i it is continuous.

Denition 5.1.3. Let T : X → Y a linear operator. We say that T is closed if its graph

GT := (x, T (x)) : x ∈ X

is closed in the topology of (X × Y, ‖ · ‖X×Y ).

Theorem 5.1.4 (Closed graph theorem). Let be E and F two Banach spaces. If T ∈L(E,F ) is linear and closed, then T ∈ L(E,F ), i.e. T is continuous.

We now state a fundamental denition, the one of adjoint operator. We will avoid tostate the general denition, conning ourselves to the case of an operator from a Hilbertspace to itself.

Denition 5.1.5 (Hilbert space adjoint). Let be H be a Hilbert space and A ∈ L(H). Wewill call A∗ the adjoint of A the unique operator A∗ ∈ L(H) such that

〈y,Ax〉 = 〈A∗y, x〉 , ∀x, y ∈ H .

Furthermore an operator is called self-adjoint if A = A∗.

167

168 5. Appendix

Denition 5.1.6 (Projection). Let T ∈ L(H) such that T 2 = T , then T is called a projec-tion. If in addition T is self-adjoint in the sense of Def. 5.1.5, then T is called orthogonalprojection.

We will not develop here the whole theory of bounded and linear operators and theirspectrum, we refer to [KJF77, Mor13, RS79] for details.

5.1.1 Dual space and weaker topologies

Denition 5.1.7 (Dual space). Let us consider a Banach space (X, ‖ · ‖X), we will denoteits dual X ′ the linear space dened by

X ′ := f : X → R : f is linear and continuous

If we equip X ′ with the norm

‖f‖X′ := supx∈X\0

|f(x)|‖x‖X

<∞

we have that (X ′, ‖ · ‖X′) is a Banach space.

Let (X, ‖ · ‖X) a Banach space we can thus dene the weak topology σ(X,X ′) (weakerthan the one induced by the norm) as the smallest topology on X with respect to whicheach f ∈ X ′ is continuous. We can further endow such a topology with a condition ofconvergence.

Denition 5.1.8 (weak convergence). Let (xh)h ⊂ X. Then

xh x as h→∞ ⇔ X′ 〈f, xh〉X → X′ 〈f, x〉X as h→∞, ∀f ∈ X ′

where we have denoted by the weak convergence and by X′ 〈·, ·〉X the canonical bilinearform.

It is sometimes useful to introduce an even weaker topology. Let us then dene for anyx ∈ X the linear continuous functional

φx(f) := X′ 〈f, x〉X , f ∈ X ′ .

Denition 5.1.9. We dene the weak∗ topology on a dual space X ′ dened as σ(X ′, X)to be smallest topology on X ′ s.t. φx is continuous for any x ∈ X.

Denition 5.1.10 (weak∗ convergence). Let (fh)h ⊂ X ′. Then

fh ∗ f as h→∞ ⇔ X′ 〈fh, x〉X → X′ 〈f, x〉X as h→∞, ∀x ∈ X

where we have denoted by the weak convergence and by X′ 〈·, ·〉X the canonical bilinearform.

Proposition 5.1.11. If X is a reexive space then σ(X,X ′) = σ(X ′, X).

5.1 Recall on functional analysis 169

Let T ∈ L(X), besides the uniform operator topology on L(X), which is the one inducedby the norm dened in Def. 5.1.1, we have two more possible topologies, the strong operatortopology and the weak operator topology.

We will write Ls(X) for the strong operator topology, which is the topology of pointwiseconvergence on (X, ‖ · ‖X).

We will write Lσ(X) the weak operator topology which is the topology of pointwiseconvergence on (X,σ(X,X ′))

Thus we have that given (Th)h ⊂ L(X), it converges to T ∈ L(X) i:

(uniform operator topology) ‖Th − T‖ → 0 as h→∞;

(strong operator topology) ‖Thx− Tx‖ → 0 ∀x ∈ X as h→∞;

(weak operator topology) X′ 〈x′, Thx− Tx〉X → 0 ⇔ Thx − Tx 0 ∀x ∈ X, x′ ∈ X ′as h→∞.

5.1.2 Spectrum and resolvent

As it will be seen spectral analysis of an operator plays a fundamental role when we are tocharacterizer the action of a the operator itself. We want in the present section to generalizethe concept of eigenvector for a matrix to a more general case of a linear operator betweeninnite dimensional Banach spaces.

It is well known from linear algebra that any N × N symmetrix matrix admits N realvalued eigenvalues λ1, . . . , λN and that furthermore it exist a basis e1, . . . , eN ∈ RNsuch that any ei is an eigenvectors associated to the eigenvalue λi.

One of our rst tasks is to generalize the concept of eigenvalues and eigenvectors toaccommodate operators acting on Banach space which may be innite-dimensional spacesand the operator may be unbounded.

We recall here a denition we already gave previously.

Denition 5.1.12 (Resolvent). Let X be a Banach space and let A be an operator on X.For any λ ∈ C we dene the operator Aλ on X by

Aλ := λ− IA

where I is the identity operator on X. If Aλ has an inverse we denote the inverse by R(λ,A)and call it the resolvent of A dened as

R(λ,A) := (λ− IA)−1 (5.1)

We then decompose the complex plane C into two dierent sets:

the resolvent set The resolvent set of the operator A is dened as

ρ(A) := λ ∈ C : λ− IA is invertible =

= λ ∈ C : R(λ,A) exists , R(λ,A) is bounded and the domain of R(λ,A) is dense in X .

Elements λ ∈ ρ(A) are called regular values of the operator A;

170 5. Appendix

the spectrum The spectrum of the operator A if the complement of the resolvent set

σ(A) := C \ ρ(A) .

The spectrum can be further decomposed into three disjoint sets:

the point spectrum The point spectrum is the set

σP (A) := λ ∈ σ(A) : R(λ,A) does not exist .

Elements of the point spectrum are called eigenvalues. If λ ∈ σP (A), an elementx ∈ Ker(Aλ) is called eigenvector or eigenfuction for A;

the continuous spectrum The continuous spectrum is the set

σC(A) := λ ∈ σ(A) : R(λ,A) exists and the domain of R(λ,A) is dense in X

but R(λ,A) is not bounded ;

the residual spectrum The residual spectrum is the set

σR(A) := λ ∈ σ(A) : R(λ,A) exists but R(λ,A) is not dense in X ;

Remark 5.1.13. For a linear operator on nite dimensional space the continuousand the residual spectrum are empty sets and therefore the complex plane canbe decomposed into regular values and eigenvalues.

Theorem 5.1.14. If λ1, . . . , λn ∈ σP (A) are distinct eigenvalues of the op-erator Aλ and xi ∈ Ker(Aλi) are corresponding eigenfunctions we have that theset

x1, . . . , xn

is linearly independent.

Proposition 5.1.15 (Prop. 6.7 [Bre11]). Let be A ∈ L(H), then the spectrum σ(A) of Ais compact and

σ(A) ⊂ [−‖A‖,+‖A‖] .

Proposition 5.1.16 (Prop. 6.9 [Bre11]). Let be A ∈ L(H) self-adjoint. Set then

m = inf‖u‖=1

〈Tu, u〉 , m = sup‖u‖=1

〈Tu, u〉 .

Then the spectrum σ(A) of A is compact and

σ(A) ⊂ [m,M ] .

Moreover

‖A‖ = max|m|, |M | .

5.2 On some particular classes of operators 171

5.2 On some particular classes of operators

Compact operators play an important role in mathematical physics for many reasons. Firstof many problems, in particular boundary value problems, are handled by reformulatingthem in terms of integral equations. A well known example is the so called Dirichlet problem.

Example 5.2.1. where one has to nd, give a certain domain Ω ⊂ R3, a function u ∈ C2(Ω)that satises

∆u(x) = 0, x ∈ Ω,

u(x) = f(x), x ∈ ∂Ω.

The main idea is to write the solution as u(x) = (TKϕ)(x) =∫∂ΩK(x, y)ϕ(y)dS(y), for

a given kernel K, a continuous function ϕ on ∂Ω and dS is the surface measure. We arenot entering the problem in details, we refer to [RS79],[Sec. VI.5] for details on Dirichletproblem.

Furthermore there is a particular subclass of compact operators, the so called trace classoperators (or nuclear) of paramount importance in quantum physics in the denition ofquantum state. We are mainly interest in this particular class of operators in order to beable to dene the Gaussian measure in an innite dimensional setting.

Among properties that make compact operators useful, that emerges also from Example5.2.1 is that the image of a bounded set under the operator K is a precompact set , i.e. aset whose closure is compact. This property is the one we are interested in and the presentsection is devoted to studying such operators. Furthermore a notable theory, closely relatedto previous integral equation, where compact operators are deeply used is the so-calledFredholm alternative, see, e.g. [Bre11, RS79, Mor13] for a brief introduction on Fredholmintegral theory. The main references of this section are [Bre11, RS79, Mor13, DPZ02], otherreferences for a deeper development of correlated topics will be [DPZ08, DP06, DP07, EN00,KJF77]. The present section is so structured, Sec. 5.2.1 is devoted to the introduction ofcompact operators with particular attention to spectral properties of such operators, Sec.?? is devoted to denitions and basic results on trace class operators and Hilbert-Schmidtoperators.

5.2.1 Compact operators

We will rst state results for the general case of A ∈ L(X,Y ), with X and Y Banach spaces,then we will limit ourselves to the particular case of A ∈ L(H), with H a Hilbert space.Even if not stated H will always be assumed to be a innite dimensional separable Hilbertspace with inner product 〈·, ·〉.

Let us rst state what we precisely mean by compact operator between two Banachspaces.

Denition 5.2.1 (Compact operator). Let be X and Y two Banach space. An operatorA ∈ L(X,Y ) is called compact (or completely continuous) if A takes bounded sets of X intoprecompact sets of Y . Equivalently, A is compact if and only of for every bounded sequencexk ⊂ X, Axk has a subsequence convergent in Y .

The integral operator TK dened in Example 5.2.1 is a rst easy example of compactoperator. Another classical example of compact operators that are useful are the nite rankoperators to be introduced later on.

172 5. Appendix

Following the notation introduced in [Bre11], the set of all compact operator A from Xto Y will be denoted by K(X,Y ). For short a compact operator A from a Banach space Xto itself will be denoted by K(X).

Theorem 5.2.2 (Th. 6.1 [Bre11]). The set K(X,Y ) is a closed linear subspace of L(X,Y ),w.r.t. the topology associated to the operator norm.

A rst important property of compact operators is the following.

Theorem 5.2.3 (Th. VI.11 [RS79]). A compact operator A ∈ L(X) maps weakly convergentsequences into norm convergent sequences. Furthermore if X is reexive, the converse holds,i.e. if A maps weakly convergent sequences into norm convergent sequences, then A is acompact operator.

We have since now given the denition of compact operator, but such a denition doesnot look easy to verify. The next result is in such direction, giving some easier condition ifone has to prove a certain operator to be compact.

Theorem 5.2.4 (Th. VI.12 [RS79]). Let be X, Y and Z three Banach spaces and letA ∈ L(X,Y ) and B ∈ L(Y, Z). Then:

(i) if Ak is a sequence of compact operators such that Ak → A in the norm topology, thenA is compact;

(ii) A is compact i A∗ is compact;

(iii) if either of A or B is compact, then AB is compact.

We can actually say something more on approximating sequence of compact operatorsintroducing an important class of compact operators, called operator of nite rank.

Denition 5.2.5 (Finite rank operator). An operator A ∈ L(X,Y ) is said of nite rank ifthe range of A, denoted by RgA, if nite dimensional.

Having a nite dimensional range means that, given A ∈ L(X,Y ), every vector in therange of A can be written as a nite dimensional sum, namely

Ax =

N∑k=1

αkyk, x ∈ X, ykk ⊂ Y, αk ∈ R .

Then we have the following result.

Corollary 5.2.6 (Cor. 6.2 [Bre11]). Let A ∈ L(X,Y ) and let Akn, Ak ∈ L(X,Y ) ∀k, asequence of nite rank operators such that Ak → A in the norm topology, then A is compact.

We close the section with an elementary proposition for an operator A from an Hilbertspace to itself that will be useful when introducing trace class operators.

Proposition 5.2.7 (Prop.4.14 [Mor13]). Let be A ∈ L(H), then A is compact i |A| :=√A∗A is compact.


Spectral theory for compact operators

We have briey mentioned that compact operators enjoy many nice properties particularlysuitable in applications. What makes compact operator so widely studied and used are themany results we have for spectra of such operators, in particular it happens that self-adjointcompact operators are the natural extension to the innite dimensional case of the nitedimensional Hermitian matrices.

Since we are mainly interest in the particular case of operator from a Hilbert space toitself we will from now on avoid to state the theory in its whole generality focusing only onA ∈ L(H). We recall rst of all a fundamental result concerning spectral properties of aself-adjoint operator.

Theorem 5.2.8 (Th.VI.8 [RS79]). Let be A a self-adjoint operator on a Hilbert space H.Then

(i) σ(A) = σP (A) ⊂ R;

(ii) eigenvectors corresponding to distinct eigenvalues of A are orthogonal.

The following result is a fundamental result on compact operators which implies manyother nice properties.

Theorem 5.2.9 (Th. VI.14 [RS79]). [Analytic Fredholm result] Let be Ω ⊂ C open andconnected. Let be f : Ω → L(H) be an analytic operator-valued function such that f(z) iscompact for each z ∈ Ω. Then either

(i) (I − f(z))−1 exists for no z ∈ Ω;

(ii) (I − f(z))−1 exists for all z ∈ Ω \S where S is a discreet subset of Ω and if z ∈ S, thenf(z)ψ = ψ has a nonzero solution in H.

Proof. See,e.g. Th. VI.14 [RS79].

Th. 5.2.9 has many important results concerning existence and solution of given opera-tors. We will not deal with the so called Fredholm theory, there are many good introductorybook on this theory, we refer among others to [Bre11, Mor13, RS79] and references therein.We will now just state some more or less straightforward corollary of the previous theoremconcerning mainly spectral properties of compact operators.

The Fredholm alternative states that if A is compact, then either Aψ = ψ has a solutionor (I−A)−1 exists. This properties is particularly important when trying to solve equations.In fact it tells us that if for any φ there is at most one ψ with ψ = φ+Aψ, then there is alwaysexactly one of such ψ. Or in a better way, compactness and uniqueness imply existence. Anexample of this principle arise i the Dirichlet problem stated before.

Corollary 5.2.10 (Cor. Sec. VI [RS79]). [The Fredholm alternative] If A is a compactoperator on H, then either (I −A)−1 exists or Aψ = ψ has a solution.

Proof. From Th. 5.2.9 with f(z) = zA and z = 1.

The next result is a rst (of many) result that generalize a property typical of linearoperators in nite dimension to the innite dimensional scenario.

174 5. Appendix

Theorem 5.2.11 (Th. 6.6 [Bre11]). Let be A a compact operator A ∈ K(H). Then thefollowing holds:

(i) Ker(I −A) is nite dimensional;

(ii) Rg(I −A) is closed and Rg(I −A) = Ker(I −A∗)⊥;

(iii) Ker(I −A) = 0 ⇔ Ker(I −A) = H;

(iv) dimKer(I −A) = dimKer(I −A∗).

It is well know that in the nite dimensional case, a linear operator A is injective if andonly if it is surjective. This property does not immediately generalize to the case H to bean innite dimensional space. A standard counterexample is the translation in l2. Thereforepoint (iii) tells us that in the case of A being both linear and compact A injective impliesA surjective and viceversa.

We are now nally to state some important results that characterize the spectrum of acompact operator. We refer to Sec. 5.1 for details and denitions.

Lemma 5.2.12 (Lem. 6.2 [Bre11]). Let A be a compact operator on H and let λkk ⊂σ(A) \ 0, be a sequence of distinct real numbers so that

λk → λ ,

then λ = 0.

Lem. 5.2.12 tells us that any λ ∈ σ(A) is an isolated point on the real line, and that theonly limit point, if it exists, is λ = 0.

In general, given an operator A ∈ L(H), we have that σP (A) ⊂ σ(A), with the inclusionbeing strict. A trivial example is again the translation in l2. Let us for instance considerl2 3 u = (u1, u2, . . . ), A : l2 → l2, u 7→ u = (0, u1, u2, . . . ). Then σP (A) = 0 butσ(A) 6= 0. The following result in another property typical of the nite dimensional casethat does not hold in general for the innite dimensional case.

Theorem 5.2.13 (Th. 6.8 [Bre11] ). Let A be a compact operator on H, then the spectrumof A σ(A) \ 0 ≡ σP (A) \ 0 and 0 ∈ σ(A). Furthermore either σ(A) = 0, σ(A) \ 0is a nite set or σ(A) \ 0 is a sequence converging to λ = 0.

Proof. We will follow the proof in [Bre11]. By contradiction we assume 0 6∈ σ(A). Then Ais bijective and the identity operator I = AA−1 is compact. Then by a well known resultsof Riesz we know that the closed ball BH is compact i dimH <∞, but this contradict ourhypothesis so that 0 ∈ σ(A).

Again by contradiction we prove that any element λ ∈ σ(A) is an eigenvalue of A. In factif λ 6∈ σP (A), then Ker(A−λI) = 0 and by Th. 5.2.11(iii) we have that Rg(A−λI) = Hand therefore λ ∈ ρ(A), so that we have reached a contradiction having assumed λ ∈ σ(A).

Eventually exploiting Lem. 5.2.12, for any k ≥ 1 we have that

σ(A) ∪ λ ∈ R : |λ| ≥ 1

k ,


is either empty or nite. Hence if σ(A) \ 0 has innitely many distinct points we mayorder them as a sequence converging to λ = 0.

Lem. 5.2.12 and Th. 5.2.11 tell us that the point spectrum is a discreet set having nolimit points excepts perhaps λ = 0. Furthermore any 0 6= λ ∈ σ(A) has nite multiplicity.

Before stating the main result of the section, the spectral decomposition, let us state aresult that in a certain sense summarize the many properties stated till now.

Theorem 5.2.14 (Th. 4.17 [Mor13]). Let be A a self-adjoint compact operator on theHilbert space H. Then the following hold:

(i) every eigenspace of A with associated eigenvalue λ 6= 0 has nite dimension;

(ii) the point spectrum σP (A) is a non empty subset at most countable of the real line.Furthermore it has at most one limit point, and, if it exists, it has to be λ = 0.Eventually is satises the estimates

‖A‖ = sup|λ| : λ ∈ σP (A) .

Remark 5.2.15. The operator A self-adjoint is the null operator i λ = 0 is the only eigen-value.

We are now to state a celebrated result by Hilbert on the expansion of self-adjointoperators in term of a (suitable) basis made of eigenvalues. The following theorem impliesthat every self-adjoint compact operator may be diagonalized w.r.t. some suitable basis ofH.

Theorem 5.2.16 (Th. 4.18 [Mor13], Th. 6.11 [Bre11]). , Th. VI.16 [RS79][Spectraltheorem] Let be A a self-adjoint compact operator on the Hilbert space H. Then it existsa hilbertian basis composed of eigenvalues of A, i.e. there is a complete orthonormal basisekk for H so that Aek = λkek and λk → 0 as n→∞.

Th. 5.2.16 says that the operator A can be diagonalized, where the element on thediagonal are the eigenvalues λkk so that the action of A on an element x ∈ H reads asfollows

Ax =∑k∈N

λk 〈ek, x〉 .

In this sense we see that we have generalized Hermitian matrices to the innite dimensionalcase.

Theorem 5.2.17 (Th. VI.17 [RS79], Th. 4.21 [Mor13]). [Canonical form for compactoperators] Let A be compact operator on H, then there exist orthonormal sets enNn=1 andenNn=1 and positive real numbers λkNn=1 with λk → 0 so that

A· =N∑k=1

λk 〈ek, ·〉 ek . (5.2)

The numbers λk are called singular values of A.

Previous theorem, Th. 5.2.17 allows us to properly introduce in the next section twoparticular subclasses of compact operators.

176 5. Appendix

5.2.2 Trace class and Hilbert-Schmidt operators

Through Sec. 5.2.1 we have said many times that compact operators have many nice proper-ties useful in application. We are anyhow interested in two particular subclasses of compactoperators, namely, trace class operators (or nuclear) and Hilbert-Schmidt operators. Thepresent section we will briey introduce and state some basic results concerning these twoclasses of operators. We will rst dene trace class operators, that are operators for whichthe trace, in a sense to be specied later on, can be dened. We will further introduce theconcept of trace, which is generalization to innite dimension of the standard trace of amatrix. Eventually we will introduce Hilbert-Schmidt operators, that in a certain sense aretrace class operators of order two. Denitions are given according to [DPZ02] and othersbook of the same author, the theory is developed following [RS79, Mor13].

Denition 5.2.18 (Trace class operator). Let be A ∈ L(H), then A is called of trace classif there exists two sequences ak and bk ⊂ H such that

Ay =

∞∑k=1

〈y, ak〉 bk, y ∈ H , (5.3)

and∞∑k=1

|ak||bk| <∞ . (5.4)

The family of all operators of trace class is denoted by L1(H).

Let us notice that if eq. (5.4) holds, then the series in eq. (5.3) converges in norm.Moreover can be shown that A is compact. It now clear how Th. 5.2.17 is fundamental inthe denition of trace class operators.

Theorem 5.2.19. The set of trace class operators L1(H) equipped with the norm

‖A‖L1(H) := inf

∞∑k=1

|ak||bk| : Ay =

∞∑k=1

〈y, ak〉 bk, y ∈ H, ak, bk ⊂ H

is a Banach space.

Denition 5.2.20 (Trace). Let H be a separable Hilbert space and let ek be an orthonor-mal basis. Then for any positive operators A ∈ L1(H) we dene

TrA :=∞∑k=1

〈Aek, ek〉 . (5.5)

The number TrA is called trace of A, it converges absolutely and it is independent of theorthonormal basis chosen.

In particular the trace dened in Def. 5.2.20 has all the properties of the trace in thenite dimensional case such has the linearity.

Let us notice that if A is given by eq. (5.3), we have that

TrA =

∞∑k=1

〈ak, bk〉 .


Theorem 5.2.21 (Th. VI. 19 [RS79], Prop. 1.11 [DPZ02]). L1(H) is a ∗−ideal, that is

(i) L1(H) is a vector space;

(ii) if A ∈ L1(H) and T ∈ L(H), then AT ∈ L1(H) and TA ∈ L1(H). Furthermore wehave

‖AT‖L1(H) ≤ ‖A‖L1(H)‖T‖, ‖TA‖L1(H) ≤ ‖A‖L1(H)‖T‖ ,

and TrAT = TrTA;

(iii) if A ∈ L1(H), then A∗ ∈ L1(H). Furthermore we have ‖A‖L1(H) = ‖A∗‖L1(H).

The following is a theorem that connects trace class operators and compact operators.We have already stressed that a trace class operator is also a compact operator. Furthermoreit emphasize why compact operators are so important, that is the sum of the singular valuesof a compact operator A is nite. This property is necessary when we are to dene aGaussian measure in innite dimension.

Theorem 5.2.22 (Th. VI.21 [RS79]). Let A be a self-adjoint compact operator. ThenA ∈ L1(H) if and only of

∑∞k=1 λk < ∞, where λk are the singular values of A (counted

with their multiplicity).

Proof. The proof easily follows from the canonical form for compact operators Th. 5.2.17.

previous theorem, Th. 5.2.22 comes from Th. 5.2.17. Its importance lies in the factthat it underlines why we are studying trace class operators rather than compact operators.If fact for compact operators the sum in eq. (5.2) in Th. 5.2.17 may be nite of innite,requiring the operator A ∈ L1(H) means the that sum (5.2) to converge.

Corollary 5.2.23 (Cor. Sec. VI [RS79]). The nite rank operators are dense in the traceclass operators.

Eventually we state an important proposition that highlight a fundamental propertyof compact operators. From Th. 5.2.13 we can deduce that the singular values of A areexactly the eigenvalues of |A| so that we can give the following characterization of trace classoperators.

Proposition 5.2.24. An operator A ∈ L1(H) if and only if Tr|A| < ∞, and ‖ · ‖L1(H) =Tr|A|.

Proof. The proof follows from Prop. 5.2.7 and Th. 5.2.13.

Therefore we can see that a trace class operator A, being compact, can be diagonalizedwith the element on the diagonal to be its eigenvalues. If furthermore the sum of all of itseigenvalues converges absolutely, then A admits trace, in the sense stated above and thesum of all of its eigenvalues counted together with their multiplicity. Let us notice that wehave stated previously that any eigenvalue of a compact operators has nite multiplicity sothat previous statement makes sense.

178 5. Appendix

Trace class operators are not enough for our purposes, we need in fact to restrict theclass of operators under consideration a bit more. In particular we have to introduce theclass of Hilbert-Schmidt operators L1(H) ⊂ L2(H) ⊂ K(H) ⊂ L(H). Previous inclusionsexplain why K(H) is sometimes denoted by L∞(H).

Denition 5.2.25 (Hilbert-Schmidt operators). Let be S ∈ L(H). Then we will say thatA is of Hilbert-Schmidt class if there exists an orthonormal and complete basis ek in Hsuch that

∞∑k=1

| 〈Sek, ej〉 |2 <∞ .

We will thus denote the class of Hilbert-Schmidt operators by L2(H).

Theorem 5.2.26. The set of trace class operators L2(H) equipped with the scalar product

〈A,B〉L2(H) :=

∞∑k=1

〈Aek, Bek〉 ,

is a Hilbert space.

Proposition 5.2.27 (Prop. 4.27 [Mor13]). Let A ∈ L(H). If√|A| ∈ L2(H), then A ∈

L1(H).

Proposition 5.2.28 (Prop. 1.1.2 [DPZ02]). Let A,B ∈ L2(H). Then AB ∈ L1(H) and

‖AB‖L1(H) ≤ ‖A‖L2(H)‖B‖L2(H) .

In particular Hilbert-Schmidt operators has the property that TrA∗A <∞.

Proposition 5.2.29 (Prop. 1.1.3 [DPZ02]). Let A be a compact self-adjoint operator, letfurther λkk its eigenvalues (counted with theri multiplicity). Then A ∈ L2(H) if and onlyof∑∞k=1 |λk|2 <∞. Furthermore

‖A‖L2(H) =

( ∞∑k=1

|λk|2) 1

2

= (Tr|(A∗A)|)12

5.3 Introduction on semigroup theory

Evolution equations govern the dynamic behavior of deterministic systems. Only few stan-dard assumptions are needed in order to be able to mathematically model a dynamic system.

Let us rst of all (X, ‖ · ‖X) be a Banach space. As a rst assumptions of course we needuniqueness, i.e. we require that at any time t ≥ 0 the state of a system can be uniquelydescribed by an element of X. Given an initial state x ∈ X, we will further denote the stateof the system at time t > 0 by u(t, x) ∈ X if the initial datum at time t = 0 is u(0) = x. Asa second assumption we require that the change of the state at time t uniquely depends onthe present state u(t), and that the law which links the present state to the present changeis given by a map A : D(A) ⊂ X → X.

Under these assumptions, the evolution of the system is determined by the abstractCauchy problem (equivalently initial value problem)

5.3 Introduction on semigroup theory 179

Denition 5.3.1 (Abstract Cauchy problem). Let (X, ‖ · ‖X) a Banach space. We willdenote abstract Cauchy problem the following

u(t) = Au(t), t ≥ 0 ,

u(0) = x .(ACP)

where A is a linear operator on X.

Of course it is not know a priori what the Banach space should be and furthermore thechoice of the state space X will the properties of the operator A. We will assume anywaythat A is linear (a general although less powerful theory for non linear operator does existas well).

We would like now to explain the central role that semigroup theory plays for evolutionequations of the form (ACP)

Denition 5.3.2 (Dynamical system). Let (X, ‖ · ‖X) a Banach space. A dynamical systemis a family of mappings (T (t))t≥0, T (t) : X → X such that

T (t+ s) = T (t)T (s), ∀ t, s ≥ 0 ,

T (0) = 1 .(DS)

Here T (t) is seen as the map describing the change of a state x ∈ X at time 0 into thestate T (t)x ∈ X at time t. Therefore we have that each T (t) is a linear operator from Xinto itself and (T (t))t≥0 is called one-parameter semigroup.

Remark 5.3.3. If t ∈ R we will talk about one-parameter group.

We try now to give a concrete idea of how the semigroup (FE) arise naturally from theCaychy problem (ACP). In general in fact we have that from a solution of (ACP) we canfound a solution of (DS). Thus (ACP) ⇒ (DS).

Let us assume rst that for each x ∈ D(A)dense⊂ X we have the existence of a unique

solution u(·) = u(·, x) of problem (ACP), thanks to uniqueness we can dene the map

T (t) : D(A)→ D(A) T (t)x := u(t, x), t ≥ 0 .

Let us now further assume that u(t, x) depends continuously (w.r.t. the norm of X) on theinitial value x. By a standard density argument one can then extend T (t) to a boundedlinear operator on X such that the orbit map

ξx : R+ → X, t 7→ T (t)x

is continuous for each x ∈ X. We then denote the resulting one parameter family of boundedlinear operators on X by (T (t))t≥0.

Eventually considering two dierent times t, s ≥ 0 and taking problem (ACP) with initialdatum to be u(s) we have two solutions, namely

t 7→ u(t, u(s)), t 7→ u(t+ s, x)

which have to coincide by uniqueness. Therefore we have that the semigroup property

T (t+ s) = T (t)T (s)

180 5. Appendix

is satised. Therefore if for each initial value x ∈ X a unique solution u(·, x) does exist,then

T (t)x := u(t, x), t ≥ 0, x ∈ X

denes an operator semigroup.

We are now interested in the opposite question, i.e. given an operator semigroup(T (t))t≥0 which solves (DS), under what conditions can it be described by a problem ofthe form (ACP) and can the operator A be found? Loosely speaking, does (DS) ⇒ (ACP)?

We will see that in some trivial, see, e.g. section 5.3.1 the relation between (ACP) and(DS) is given by

T (t) = etA, A =d

dtT (t)

∣∣∣∣t=0

(5.6)

In general relation (5.6) is not satised. If we require a further condition, i.e. the semi-group to be strongly continuous then we have the so called C0−semigroup. An astonishinglyrich and elegant theory on C0 − semigroups has been developed over the years for linearevolution equations. Its foundation is the theorem of Hille-Yosida. This result character-izes those operators A which generate a strongly continuous semigroup that solves (ACP)for A. We will also show that the linear problem (ACP) admits a unique solution whichcontinuously depends on the initial datum, i.e. we will give a positive answer to previousquestion whether (DS) ⇒ (ACP) for a densely dened linear operator A if and only if Agenerates a C0 − semigroup, which then solves (ACP). In this sense, semigroup theoryprovides the natural framework for linear evolution equations and the Hille-Yosida theoremcloses the circle between the given linear operator A, the evolution equation (ACP) and thecorresponding solution semigroup (T (t))t≥0.

Eventually we give an example of problem (ACP) and the corresponding semigroup maybe linked.

Example 5.3.1 (The N × 1−dimensional heat equation). Let be Ω ⊂ RN a domain withboundary given by ∂Ω. On such domain let us consider the N−dimensional (in space) heatequation

∂u(t,x)∂t = ∆u(t, x), t ∈ [0, T ], x ∈ Ω,

u(t, x) = 0, t ∈ [0, T ], x ∈ ∂Ω,

u(0, x) = u0, x ∈ Ω

(5.7)

with ∆ :=∑Ni=1

∂2

∂x2ithe Laplacian operator.

For an initial value u0 ∈ Lp, 1 ≤ p <∞ the heat equation can be rewritten in the formof (ACP) by taking X = Lp and setting

Af := ∆f, f ∈ D(A),

D(A) = W 2,p(Ω) ∩W 1,p0 (Ω) = f ∈W 2,p(Ω) : f |∂Ω = 0 .

where we denoted byW a Sobolev space and byW0 the closure of the test functions D w.r.t.the Sobolev norm.

The main idea is now instead of looking for a standard solution

u : [0, T ]× Ω→ R


of the heat equation (5.7), we look for a solution

u : [0, T ]→ Lp(Ω)

of problem (ACP) (of course rewritten such that it corresponds to the N.dimensional heatequation). Since we do not have enough tool at the moment to deal with the innitedimensional case of Lp we restrict aourselves to nite dimensional case of X = RN so thatA : D(A) → D(A) is a N × N matrix. In the present case a unique solution of (ACP) isgiven by

u(t) = xetA, t ∈ [0, T ] .

The matrices etA may be thought of as solution operators which maps the initial datumx ∈ RN to the solution xetA at any time t. The semigroup property (DE) is further easilysatised and the map t 7→ etA is continuous.

5.3.1 Strongly continuous semigroups

Denition 5.3.4 (Strongly continuous semigroup). A family (T (t))t≥0 of bounded linearoperators on a Banach space (X, ‖ · ‖X) is called strongly continuous (one-parameter) semi-group (in short C0semigroup) if it satises the functional equation

T (t+ s) = T (t)T (s), ∀ t, s ≥ 0 ,

T (0) = 1 .(FE)

and is strongly continuous in the sense that for every x ∈ X the orbit maps

ξx : t 7→ ξx := T (t)x (SC)

are continuous from R+ into X for every x ∈ X.

Heuristically speaking ξx := T (t)x : t 7→ T (t)x gives us the trajectory along which thestarting point x ∈ X evolves.

Remark 5.3.5. For brevity reason we will always denote the Banach space (X, ‖ · ‖X) justby X. Further in order to avoid confusion we will denote by |·| := ‖ · ‖X and simply by ‖ · ‖the norm on the operator.

We start by given some introductory result that we will need in order to proper developthe theory of strongly continuous semigroups.

Proposition 5.3.6. Property (SC) holds i the map t 7→ T (t) is continuous from R+ intothe space Ls(X) of all bounded linear operators on X endowed with the strong operatortopology.

Thus we have that T (t) ∈ L(X) and that we are looking for a pointwise convergence ofthe operator on the Banach space i.e. we require

limt→s|T (t)x− T (s)x| = 0, ∀x ∈ X . (5.8)

Conditions (5.8) and (SC) are quite weak conditions for a convergence but anyway theyare not so easy to verify. We therefore try to facilitate the verication of the strong continuity.In order to this we introduce the following lemma, which follows from Th. ??.

182 5. Appendix

Lemma 5.3.7. Let (X, ‖ · ‖X) be a Banach space and let F : K → L(X) with K ⊂⊂ R+.Then the following are equivalent:

(i) F is continuous in the strong operator topology i.e. eq. (5.8) holds;

(ii) F is uniformly bounded on K and the mapping K 3 t 7→ F (t)x ∈ X are continuous for

all x ∈ X in Ddense⊂ X;

(iii) F is continuous for the topology of uniform convergence on compact subset of X i.e.the map

K × C 3 (t, x) 7→ F (t)x ∈ X

is uniformly continuous for every C ⊂⊂ X.

Proof. See, Lemma 1.2 [EN06].

From lemma 5.3.7, together with equation (FE), we can see taht the continuity of theorbit maps (SC) at each time t ≥ 0 and for each x ∈ X is implied by much weaker conditions.

Proposition 5.3.8. For a semigroup (T (t))t≥0 on a Banach space X the following areequivalent:

(i) (T (t))t≥0 is a C0-semigroup;

(ii) limt↓0 T (t)x = x for all x ∈ X ;

(iii) ∃δ > 0, M ≥ 1, Ddense⊂ X such that:

(iiia) ‖T (t)‖ ≤M for all t ∈ [0, δ];

(iiib) limt↓0 T (t)x = x for all x ∈ D.

Proof. See e.g. Proposition 1.3 [EN06]

Example 5.3.2 (The left translation semigroup). We now give an example of how importantcould be proposition 5.3.8. In fact it often happen that the uniform boundedness of theoperator T (t) for t ∈ [t, δ] is easily satised. It is therefore enough to check right continuityof the orbit at t = 0 on some nice dense subset of X.

Let be Lp(R) and let us consider the left translation group (Tl(t))t∈R on Lp(R) denedas

(Tl(t)f) (s) := f(s+ t), s ∈ R, f ∈ Lp . (5.9)

We claim that (Tl(t))t∈R is strongly continuous. It is easy to show that T (0) = I and thatT (t) is a linear isometry on X, in fact∫

R|f(s)|pds =

∫R|f(s+ t)|pds r=s+t=

∫R|f(r)|pdr .


We have therefore that ‖T (t)‖ = 1. Further we show that

T (t)T (s) = T (s)f(·+ t) = f(·+ t+ s) = T (s+ t)f ,

so that the semigroup property holds. Exploiting proposition 5.3.8 if we show that limt↓0 T (t)x =x holds on since dense subset D ⊂ X we are done.

Then we consider f ∈ Cc(R) a continuous function with compact support and we observethat

limt↓0‖T (t)f − f‖∞ = lim

t↓0sups∈R|f(t+ s)− f(s)| = 0 .

Since we have that ¯Cc(R) = Lp(R) we have thus shown that the left translation is a stronglycontinuous semigroup on Lp. The same holds for X = C0

For a strongly continuous semigroup the nite orbits are continuous images of a compactinterval, hence compact and thus bounded for each x ∈ X. So by Th. ?? each C0-semigroupis uniformly bounded on each compact interval. Therefore we have the following exponentialboundedness.

Proposition 5.3.9. For every C0 − semigroup (T (t))t≥0 ∃ w ∈ R and M ≥ 1 s.t.

‖T (t)‖ ≤Mewt, ∀t ≥ 0 (5.10)

Proof. Since we are considering a C0 − semigroup, ∃ M s.t. ‖T (s)‖ ≤ M ∀s ∈ [0, 1].Therefore let us consider such an M and let us write t = s + n for a t ≥ 0, n ∈ N ands ∈ [0, 1]. Thus

‖T (t)‖ ≤ ‖T (s)‖‖T (1)‖n ≤Mn+1 = Men logM ≤Mewt

holds for w := logM at each t ≥ 0.

Remark 5.3.10. From previous proposition can be seen why we need M ≥ 1. In fact sincethe upper bound must hold for any t ≥ 0 we have that in t = 0 we have

1 = ‖T (0)‖ ≤Mew0 = M .

The inmum of all w plays an important role.

Denition 5.3.11 (Growth bound). For a C0 − semigroup (T (t))t≥0 we dene

w0 := infw ∈ R : ∃Mw ≥ 1 : ‖T (t)‖ ≤Mwewt ∀t ≥ 0

its growth bound. Moreover a semigroup is called bounded if we can take w = 0 and con-tractive if w = 0 and M = 1 are feasible. Eventually a semigroup is called isometric if‖T (t)x‖ = |x| for all t ≥ 0 and x ∈ X.

Remark 5.3.12. (i) w0 < +∞;

(ii) it could be possible that w0 = −∞;

184 5. Appendix

(iii) it is not always M = 1;

(iv) the inmum may not be attained.

Eventually we show that nothing change if instead of the strong operator topology indenition 5.3.4 we use the use the weak operator topology we do not change the class of thesemigroup.

Theorem 5.3.13. A semigroup (T (t))t≥0 on a Banach space (X, ‖ · ‖X) is C0 i it is weakcontinuous i.e. if the mapping

R+ 3 t 7→ X′ 〈x′, T (t)x〉X

are continuous for each x ∈ X and x′ ∈ X ′.

Standard construction

We devote this section to the construction of new C0 − semigroups from a given one.

Similar semigroup Let (T (t))t≥0 be a C0 − semigroup on X Banach space and let V :X → Y an isometry with Y another Banach space.

Thus

S(t) := V −1T (t)V, t ≥ 0

is a C0 − semigroup on Y . Furthermore we have that w0(T ) = w0(S).

Rescaled semigroup Again let (T (t))t≥0 be a C0− semigroup on X. Then for any µ ∈ Rand α > 0 the semigroup

S(t) := eµtT (αt), t ≥ 0

is a C0 − semigroup on X. Furthermore we have that w0(S) = µ+ w0(T ).

Product semigroup Let (T (t))t≥0 and (S(t))t≥0 two C0 − semigroups on X s.t. theycommute, i.e. S(t)T (t) = T (t)S(t) for all t ≥ 0. Then

U(t) := S(t)T (t)

is a C0 − semigroup.

Adjoint semigroup In general if (T (t))t≥0 is a C0 − semigroup on a Bancah space X itdoes not imply that the adjoint semigroup T (t)′ is a C0 − semigroup as well. It willbe of course weak∗ continuous in the sense that the maps

t 7→ X′ 〈T (t)′x′, x〉X = X′ 〈x′, T (t)x〉X

are continuous for allx ∈ X and x′ ∈ X ′. Then if X is reexive we know that weakand weak∗ convergence coincide and thus from Th. 5.3.13 we have that it is stronglycontinuous as well. We give two examples.


1. Let X = L1(R) endowed with the standard ‖ · ‖L1 . Let us then consider the righttranslation semigroup

(Tr(t)f) (s) := f(s− t)

The (T (t))t≥0 is a C0 − semigroup. Since L1 is not reexive we have that thedual

T (t)′ = (Tl(t)f) = f(s+ t)

is not a C0 − semigroup on L∞.

2. Conversely if we consider a reexive space such as Lp with 1 < p < ∞ endowedwith the norm ‖ · ‖Lp we have that T ∈ C0 ⇔ T ′ ∈ C0

Finite dimensional semigroups

For the whole section X = Cn and L(X) = Mn(C) the complex n × n matrices. Since weare nite dimensional space we recall that strong and weak topology coincide. Therefore wewill simply talk about continuity of the semigroup.

Proposition 5.3.14. For any A ∈Mn(C) and t ≥ 0 the series

etA :=∑n≥0

tnAn

n!

converges absolutely. Moreover the mapping

R+ 3 t 7→ etA ∈Mn(C)

is continuous and it satises e(t+s)A = etAesA, s, t ≥ 0 ,

e0A = 1 .

Proof. The only real problem is only to show the continuity of the map t 7→ etA. We rstof all observe that

e(t+h)A − etA = etA(ehA − I)

Thus if we can show that limh→0 ehA = I we are done. In order to do this we estimate

‖ehA − I‖ = ‖∑n≥1

hnAn

n!‖ ≤

∑n≥1

|h|n‖A‖n

n!= e|n|‖A‖ − 1→ 0 as h→ 0

Denition 5.3.15. We call(etA)t≥0

the one-parameter semigroup generated by the matrix

A ∈Mn(C).

Actually we can extend the previous semigroup to the whole R dening the one-parametergroup generated by the matrix A.

We now to state some examples.

186 5. Appendix

Example 5.3.3. 1. Let be A = diag(λ1, . . . , λn) be a diagonal matrix. Therefore wehave the semigroup to be

T (t) = etA = diag(etλ1 , . . . , etλn) ;

2. Let be A = λI +N with N an upper triangular matrix of all 1. Therefore we have

A =

λ 1 . . . 10 λ 1 1...

.... . .

...0 0 0 λ

Then we can prove that, being n× n, the n− th power is 0. Therefore we get

etA = eλtIn∑i=0

tiN i

i!;

Let be S ∈Mn(C) be an invertible matrix and let A ∈Mn(C). Let us then dene

B := SAS−1

Therefore we have

etB =∑n≥0

tn

n!(SAS−1)n

At the beginning of the section we ask whether (DS) ⇒ (ACP), namely if a given semi-group can describe a dierential equation. We will now show rst of all that in the case ofT (t) = etA we even have dierentiability of the map t 7→ T (t) and that U(t) := etA solvesthe dierential equation

ddtU(t) = AU(t), t ≥ 0 ,

U(0) = I .(DE)

Next we will show that a general continuous operator semigroup is even dierentiable int = 0.

Theorem 5.3.16. Let T (t) := etA for some A ∈ Mn(C). Then the function T (·) : R+ →Mn(C) is dierentiable and it satises the dierential equation (DE). COnversely, everydierentiable function T (·) : R+ →Mn(C) such that is satises (DE) is already of the formT (t) = etA for A := T (0) ∈Mn(C).

Proof. From (FE) implies

T (t+ h)− T (t)

h=T (h)− I

hT (t)

for all t, h ∈ R. Then from

‖T (h)− Ih

−A‖ ≤ sum∞n=2

|h|n−1‖A‖n

n!=e|h|‖A‖ − 1

|h|− ‖A‖ → 0 as h→ 0


Theorem 5.3.17. Let T (·) : R+ →MN (C) be a continuous function satisfying (FE). Thenit exists A ∈Mn(C) s.t.

T (t) = etA, t ≥ 0

Proof. From continuity of T and the fact that T (0) = I is invertible we have that

V (t0) :=

∫ t0

0

T (s)ds

are invertible for t0 suciently small. Then from (FE) follows that

T (t) = V (t0)−1V (t0)T (t) = V (t0)−1

∫ t0

0

T (t+s)ds = V (t0)−1

∫ t+t0

t

T (s)ds = V (t0)−1 (V (t+ t0)− V (t))

Hence T is dierentiable with derivative

d

dtT (t) = T (0)T (t)

Thus T satises (DE) with A = T (0).

Remark 5.3.18. If now X is an innite dimensional Banach space we cannot extend previousresults a priori. In order to be able to do this we have to assume the map t 7→ T (t) ∈ L(X)to be continuous in the operator norm. If it is so then it is enough to replace the matrixA ∈Mn(C) by a bounded operator A ∈ L(X) and arguing as done previously we can extendtheorems 5.3.16 and 5.3.17.

5.3.2 Semigroups, generators and resolvents

We have seen that if X is a nite dimensional Banach space or if X is an innite dimensionalBanach space and the orbit ξx : R+ → L(X) t 7→ T (t)x =: ξx is continuous in the operatornorm and then the semigroup is uniformly continuous and we have a complete characteri-zation of the semigroup. Then we can dene the semigroup as T (t) = etA with A ∈ L(X)and we have everything we need. We are now interested in the case we are not in any of theprevious cases. In particular we will consider an innite dimensional Banach space X anda linear unbounded operator A and we will study strongly semigroups.

In general in order to uniquely characterize a semigroup we need three objects, namelythe semigroup T (t), its generator A and a third new one called the resolvent.

Denition 5.3.19 (Resolvent). Let X be a Banach space and let A be an operator on X.For any λ ∈ C we dene the operator Aλ on X by

Aλ := λ− IA

where I is the identity operator on X. If Aλ has an inverse we denote the inverse by R(λ,A)and call it the resolvent of A dened as

R(λ,A) := (λ− IA)−1 (5.11)

The resolvent R(λ,A) is dened for all complex numbers in the resolvent set ρ(A). Inthe present section we will try to connect these three object.

We recall the notation | · | for the norm of X and the notation ‖ ·‖ for the operator norm.

188 5. Appendix

(T (t))t≥0

(A,D(A)) R(λ,A)

”etA” ?

R(λ,A):=(λ−A)−1

Generators of semigroup and their resolvent

We are now in the case of strongly continuous semigroups. We look for some criteria ofdierentiability of the orbit map

ξx : t 7→ T (t)x ∈ X

The rst result is the following lemma.

Lemma 5.3.20. Let us consider a strongly continuous semigroup (T (t))t≥0 and an elementx ∈ X. For the orbit map ξx the following properties are equivalent:

(i) ξx(·) is dierentiable on R+;

(ii) ξx(·) is right dierentiable at time t = 0.

Therefore we can reduce the problem to only look at right derivatives at time t = 0instead of look for at the whole set R+. Then on the subspace of the space X for which theorbits are dierentiable, from the right derivative alone we can obtain the operator A fromwhich we can deduce the operator T (t).

Denition 5.3.21 (Innitesimal generator). The generator A : D(A) ⊂ X → X of astrongly continuous semigroup (T (t))t≥0 on a Banach space X is the operator

Ax := ξx(0) = limh↓0

1

h(T (h)x− x)

dened for every x in its domain

D(A) := x ∈ X : ξx is dierentiable in R+ = x ∈ X : limt↓0

1

h(T (h)x− x) exists

Therefore we have nd the rst arrow namely

A := limt↓0

1

t(T (t)x− x) (5.12)

The generator A and its domain D(A) should always go in pair (A,D(A)) but for brevityreason we will only denote the generator A.


(T (t))t≥0

(A,D(A)) R(λ,A)

A:=limt↓01t (T (t)x−x) ?

R(λ,A):=(λ−A)−1

Lemma 5.3.22. For a generator A of a strongly continuous semigroup (T (t))t≥0 the fol-lowing properties hold:

(i) A : D(A) ⊂ X → X is a linear and close operator with D(A)dense⊂ X;

(ii) if x ∈ D(A) then T (t)x ∈ D(A) and

d

dtT (t)x = T (t)Ax = AT (t)x, t ≥ 0 ; (5.13)

(iii) for every t ≥ 0 and x ∈ X one has∫ t

0

T (s)xds ∈ D(A) ;

(iv) for every t ≥ 0 one has

T (t)x− x = A

∫ t

0

T (s)xds, if x ∈ X ; (5.14)

Further we have

T (t)x− x = A

∫ t

0

T (s)xds =

∫ t

0

T (s)Axds, if x ∈ D(A) ; (5.15)

(v) the innitesimal generator A identies uniquely the semigroup (T (t))t≥0.

Sketch of the proof. (ii) we have that

1

h(T (t+ h)x− T (t)x) = T (t)

T (h)x− xh

h→0→ T (t)A =

Therefore

limh↓0

1

h(T (h)T (t)x− T (t)x)

exists and hence T (t)x ∈ D(A) from the denition on D(A) with AT (t)x = T (t)Ax.

(iv) equation (5.14) follows from (iii) whether equation (5.15) follows from the fact thatbeing x ∈ D(A) I can take it inside the integral and then applying (ii) the commute.

190 5. Appendix

Therefore despite being unbounded in general from lemma 5.3.22, in particular points(i) and (v) we can see that the generator introduced in denition 5.3.21 has nice properties.

Now lemma 5.3.22 together with the closed graph theorem 5.1.4 we have the followingresult.

Corollary 5.3.23. For a strongly continuous semigroup (T (t))t≥0 on a Banach space Xwith generator A, the following assertions are equivalent:

(i) the semigroup (T (t))t≥0 is uniformly continuous;

(ii) D(A) = X;

(iii) D(A) is closed in X;

(iv) the generator A is bounded , i.e. ∃ M > 0 s.t. |Ax| ≤M |x| for all x ∈ D(A).

In each of the previous cases the semigroup is given by

T (t) = etA :=∑n≥0

tnAn

n!, t ≥ 0 .

We have thus add a new arrow

(T (t))t≥0

(A,D(A)) R(λ,A)

A:=limt↓01t (T (t)x−x)

etA?

R(λ,A):=(λ−A)−1

It may happen in practice that it is hard to nd explicitly the domain of the generatorD(A). In fact often in application, for instance when dealing with certain stochastic dieren-tial equations, it is easier to identify a particular subspace of the domain D(A). Furthermoreit often happen that this particular subspace is particularly nice.

Denition 5.3.24. A subspace D of the domain D(A) of a linear operator A : D(A) ⊂X → X is called a core for A if D is dense in D(A) for the graph norm ‖x‖A := |x|+ |Ax|.

The following is a useful criterion for subspaces to be a core for the generator.

Proposition 5.3.25. Let A be the generator of a strongly continuous semigroup (T (t))t≥0

on a Banach space X. A subspace D of D(A) that is dense in the graph norm in X andinvariant under the semigroup (T (t))t≥0, namely

T (t)D ⊂ D

is always a core for A.


Some important examples of cores are given by the domains of the n−th powers An ofa generator A.

Proposition 5.3.26. Let A be a generator of a strongly continuous semigroup (T (t))t≥0.Let be

D(An) := x ∈ D(An−1 : An−1x ∈ D(A), n ∈ N

and

D(A∞) :=⋂n∈N

D(An) .

Then for all n ∈ N D(An) and D(A∞) are cores.

Proof. See, e.g. [EN06], Proposition II.1.8.

Proposition 5.3.26 is important since, let for instance A be a dierentiable operator, thenD(A∞) is the space of the function that are innitely dierentiable. Therefore we can studythe operator on D where functions are particularly nice and then extend everything to D(A)by density.

We are now to introduce some basic spectral properties for generators of strongly con-tinuous semigroups. For a deeper treatment see, e.g. [EN06] section V.1.a.

Denition 5.3.27. Let us dene

spectrum σ(A) := λ ∈ C : λ−A is not invertible ;

resolvent set ρ(A) := λ ∈ C : λ−A is invertible = C \ σ(A);

resolvent R(λ,A) := (λI −A)−1

at λ ∈ ρ(A).

Heuristically the resolvent set of a linear operator is a set of complex numbers for whichthe operator is in some sense "well-behaved". The resolvent set plays an important role inthe resolvent formalism. In general it is not an easy task to nd the spectrum σ(A) but itis enough to nd a space that contains the spectrum.

Eventually we can nd a relation between the semigroup and the resolvent of its gener-ator.

Theorem 5.3.28. Let (T (t))t≥0 be a strongly continuous semigroup on the Banach spaceX and take two constants ω ∈ R and M ≥ 1 such that

‖T (t)‖ ≤Meωt, t ≥ 0 .

For the generator A of the semigroup (T (t))t≥0 the following properties hold.

(i) if λ ∈ C s.t.

R(λ)x :=

∫ ∞0

e−λsT (s)xds

exists for all x ∈ X, then λ ∈ ρ(A) and R(λ,A) = R(λ);

192 5. Appendix

(ii) if Reλ > ω, then λ ∈ ρ(A) and the resolvent is given by the Laplace transform of thesemigroup given in (i);

(iii) ‖R(λ,A)‖ ≤ MReλ−ω for all Reλ > ω.

The formula given in (i) for the resolvent is called the integral representation of theresolvent.

Thus we have add one more arrow

(T (t))t≥0

(A,D(A)) R(λ,A)

A:=limt↓01t (T (t)x−x)

etA

R(λ,A)=∫∞0e−λsT (s)ds

R(λ,A):=(λ−A)−1

Therefore we have that the half plane that is to the right of some ω ≥ ω0 s.t. Reλ > ωcontains the resolvent ρ(A). In fact we have that ρ(A) ⊃ λ : Reλ > ω0. On the contrarydening the spectral bound

S0 := supReλ : λ ∈ σ(A)we have that

−∞ ≤ S0 ≤ ω0 <∞

Re(λ)

Im(λ)ω0 ωS0

σ(A)

ρ(A)

We thus we have uniquely characterized a strongly continuous semigroup (T (t))t≥0, itsgenerator and the resolvent. The characterization is shown in diagram 5.3.2.

Example 5.3.4 (Diusion semigroup-part I). Let us consider the Banach space X :=C ([0, 1]) and the dierential operator together with its domain

Af := f ′′,

D(A) := f ∈ C2 ([0, 1]) : f ′(0) = f ′(1) = 0


In particular this is linked to the heat equation with Neumann condition (in particular wehave no dissipation).

The domain D(A) is dense in X that is compete for the graph norm, hence A is closed,densely dened operator.

We can dene the following family of function

x 7→ en(x) :=

1 n = 0√

2 cos(πnx) n ≥ 1 .

Each of these functions belongs to D(A) and satises Aen = −π2n2en. Therefore the space

Y := Spanen : n ≥ 0dense⊂ X

by classical result on trigonometric series. Let us further consider

en ⊗ en : f 7→ 〈f, en〉en :=

(∫ 1

0

f(x)en(x)dx

)en ,

then it satises(en ⊗ en) em = δnmem . (5.16)

Then we can dene the operators

T (t) :=∑n≥0

e−π2n2ten ⊗ en

We can easily see from equation (5.16) that the semigroup (T (t))t≥0 satises the semigroupcondition (DS). Similarly the strong continuity holds on Y and by density on the whole X.

Example 5.3.5 (Diusion semigroup-part II). Let now X = Lp(RN ), 1 ≤ p < ∞. Let usdene

gt(x) :=1√

(2πt)Ne−|x|N2t

the gaussian kernel. Let us then dene

T (t)f(x) :=

∫RN

gt(x− y)f(y)dy (5.17)

Then the operator dened in equation (5.17) together with the initial condition T (0) = Iform a C0 − semigroup on Lp and its generator is

Af :=1

2∆f

where we have denoted ∆ the Laplacian operator. Furthermore we can see that the core Dis the so called Schwartz space S(RN ) of rapidly decreasing functions

S(RN ) := f ∈ C∞(RN ) : lim|x|→∞

|x|kDαf(x)→ 0, ∀k ∈ R, α ∈ NN

Therefore we have that the domain of the generator A is given by D(A) = S(RN ), closurein the graph norm of A.

194 5. Appendix

Example 5.3.6. Let us consider the space

X = f ∈ C0(R+) ∩ C1[0, 1]

endowed with the graph norm

‖f‖ := ‖f‖∞ + ‖f ′‖∞ .

Let us further consider the operator dened by

Af = f ′, D(A) = f ∈ C10 (R+) : f ′ ∈ X

Then A is closed, densely dened and its resolvent exists for Reλ > 0 can be dened as

R(λ,A)f(x) =

∫ ∞s

e−λ(y−x)f(y)dy .

Assumes now that A generates a strongly continuous semigroup (T (t))t≥0 on X. For f ∈D(A) we can dene

ξ(y) := (T (t− y)f) (x+ y), 0 ≤ y ≤ t

which is dierentiable. Then its derivatives

ξ(y) := − (T (t− y)Af) (x+ y) + (T (t− y)f ′) (x+ y) = 0

and hence(T (t)f) (x) = ξ(0) = ξ(t) = f(x+ t)

Then the semigroup (T (t))t≥0 is the left translation. But the left translation semigroup doesnot map X into itself

From example 5.3.6 we can see that we need more assumptions on A and the estimate

‖R(λ,A)‖ ≤ M

Reλ− ω

may serve to this purpose.We have since now dened the exponential function as the convergent series

etA :=∑n≥0

tnAn

n!

but it can be proved that if A is unbounded the series may not converges. So we need adierent approach. We will do now a quick overview on some of these dierent approaches.

Cauchy integral formula

etA :=1

2πi

∫∂U

eλtR(λ,A)dλ

where ∂U is a path that surround the spectrum σ(A).

Problem: the spectrum may be unbounded so we need an extra condition in order theintegral to converge.

Good feature: the Cauchy integral formula works nicely for analytic semigroups (whichis a subclass of C0 − semigroups).


Hille formula we can dene, at least in the one dimensional case, the exponential as thelimit

eta := limn→∞

(1 +

t

na

)n= limn→∞

(1− t

na

)−n= limn→∞

(t

n

(nt− a))−n

= limn→∞

(ntR(nt, a))n

;

Yosida approximants if A is an unbounded operator, let us try to dene A as the limitof some bounded operators An, called the Yosida approximants

etA := limn→∞

etAn

The rst of all give the characterization theorem for contractions semigroups.

Theorem 5.3.29 (Contraction case, Hille-Yosida). For a linear operator A on a Banachspace X, the following are equivalent:

(i) A generates a strongly continuous contraction semigroup;

(ii) A is closed, densely dened and for every λ > 0 one has λ ∈ ρ(A) and

‖λR(λ,A)‖ ≤ 1 ;

(iii) A is closed, densely dened and for every λ ∈ C with Reλ > 0 one has λ ∈ ρ(A) and

‖R(λ,A)‖ ≤ 1

Reλ;

We can generalize Th. 5.3.29 with the following.

Theorem 5.3.30 (General case, Feller-Philips-Miyadera). Let A a linear operator on aBanach space X and let ω ∈ R and M ≥ 1 be constants. Then the following are equivalent:

(i) A generates a strongly continuous semigroup (T (t))t≥0 satisfying

‖T (t)‖ ≤Meωt, t ≥ 0 ;

(ii) A is closed, densely dened and for every λ > ω one has λ ∈ ρ(A) and

‖ [(λ− ω)R(λ,A)]n ‖ ≤M, ∀n ∈ N ;

(iii) A is closed, densely dened and for every λ ∈ C with Reλ > ω one has λ ∈ ρ(A) and

‖R(λ,A)n‖ ≤ M

(Reλ− ω)n, ∀n ∈ N ;

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Mathematical Finance - · PDF fileMathematical Finance. Contents I Stochastic calculus7 ... De nition 1.1.3 (Stochastic process) . A stochastic pressco is a collection fX t: t 0gof