Tankov Jump Processes Ch3

8/12/2019 Tankov Jump Processes Ch3

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Contents

1. Levy processes: denition and properties 11.1. Compound Poisson processes 31.2. Jump measure of compound Poisson process 41.3. Innite activity Levy processes 61.4. Pathwise properties of Levy processes 121.5. Distributional properties 151.6. Stable laws and processes 181.7. Levy processes as Markov processes 201.8. Levy processes and martingales 22

1. Levy processes: definition and properties

1.0.1. From random walks to Levy processes.

Denition 1.1. Levy process A cadlag stochastic process (X t )t 0 on (, F , P )with values in R d such that X 0 = 0 is called a Levy process if it posses the following properties:

(1) Independent increments: for every increasing sequence of time t0, . . . , t n ,the random variables X t 0 , T t 1 X t 0 , . . . , X t n X t n 1(2) Stationary increments: the law of X t+ h X t doe snot depend on t.(3) Stochastic continuity: > 0, limh0P (X t+ h X t | ) = 0 .

The third condition does not mean that sample paths are continuous, it meansthat for a given time t, the probability of seeing a jump at t is zero: discontinuities

occur at random times.If we sample a Levy process at regular time intervals 0 , , 2 , . . . we obtain arandom walk: dening S n ( ) = X n , we can write S n ( ) = k = 0

n 1Y k where

Y k = X (k+1) X kare i.i.d. random variables whose distribution is the same as the distribution of X . Choosing n = t, we see that for any t > 0 and any n 1, X t = S n ( ) canbe represented as a sum of n i.i.d random variables whose distribution is that of

1


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X t/n : X t can be divided into n i.i.d parts. A distribution having this property issaid to be innitely divisible :

Denition 1.2. Innite divisibility A probability distribution F on R d

is said tobe innitely divisible if for any integer n 2, there exists n i.i.d random variables Y 1, . . . , Y n such that Y 1 + Y 2 + + Y n has distribution F .Since the distribution of i.i.d. sums is given by convolution of the distribution

of the summands, if we denote by the distribution of the Y k-s in the denitionabove, then F = is the n-th convolution of . So an innitelydivisible distribution can also be dened as a distribution F for which the n-thconvolution root is still a probability distribution, for any n 2.If X is a Levy process, for any t > 0 the distribution of X t is innitely divisible.this puts a constraint on the possible choices of distributions of r X t : whereas theincrements of a discrete time random walk can have arbitrary distribution, thedistribution of increments of a Levy process has to be innitely divisible.

Examples of innitely divisible laws are: the Gaussian distribution 1, the gammadistribution, the Poisson distribution.

Given an innitely divisible distribution F , it is easy to see that for any n 1by chopping it into n i.i.d. components we ca construct a random walk model ona time grid with step size 1 /n such that the law of the position at t = 1 is given byF . In the limit, this procedure can be used to construct a continuous time Levyprocess (X t )t 0 such that the law of X 1 if given by F :

Proposition 1.3. Innite divisibility and Levy processes Let X t )t 0 be a Levy process. Then for every t, X t has a innitely divisible distribution. Con-versely, if F is an innitely divisible distribution then there exists a Levy process (X t ) such that the distribution of X 1 is given by F .

Dene the characteristic function of X t :

t (z ) = X t (z ) = E [eiz.X T ], z R d (1)For t > s , by writing X t + s = X s + ( X t + s X s ) and using the fact thatX t+ s X s is independent of X s , we obtain that t t (z ) is a multiplicativefunction:

t+ s (z ) = X t + s (z ) = X s (z )X t + s X s (z )

= X s (z )X t (z )= st .

The stochastic continuity of t X t implies in particular that X t X s in distribu-tion when s t. Therefore, from the fact that X n (z ) X (z ), X s (z ) X t (z )1If X sin N (, 2 ) then one can write X =

k =0n 1 Y k where Y k are i.i.d.with law N (/n,

2 /n ).


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when s t so t t (z ) is a continuous function t. Together with the multiplica-tive property s+ t (z ) = s (z )t (z ) this implies that t t (z ) is an exponentialfunctionProposition 1.4. Characteristic function of a Levy process Let (X t )t 0 be a Levy process on R d. There exists a continuous function : R d R called the characteristic exponent of X , such that

E [eiz.X t ] = et (z) , z R d (2)Recalling the denition of the cumulant generating function of a random vari-

able, we see that is the cumulant generating function of X 1 : = X 1 and thatthe cumulant generating function of X t varies linearly in t : X t = tX 1 = t.The law of X t is therefore determined by the knowledge of the law of X 1: the onlydegree of freedom we have in specifying a Levy process is to specify the distribution

of X t for a single time (say, t = 1).1.1. Compound Poisson processes.

Denition 1.5. Compound Poisson process A compound Poisson process with intensity > 0 and jump size distribution f is a stochastic process X t dened as

X t =N t

i=1

Y i (3)

where jumps sizes Y i are i.i.d. with distribution f and (N t ) is a Poisson process with intensity , independent from (Y i)i 1.

The following properties of a compound Poisson process are easily deductedfrom the denition:

(1) The sample paths of X are cadlag piecewise constant functions.(2) The jump times ( T i)i 1 have the same law as the jump times of the Poisson

process N t : they can be expressed as partial sums of independent exponen-tial random variables with parameter .

(3) The jump sizes ( Y i)i 1 are independent and identically distributed with lawf .

The Poisson process itself can be seen as a compound Poisson process on Rsuch that Y = 1. Let R(n), n 0 be a random walk with step size distributionf : R(n) =

ni=0 yi . The compound Poisson process X t can be obtained bychanging the time R with an independent Poisson process N t : X t = R(N t ). X t

thus describes the position of a random walk after the random number of steps,given by N t . This operation is similar to the subordination of Levy processes.

Proposition 1.6. (X t)t 0 is compound Poisson process if and only if it is a Levy process and its sample paths are piecewise constant functions.

See Tankov and Cont (2004), pp.72 for a proof.


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Proposition 1.7. Characteristic function of a compound Poisson pro-cess Let (X t )t 0 be a compound Poisson process on R d. Its characteristic function has the following representation:

E [eiu.X t ] = exp t R d (eiu.x 1)f (dx)}, u R d , u R d (4)where denotes the jump intensity and f the jump size distribution.

Comparing (4) with the characteristic function of a Poisson process E [eiuN t ] =exp {t (eiu 1)}, u R we see that a compound Poisson random variable canbe represented as a superposition of independent Poisson processes with different jump sizes. The total intensity of Poisson processes with jump sizes in the interval[x, x + dx] is determined by the density f (dx).

Proof. Conditioning the expectation on N t and denoting the characteristic functionof f by f , we nd

E [exp(iu.X t )] = E [E [exp(iu.X t )]|N t ] = E [( f (u))N t ]=

n =0

e t (t )n ( f (u))n

n!

= exp{t ( f (u) 1}= exp t R d (eiu.x 1)f (dx).

For one-dimentional compound Poisson processes the characteristic function hasa simpler form:

E [exp{iuX t}] = exp t

(eiux 1)f (dx) , u R .

Introducing a new measure v(A) = f (A), we can rewrite the formula (4) as:

E [exp(iu.X t )] = exp t R d (eiu.x 1)v(dx) , u R d (5)1.2. Jump measure of compound Poisson process. We will use the notion of

random measure to study the behaviour of jumps of a compound Poisson process.To every cadlag process and in particular to every compound Poisson process(X t )t 0 on R d one can associate a random measure on R d [0, ] describing the jumps of X : for measurable set B R d [0, [

J X (B) = # {t, X t X t ) B} (6)For every measurable set A R d, J X ([t1, t2]A) counts the number of jumps of X between t1 and t2 such that their jump sizes are in A. The following proposition


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shows that J X is a Poisson random measure (in the sense of proposition 2.18 TC pp. 57.)

Proposition 1.8. Jump measure of a compound Poisson process Let (X t )t 0 be a compound Poisson process with intensity and jump size distribu-tion f . Its jump measure J X is a Poisson random measure on R d [0, [ with intensity measure (dx dt) = v(dx)dt = f (dx)dt.

This represents an alternative interpretation of the Levy measure of a compoundPoisson process as the average number of jumps per unit of time.

Denition 1.9. Levy measure Let (X t)t 0 be a Levy process on R d. The measure v on R d dened by:

v(A) = E [# {t [0, 1] : X t A}], A B (R d) (7)is called a Levy measure of X : v(A) is the expected number, per unit time, of jumps whose size belong to A.

Proof of Proposition 1.8, see Tankov and Cont (2004) pp. 76.Proposition 1.8 implies that every compound Poisson process can be represented

in the following form:

X t =s [0,t ]

X s = [0,t ] R d xJx (ds dx) (8)where J X is a Poisson random measure with intensity measure v(dx)dt. This isa special case of the Levy-Ito decomposition for Levy processes. Here we haverewritten the process X as the sum of its jumps. A compound Poisson process hasalmost surely a nite number of jumps in the interval [0 , t ], the stochastic integralappearing in (8) is a nite sum, so there are no convergence problems.

Lemma 1.10. Let M be a Poisson random measure with intensity measure and let A be a measurable set much that 0 < (A) < . Then the following tworandom measures on the subsets of A have the same distribution conditionally on M (A):

(1) M |A , the restriction of M to A.(2) M A dened by M A(B) = # {X i B} for all measurable subsets B of A,where X i , i = 1, 2, . . . , M (A) are independent and distributed on A with the

law (dx )(A) . In other words, M A is the counting measure of M (A) independent random points, identically distributed on A.

Proof see Tankov and Cont (2004) pp. 78.As an application of Lemma 1.10, consider the following

Proposition 1.11. Exponential formula for Poisson random measuresLet M be a Poisson random measure with intensity measure . Then the following


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formula holds for every measurable set B such that (B) < and for all functions f such that

B e

f (x)(dx) < :E exp B f (x)M (dx) = exp B (e

f (x) 1)(dx) (9)Proof. Condition the expectation on (B) and use Lemma 1.10.

We will see that to obtain this formula we do not need the assumption that both(B) and B e

f (x)(dx) be nite, it suffice only to require B |ef (x) 1|(dx) < .Proposition 1.11 allows to establish a one-to-one correspondence between com-

pound Poisson processes and Poisson random measures with intensity measures of the form v(dx)dt) with v nite. Indeed, let v be a nite measure on R D and letM be a Poisson random measure on R d [0, [ with intensity measure v(dx)dt.Then one can show using Proposition 1.11 that equation (8) denes a compoundPoisson process with Levy measure v.1.3. Innite activity Levy processes. in the previous section, every piecewiseconstant Levy process X 0t can be represents in the form (8) for some Poissonrandom measure with intensity of the form v(dx)dt where v is a nite measure,dened by

v(A) = E [# {t [0, 1] : X 0t = 0 , X 0t A}], A B (R d). (10)Given a Brownian motion with drift t + W t , independent from X 0, the sum

X t = X 0t + t + W t denes another Levy process, which can be decomposed as:

X t = t + W

t +

s [0,t ]

X s = t + W

t +

[0,t ]R d

xJ X

(ds

dx),

where J X is a Poisson random measure on [0, [R d with intensity v(dx)dt.Can every Levy process be represented in this form ? Given a Levy process X t ,can still dene its Levy measure v above. v(A) is still nite for any compact setA such that 0 / A: if this is not true, the process would have an innite numberof jumps of nite size on [0, T ], which contradicts the cadlag property.

So v denes a Radon measure on R d\{0}. But v is not necessarily a nitemeasure: the above restriction still allows it to blow up at zero and X may havean innite number of small jumps on [0 , T ]. In this case the sum of the jumpsbecomes an innite series and tis convergence imposes some conditions on themeasure v under which we obtain a decomposition of X similar to the one above:Proposition 1.12. Levy-Ito decomposition Let (X t )t 0 be a Levy process on R d and v its Levy measure, given by Denition 1.9

(1) v is a Radon measure on R d\{0} and veries:

|x | 1 |x|2v(dx) < |x | 1 v(dx) <


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(2) The jump measure of X , denoted bby J X , is a Poisson random measure on [0, [R d with intensity measure v(dx)dt.(3) There exist a vector and a d-dimensional Brownian motion 2 (B t )t 0 with covariance matrix A such that

X t = t + Bt + X lT + lim0 X t , where

X lt = |x | 1,s [0,t ] xJ X (ds dx) andX t = | x|< 1,s [0,t [ x{J X (ds dx) v(dx)ds}

= | x|< 1,s [0,t [ x J X (ds dx)(11)

The terms in X t = t + B t + X lT + lim0 X t are independent and the conver-

gence in the last term is almost sure and uniform in t on [0, T ].

The Levy-Ito decomposition entails that for every Levy process there exist avector , a positive denite matrix A and a positive measure v that uniquelydetermine its distribution. the triplet ( A,v, ) is called characteristic triplet orLevy triplet of the process X t .

Given the importance of this result, let us comment a bit on the meaning of the

terms in

X t = t + Bt + X lT + lim0 X t

First t + B t is a continuos Gaussian Levy process and every Gaussian Levy process is continuous and ca be written in this form and can be described by twoparameters: the drift and the covariance matrix of Brownian motion, denotedby A.

The other two terms are discontinuous processes incorporating jumps of X t andare described by the Levy measure v. The condition

|y| 1 v(dy) < means that

X has a nite number of jumps with absolute value larger than 1. So the sum

X lt =| X s | 1

0 s t

X s

contains almost surely a nite number of terms and X lt is a compound Poissonprocess. There is nothing special bout the threshold X = 1: for any > 0 the


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sum of jumps with amplitude between and 1:

X t =

1> | X s |

o s t X S = | x|< 1,s [0,t ] xJ X (ds dx) (12)

is again a well dened compound Poisson process. Contrary to the compoundPoisson case, v can gave a singularity zero: there can be innitely many small jumps and their sum does not necessarily converge. This prevents us from making

go to 0 directly in expression (12). In order to obtain convergence we haveto center the remainder term i.e. replace the jump integral by its compensatedversion (dened in 2.6.2)

X t =

|| x< 1,s [0,t ]

x J X (ds dx) (13)which (as in proposition 2.16) is a martingale. While X can be interpreted as aninnite superposition of independent Poisson processes, X t should be seen as aninnite superposition of independent compensated, i.e. centred Poisson processesto which a central-limit type argument can now be applied to show convergence.

1.3.1. Implications. An important implication is that the Levy-Ito decompositionis that every Levy process is a combination of a Brownian motion with drift anda possibly innite sum of independent compound Poisson processes. This alsomean that every Levy process can be approximated with arbitrary precision by adiffusion process, that is the sum of Brownian motion with drift and a compoundPoisson process, a point which is useful both in theory and in practice.

Proof. of the Levy-Ito decomposition (outline) We construct a Poisson ran-dom measure J X on [0, t]R d from the jumps of (X t). Since (X t ) is cadlag, for anypositive the set {t : |X t X | } is nite and the Poisson random measure (of any closed set non containing 0) can be constructed using 1.8.

The intensity measure J X is homogeneous and equal to v(dx)dt. Throughoutthe rest of the proof we can suppose without loss of generality that all jumps of (X t ) are smaller that 1 in absolute value.

Lemma 1.13. Let (X T , Y T ) be a Levy process. If (Y t ) is compound Poisson and (X t ) and (Y t ) never jump together, then they are independent.

For a proof see Kallenberg, Lemma 13.6.This lemma together with the exponential formula (6) allows to prove that the

Levy measure v satises the integrability condition

(|x|2 1)v(dx) < (14)


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Since the Levy measure of any closed set not containing zero is nite, it issufficient to prove that for some > 0,

|x | |X |2v(dx) < .

Let X t be as above in (13) and let Rt = X t X t . Then (X t , R t ) is a Levy processbecause (X t ) is. Clearly for some u and some t we have |E exp{iuX t}| > 0. Letus x this u and this t. Since Lemma 1.13, (X t ) and (R t ) are independent,E exp{iuX t} = E exp{iuR t }E exp{iuX t },

and this means that |E exp{iuX t }| is bounded from below by a positive numberwhich does not depend on . By the exponential formula (Proposition 1.11) thisis equivalent to

exp t |x | (eiux 1) v(dx) C > 0,which implies that

|x | (1 cos(ux))v(dx) C < .Making tend to zero, we obtain (14).

Now we can use it to show the convergence of X t . Consider a sequence {n} 0and let Y n = X n +1t X nt . All the variables Y i have zero mean and (14) entailsthat

V arY i < . Hence, by Kolmogorovs three series Theorem (Kallenberg,

Lemma 3.15), one can show that the convergence is uniform in t.To complete the proof, consider the process X ct = X t lim X t . It is a Levyprocess which is independent from lim X t by Lemma 1.13. It is continuous becauseX t converges uniformly in t and therefore one can interchange the limits. Finally,the Feller-Levy central limit Theorem (Kallenberg, Theorem 4.15) implies that itis also Gaussian.

Our knowledge of the structure of paths of a Levy process allows to obtain al-most without additional work the second fundamental result of the theory: theexpression of the characteristic function of a Levy process is terms of its charac-teristic triplet ( A,v, ).

Theorem 1.14. Levy-Khinchin representation Let (X t )t 0 be a Levy process on R d with characteristic triplet (A,v, ). Then

E [eiz.X t ] = et (z) , z R d (15)with

(z ) = 12

z Az + i.z + R d (eiz.x 1 iz.x 1 |x | 1)v(dx).


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For real-valued Levy processes, the formula 15 takes the form

E [eizX t ] = et (z) , z Zwith

(z ) = 12

Az 2 + iz +

(eiux 1 izx 1 |x | 1)v(dx)

An equivalent version of the Levy-Khinchin representation may be obtained bytruncating the jumps larger than an arbitrary number :

(z ) = 12

z Az + i .z + R d (eiz.x 1 iz x1 |x | )v(dx),where

= + R d x(1|x | 1|x | 1)v(dx).

More generally, for every bounded measurable function g : R d R satisfyingg(x) = 1 + o(|x|) as x 0 and g(x) = O(1/ |x|) as x one can write:(z ) =

12

z az + i g.z + R d (eiz.x 1 iz.xg (x))v(dx).Such a function g is called the truncation function and the characteristic triplet(A,v, g) is called the characteristic triplet of X with respect to the truncationfunction g. Different choices of g do not affect A and v which are intrinsic param-eters of the Levy process, but depends on the choice of truncation function soone should avoid calling it the drift of the process. Various version for g exists.Paul Levy used the truncation function g(x) = 11+ |x| 2 while most recent texts useg(x) = 1|x | 1. In the sequel, when we refer to the Levy triplet of a Levy processwe implicitly refer to the truncation function g(x) = 1|x | 1.

If the Levy meaure satises the additional condition |x | 1 |x|v(dx) < thereis no need to truncate large jumps and one can use the simpler form(z ) =

12

z.Az + i c.z + R d (eiz.x 1 iz.x )v(dx).In this case it can be shown that E [X t ] = ct and c is called the center of process(X t ). It is linked to by the relation

c = + |x | 1 xv(dx).Proof. of Theorem 1.14

The Levy-Ito decomposition(Proposition 1.12) shows that for every t, the ran-dom variable X ct + X lt + X t converges almost surely to X t when tends to 0.


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Since almost sure convergence implies convergence in distribution, the character-istic function of X ct + X lt + X t converges to the characteristic function of X t . SinceX ct , X lt and X t are independent,

E [exp{iz. (X ct + X lt + X t )}] = exp{12

tz.Az + it.z }exp{t |x | 1(eiz.x 1)v(dx)}exp{t | x| 1(eiz.x 1 iz.x )v(dx)

and this expression converges to (15) for every z when tends to 0.

When v(R d = (innite activity case), the set of jump times of every trajectoryof the Levy process is countable innite and dense in [0, [. The countabilityfollows directly from the fact that the paths are cadlag. to prove that the set of jump times is dense [0,

[, consider a time interval [a, b] and let

(n) = sup r : |x | r v(dx) nand

Y n = (n )| x|< (n 1),t [a,b] J X (dx dt)Then, if the Levy measure has no atoms, Y i are independent and identically Poissondistributed Poisson random variables. The total number of jumps in the interval[a, b] is equal to

i=1 Y i , hence, by the law of large numbers, it is almost surely

innite. Since this is true for every nonempty time interval [ a, b] , this means

that the set of jump times is dense in [0 , [. The proof can be easily modied toinclude the case when Levy measures has atoms.Since an innitely divisible distribution is the distribution at time t = 1 of some

Levy process, the Levy-Khinchin formula also gives a general representation forthe characteristic function of any innitely divisible distribution:

Theorem 1.15. Characteristic function of innitely divisible distribu-tions Let F be an innitively divisible distribution on R d. Its characteristic func-tion can be represented as:

F (z ) = e (z ), z R d(z ) = 12z.Az + i.z + R d (e

iz.x 1 iz.x 1 |x | 1)v(dx)

where A is a symmetric positive n n matrix, R d and v is a positive Radon measure on R d\{0} verifying:

|x | 1 |x|2v(dx) < |x | 1 v(dx) < .


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v is called the Levy measure of the distribution F .

1.4. Pathwise properties of Levy processes. Using the Levy-Ito decomposi-tion, we deduce some properties of typical sample paths of a Levy process fromanalytical properties of its characteristic triplet ( A,v, ).

Piecewise constant trajectories: we saw in the above (Proposition 1.6) thatalmost all trajectories of a Levy process are piecewise constant iff it is of compoundPoisson type. Combining this with equation (4), which gives the characteristicfunction of a compound Poisson process, we obtain the following:

Proposition 1.16. A Levy process has piecewise constant trajectories if and only if its characteristic triplet satises the following conditions: A = 0, R d v(dx) < and = |x | 1 xv(dx) or equivalently, if its characteristic exponent is of the form:

(z ) =

(eiux

1)v(dx) with v(R d) 0:

T V (X t ) | x|< 1,x [0,t ] |x|J X (ds dx)= t

| x|< 1,x [0,t ] |x|v(dx) +

| x|< 1,x [0,t ] |x|(J X (ds dx) v(dx)ds).

Using the exponential formula on can show that the variance of the second termin the last line is equal to

t | x|< 1,x [0,t ] |x|2v(dx).Hence by the same argument that was used in the proof for Levy-Ito decomposition,the second term converges almost surely to something nite.

Therefore in the condition (|x| )v(dx) < is not satised, the rst term inthe last line will diverge and the variation of X t will be innite. Suppose now thatthis condition is satised. This means that X t may be written as

X t = X ct + [0,t ] R d xJ X (ds dx).where the second term is of nite variation. Since trajectories of Brownian motionare almost surely of innite variation (see Revuz and Yor (1999)), if A is nonzero,X t will also have innite variation. Therefore we must have A = 0.

The proposition shows that in the nite variation case Levy-Ito decompositionand Levy-Khinchin representation can be simplied:

Corollary 1.18. Levy-Ito decomposition and Levy-Khinchin represen-tation in the nite-variation case

Let (X t )t 0 be a Levy process of nite variation with Levy triplet given by (v,a, ). Then X can be expressed as teh sum of its jumps between 0 and t and a linear drift term:

X t = bt + [0,t ] R d xJ X (ds dx) = bt +X s =0

s [0,t ]

(17)


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and its characteristic function can be expressed as:

E [eiz.X t ] = exp t ib.z +

R d

(eiz.x 1)v(dx) (18)where b = |x | 1 xv(dx).The Levy triplet of X is not given by (b,0, ) but by ( , 0, v). As mentionedbefore, is not an intrinsic quantity and depends on the truncation function usedin the Levy-Khinchin representation while bt has an intrinsic interpretation as thecontinuous part of X .

Increasing Levy processes (subordinators). Increasing Levy processes arealso called subordinators because they can be used as time changes of other Levyprocess (see next section). They are important ingredients for building Levy-basedmodels in nance.

Proposition 1.19. Let (X t)t 0 be a Levy process on R . The following conditions are equivalent:

(1) X t 0 a.s. for some t > 0.(2) X t 0 a.s. for every t 0.(3) Sample paths of (X t ) are almost surely nondecreasing: t s X t X sa.s.(4) The characteristic triplet of (X t) satises A = 0, v((, 0]) = 0,

0 (x

1)v(dx) < and b 0, that is (X t ) has no diffusion component, only positive jumps of nite variation and positive drift.Proof. of Proposition 1.19

(1) (i iii ) For every n, X t is a sum of n i.i.d. random variables X t/n , X 2t/n X t/n , . . . , X t X (n 1)t/n . This means that the variable are almost surelynonnegative. With the same logic we can prove that for two rationals pand q such that 0 < p < q, X qt X pt 0 a.s. Since teh trajectories areright-continuous, this entails that they are nondecreasing.

(2) (iii ii ) is trivial.(3) (iv iii ) Under the conditions of ( iv) the process if of nite variation,therefore equal to the sum of jumps plus an increasing linear function. Forevery trajectory, the number of negative jumps on a xed interval is aPoisson random variable with intensity 0, hence almost surely zero. This

means that almost every trajectory is nondecreasing.(4) (iii iv) Since the trajectories are nondecreasing, they are of nite vari-ation. Therefore, A = 0 and

(x 1)v(dx) < . For trajectories to benonincreasing, there must be no negative jumps, hence v(] , 0]) = 0.If a function is nondecreasing then after removing some of its jumps, weobtain another nondecreasing function. Then we remove all jumps from atrajectory of X t , we obtain a deterministic function bt which must thereforebe non decreasing. This allows to conclude that b 0.


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The following proposition introduces an important example of subordinator.

Proposition 1.20. Let (X t )t 0 be a Levy process on R d and let f : R d [0, [be a positive function such that f (x) = O(|X |2) when x 0. Then the process (S t )t 0 dened by S t =

s t, X s =0

f ( X s ) (19)

is a subordinator.

Proof. of Proposition 1.20 Let us rst show that the sum in (19) converges tosomething nite. By truncating large jumps we can suppose that for each s,

X s for some > 0 and f ( X s ) C X 2s for some C > 0. But thenE [S t ] = [0,t ] R f (x)dsv(dx) < (20)

Since all the terms in the sum are positive, this means that it always converges andS t is almost surely nite for all t. The fact that S has independent and stationaryincrements follows directly from independence and stationarity of increments of X . To prove that its is continuous in probability one can once again suppose that jumps of X t are bounded (because the compound Poisson part is always continuosin probability). But then E [|S t S s|] 0 as s t. Therefore, S is continuous inprobability.

The choice f (x) = x2 yield the sum of squared jumps

S t =s t, X s =0

| X s|2 (21)

This process which by the above is a subordinator is usually denoted [ X.X ]d andcalled the discontinuous quadratic variation of X .

1.5. Distributional properties. If (X t )t 0 is a Levy process the for any t > 0,the distribution of X t is innitely divisible and has a characteristic function of the form (15). However, X t does not always have a density: indeed, if X t is acompound Poisson process we have

P (X t = 0) = e t (22)so the probability distribution has an atom at zero for all t. But if X is not acompound Poisson process, then X t has a continuous density; we give the followingresult for d = 1 from Orey (1968):

Proposition 1.21. Let X be a real-valued Levy process with Levy triplet (2, v, ).(1) If > 0 or v(R ) = then X t has a continuous density pt () on R .


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16 TANKOV-CONT

(2) If the Levy measure v veries

]0, 2[, lim inf

0

|x

|2dv(x) > 0 (23)

then for each t > 0, X t has a smooth density pt() such that pt () C (R ) n 1,

n ptx n

(t, x ) |x | 0. (24)These and other properties of the density may be obtained using the Levy-

Khinchin representation and the properties of the Fourier transform (see Sato(1999), Chapter 5).

Relation between probability density and Levy density In the compoundPoisson case there is a simple relation between probability distribution at time tand the jump size distribution/Levy measure. Let ( X t )t 0 be a compound Poissonprocess with intensity and jump size distribution f and (N t )t 0 be a number of jumps of X on [0, t]. Then

P {X t A} =

n =0

P {X t A|N t = n}e t (t )n

n!

= e t 0 +

n =1

f n (A)e t (t )n

n!

(25)

where f n

denotes the n th convolution power of f , and 0 is the Dirac measureconcentrated at 0. As noted above, this probability measure does not have adensity because P {X t = 0} > 0.However, if the jump size distribution has a density with respect to Lebesguemeasure, then the law of X t is absolutely continuous everywhere except at zero(because convolution of absolutely continuous distribution is absolutely continuous) i.e. the law of X t can be decomposed as

P {X t A} = e t 10 A + A pact (x)dx where pac

t (x) =

n =1

f n (x)e t (t )n

n! x = 0 .

where we denote the jump size density by f (x). pact is the density conditional onthe fact that the process has jumped at least once. This implies in particular thefollowing asymptotic relation

limt0

1t pact (x) = f (x) = v(x) x = 0 ,


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JUMP PROCESSES 17

where v(x) is the Levy density. This means that the Levy density describes thesmall time behaviour of the probability density.

This relation also gives the small time behaviour for expectations of functionsof X t : given any bounded measurable function f such that f (0) = 0 ,

limt0

1t

E [f (X t )] = limt0

1t R d f (x) pt (dx) = R d f (x)v(dx) (26)

In teh innite activity setting the classical asymptotic result for expectations (seeSato (1999), Corollary 8.9) is weaker: it states that the formula (26) holds forany bounded continuous function f vanishing in the neighbourhood of zero. Moreresults on the relation between probability density of X and Levy measure forinnite-activity processes may be found in Barndorff-Nielsen (2000) and Ruschen-dorf (2002).

Moments and cumulants The tail behaviour of the distribution of a Levyprocess and its moments are determined by the Levy measure , as shown by thefollowing proposition, which is consequence of Sato (1999) (Theorem 25.3).

Proposition 1.22. Moments and cumulants of a Levy process Let (X t )t 0be a Levy process on R with characteristic triplet (A,v, ). The n th absolute moment of X t , E [|X t |n ] is nite for some t or, equivalently, for every t > 0 if and only if |x | 1 |x|

n v(dx) < . In thsi case moments of X t can be computed from its characteristic function by differentiation. In particulars, the form of cumulants of (X t ) is especially simple:

E [X t ] = t +

|x | 1x v(dx) ,

c2(X t) = Va r X t = t A + i

nfty x2v(dx) ,

cn (X t) = t

xn v(dx) for n 3.

This entails that all innitely divisible distributions are leptokurtic since c4(X t ) >0. Also, the cumulants of the distribution of X t increase linearly with t. In particu-lar the kurtosis and skewness of X (or, equivalently, of the increments X t+ X tare given by:

s(X ) = c3(X )c2(X )3/ 2

= s(X 1) , k(X ) =

c4(X )c2(X 2 )

= k(X 1) . (27)

Therefore the increments of a Levy process, or , equivalently, all innitely di-visible distributions are always leptokurtic but the kurtosis (and skewness if thereis any) decreases with the time scale over which increments are computed: theskewness fall at 1/ 2 while the kurtosis decays as 1 / .


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18 TANKOV-CONT

Proposition 1.23. Exponential moments Let (X t )t 0 be a Levy process on Rwith characteristic triplet (A,v, ) and let u R . The exponential moment E [euX t ]is nite for some t or, equivalently, for all t

0 if and only if

|x | 1 e

ux v(dx) 0,W at a t 0

d= ( W t )t 0.

If we consider a Brownian motion with drift B t = W t + t then this property isonly veried up to a translation:

a > 0,Bat a t 0

d= ( B t + at )t 0A natural question is whether there exist other real valued Levy processes thatshare this selfsimilarity property: a Levy process X t is said to be selfsimilar if

a > 0, b(a) > 0 :X atb(a) t 0

d= ( X t )t 0

Since the characteristic function of X t has the form

X t (z ) = exp[t (z )]this property is equivalent to the following property of the characteristic function:

a > 0, b(a) > 0 : P hi X t (z )a = X t (zb(a)) z.The distribution that verify this property are called strictly stable distributions .More precisely, we have the following denition.

Denition 1.24. A random variable X R d is said to have stable distribution if r every a > 0 there exists b(a) > 0 and c(a) R d such that X (z )a = X (zb(a))eic.z , z R d. (28)

It is said to have a strictly stable distribution if X (z )a = X (zb(a)) , z R d. (29)

The name stable comes for the following stability under addition property: if X has a stable distribution and X (1) + + X (n ) are independent copies of X thenthere exists a positive number cn and a vector d such that

X (1) + + X (n ) d= cn X + d . (30)


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JUMP PROCESSES 19

This property is clearly veried if the distribution of X is that of selfsimilar Levyprocess at a given time t. It can be show (see Samorodnitsky (1994), Corollary2.1.3) that for every stable distribution there exists a constant

(0, 2] such

that (28), b(a) = a1/ . This constant is called the index of stability and stabledistributions with index are also referred to as stable distributions. the only2-stable distributions are Gaussian.

A selfsimilar Levy process therefore has strictly stable distribution at all times.For this reason, such processes are also called strictly stable Levy processes. Astrictly stable process satises:

a > 0,X ata1/ t 0

d= ( X t )t 0. (31)

In the case of the Wiener process = 2. More generally, an stable Levy processsatises this relation up to a translation:a > 0, c R d : (X at )t 0

d= ( a1/ X t + ct)t 0.A stable Levy process denes a family of stable distributions and the converseis true: every stable distribution is innitely divisible and can be seen as thedistribution at a given time of a stable Levy process. The following result givesthe form of characteristic triplet of all stable distributions (and therefore Levy)

Proposition 1.25. Stable distribution and Levy processes A distribution on R d is stable with 0 < < 2 if and only if it is innitely divisible with characteristic triplet (0,v, ) and there exists a nite measure on S , a unit sphere of R d, such that

v(B) = S (d )

01 B (r )

drr 1+

(32)

A distribution on R d is stable with = 2 if and only if it is Gaussian.A proof is given in (Sato (1999), Theorem 14.3), see also Samorodnitsky (1994).

For real valued stable variables and Levy process (d=1) the above representationcan be made explicit: if X is a real-valued stable variable with 0 < < 2 thenits Levy measure is of the form

v(x) = Ax +1

1x 0 + B

|x

| +1 1x 0 (33)

for some positive constants A and B. The characteristic function of a real-valuedstable random variable X has the form

X (z ) = exp |z | (1 i sgnz tan2

+ iz , if = 1X (z ) = exp |z |(1 + i

2

sgn z log |z |) + iz , if = 1,(34)


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20 TANKOV-CONT

where (0, 2], 0, [1, 1] and R . In the sequel , a stable distributionon R in this parametrization is denoted by S (,,v ). In this representation, isthe scale parameter (it has nothing to do with the Gaussian component if < 2), is the shift parameter (when = 1 this is not true: (see Samorodnitsky (1994),Section 1.2), determines the shape of the distribution and the skewness.

When = 0 and = 0, X is said to have a symmetric stable distribution andthe characteristic function is given by

X (z ) = exp( |z | ).The explicit form of the Levy measure (33) shows that stable distributions onR never admit a second moment, and they only admit a rst moment if > 1.The probability density of an stable law is not known in closed for except inthe following three cases

(1) The Gaussian distribution S 2(, 0, ) with density1

2 e(x )2 / 4 2

(2) The Cauchy distribution S 1(, 0, ) with density

((x )2 + 2.

(3) The Levy distribution S 1/ 2(, 1, ) with density

2

1/ 2 1(x )3/ 2

exp

2(x )1x

While the rst two distributions are symmetric around their mean, the last oneis concentrated on ( , ). Despite the fact that closed formulae for probabilitydensity are only available in these three cases, closed-form algorithms for simulat-ing stable random variables on R exists for all values of parameters (see Chapter6 Cont and Tankov, 2004).

1.7. Levy processes as Markov processes. An important property of Levyprocesses is the Markov property, which states that conditionally on X t , the evo-lution of the process after time t is independent on its past before this moment.In other words, for every random variable Y depending on the history F t of X tone must have

E [Y |F t ] = E [Y |X t ].The transition kernel of process X t is dened as follows:P s,t (x, B ) = P {X t B |X s = x}, B B (35)

The Markov property implies the following relation between transition kernels(known as the Chapman-Kolmogorov equations):

P s,t (x, B ) = R d P s,t (x,dy)P t,u (y, B ).


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JUMP PROCESSES 21

It can be seen from (35) that the transition kernels of Levy processes are homoge-neous in space and time, that is,

P s,t (x, B ) = P 0,t s (0, B x).Levy processes are completely characterised by this condition (see Sato (1999),Theorem 10.5): they are the only Markov process which are homogenous in spaceand time.

Levy processes satisfy a stronger version of the Markov property, namely, forall t, the process (X t+ s X t )s> 0 has the same law as the process (X s )s> 0 and itsindependent from ( X s )0 s t .

Finally, the strong Markov property of Levy processes allows to replace thenonrandom time t by any nonrandom time which is nonanticipating with respectto the history of X (see section 2.4.2): if is a nonanticipating random time, thethe process Y t = X t+ X t is again a Levy process, independent from F and withsame law as X t)t> 0.

The transition operator for Markov processes is dened as follows:

P f (x) = E [f (s + X t ]Chapman-Kolmogorov equations and the time homogeneity of transition kernelsimply the following semigroup relation between transition operators:

P tP s = P t+ s (36)

Let C0 be the set of continuous functions vanishing at innity. then for any t > 0,P t f C0 andx lim

t0P t f (x) = f (x).

where the convergence is in the sense of supremum norm on C0. This property iscalled the Feller property. A semigroup P t verifying the Feller property (36) canbe described by means of its innitesimal generator L which is a linear operatordened by

Lf = limt0

t 1(P tf

f ). (37)

where the convergence is int the sense of supremum norm on C0 and f should besuch that the right-hand side of 37 exists. The innitesimal generator of a Levyprocess can be expressed in terms of its characteristic triplet.

Proposition 1.26. Innitesimal generator of a Levy process Let (X t )t 0be a Levy process on R d with characteristic triplet (A,v, ). Then the innitesimal


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22 TANKOV-CONT

generator of X is dened for any f C 20 (R ) as

Lf (x) = 1

2

d

j,k =1

A jk

2f

x j x k(x) +

d

j =1

j

f

x j(x)

+ R d f (x + y) f (x) d

j =1

y jf x j

(x)1 |y| 1 d(dy),

(38)

where C 20 (R d) is the set of twice continuously differentiable functions, vanishing at innity.

For a proof see Sato, 1999 (Theorem 31.5).

1.8. Levy processes and martingales. The notion of martingale is crucial forprobability theory and mathematical nance. Different martingales can be con-structed from Levy processes using their independent increments property.

Proposition 1.27. Let (X t )t 0 be a real-valued process with independent incre-ments. Then

(1) eiuX t

E [eiuX t ] t 0is a martingale u R

(2) If for some u R , E [euX t ] < t 0 then euX t

E [euX t ] t 0is a martingale.

(3) If E [X t ]

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