The Brownian motion - HEC Montréal

BrownianMotion

Brownianmotion

Constructionof theBrownianmotion

An alternativeconstruction

The Brownian motion80-646-08

Stochastic calculus

Geneviève Gauthier

HEC Montréal

BrownianMotion

BrownianmotionDenitionRefresher:normal lawPropertiesOther properties



IntroductionBrownian motion

In 1827, Robert Brown observed that small particlessuspended in a drop of water exhibit ceaseless irregularmotions. Historically, Brownian motion was at rst anattempt to model such a phenomenon. Today, theBrownian motion process occurs in diverse areas such aseconomics, communication theory, biology, managementscience and mathematics. (Adapted from the introductoryparagraph about Brownian motion, S. Karlin and H. M.Taylor (1975).)

The mathematician Norbert Wiener is credited withrigorously analyzing the mathematics related to Brownianmotion, and that is why such a process is also known as aWiener process.

BrownianMotion




Denition IBrownian motion

Let (Ω,F ,F,P) be a ltered probability space.Technical condition. Since we will work with almost-sureequalities1, we require that the set of events which have azero-probability to occur be included in the sigma-algebraF0, which is to say that the set

N = fA 2 F : P (A) = 0g F0.

Thus, if X is Ftmeasurable and Y = XPalmost-surely, then we know that Y is Ftmeasurable.

BrownianMotion




Denition IIBrownian motion

DenitionA standard Brownian motion fWt : t 0g is an adaptedstochastic process, built on a ltered probability space(Ω,F ,F,P) such that:(MB1) 8ω 2 Ω, W0 (ω) = 0,

(MB2) 80 t0 t1 ... tk , the random variablesWt1 Wt0 , Wt2 Wt1 , ..., Wtk Wtk1 are independent,

(MB3) 8s, t 0 such that s < t, the random variableWt Ws is normally distributed with expectation 0 andvariance t s i.e. Wt Ws N (0, t s) ,(MB4) 8ω 2 Ω, the path t ! Wt (ω) is continuous.

BrownianMotion




Denition IIIBrownian motion

In general, the ltration we use is F = fFt : t 0g where

Ft = σ ffWs : 0 s tg [Ng

is the smallest sigma-algebra for which the randomvariables Ws : 0 s t are measurable and that containsthe zero-measure sets.

1X = Y Palmost-surely if the set of ω for which X di¤ers from Yhas a zero probability, i.e.

P fω 2 Ω : X (ω) 6= Y (ω)g = 0.

BrownianMotion




Reminder about normal law IIf X is a random variable following a normal law withexpectation µ and standard deviation σ > 0, then itsprobability density function is

fX (x) =1

σp2π

exp

( (x µ)2

2σ2

),

which allows us to determine 8a, b 2 R, a < b,

P [a < X b] =Z b

afX (x) dx .

Unfortunately, the integral above has no primitive. So weneed to value it numerically. The cumulative distributionfunction of X is

FX (x) =Z x

∞fX (y) dy .

BrownianMotion




Reminder about normal law II

In general, if two random variables X and Y , built on thesame probability space, are independent, then theircovariance

Cov [X ,Y ] = E [XY ] E [X ]E [Y ]

is nil.

However, it is possible that two variables have zerocovariance, but that they are not independent. Here is anillustrative example:

BrownianMotion




Reminder about normal law III

Example

ω X (ω) Y (ω) X (ω)Y (ω) P (ω)

ω1 0 1 0 14

ω2 0 1 0 14

ω3 1 0 0 14

ω4 1 0 0 14

BrownianMotion




Reminder about normal law IV

The covariance between these two variables is zero since:

EP [X ] = 0, EP [Y ] = 0, EP [XY ] = 0

) CovP [X ,Y ] = EP [XY ] EP [X ]EP [Y ] = 0

but they are dependent since

P [X = 0 and Y = 0] = 0 6= 14= P [X = 0]P [Y = 0] .

However, when variables are normally distributed (notnecessarily with the same expectation and standarddeviation), there is a result that allows us to verify whethersuch variables are independent by using covariance :

BrownianMotion




Reminder about normal law V

TheoremProposition. If X and Y are two random variables following amultivariate normale distribution, both built on the sameprobability space, then X and Y are independent if and only iftheir covariance is zero (T. W. Anderson, 1984, Theorem 2.4.4,page 28).

BrownianMotion




Properties IThe Brownian motion

TheoremLemma 1. Let fWt : t 0g be a standard Brownian motion.Then(i) For all s > 0, fWt+s Ws : t 0g (time homogeneity)(ii) fWt : t 0g (symmetry)

(iii)ncW t

c2: t 0

o(time rescaling)

(iv)nW t = tW 1

t1t>0 : t 0

o(time inversion)

are also standard Brownian motions.

Exercise. Verify it.

BrownianMotion




Properties IIThe Brownian motion

TheoremLemma 2. Brownian motion is a martingale.

DenitionDenition. On the ltered probability space (Ω,F ,F,P),where F is the ltration fFt : t 0g, the stochastic processM = fMt : t 0g is a martingale in continuous time if

(M1) 8t 0, EP [jMt j] < ∞;

(M2) 8t 0, Mt is Ft measurable;

(M3) 8s, t 0 tel que s < t, EP [Mt jFs ] = Ms .

BrownianMotion




Proof of Lemma 2 IBrownian motion properties

From the very denition of the ltration, it is obvious thatW is an adapted stochastic process.

For all time t, the random variable Wt is integrable since

EP [jWt j] =Z ∞

∞

jz jp2πt

expz

2

2t

dz

= 2Z ∞

0

zp2πt

expz

2

2t

dz

= r2tπexp

z

2

2t

∞

0

=

r2tπ< ∞.

BrownianMotion




Proof of Lemma 2 IIBrownian motion properties

The only thing left to verify is that 8s, t 0 such thats < t, EP [Wt jFs ] = Ws .

EP [Wt jFs ] = EP [Wt Ws +Ws jFs ]= EP [Wt Ws jFs ] + EP [Ws jFs ]= EP [Wt Ws ] +Ws from (MB2)

= Ws from (MB3)

The proof is complete.

BrownianMotion




Properties IBrownian motion

TheoremLemma 3. The Brownian motion is a Markov process.

Idea of the proof of Lemma 3 : For all u 2 [0, s ], therandom variables Wt Ws and Wu are independent since

Cov [Wt Ws ,Wu ]

= Cov [Wt Wu +Wu Ws ,Wu ]

= Cov [Wt Wu ,Wu ]Cov [Ws Wu ,Wu ]

= 0+ 0 par (MB2) .

As a consequence, Wt = (Wt Ws ) +Ws can be written as the sum of

two random variables: Ws which depends on information available at time

s , Fs , only through σ (Ws ) and Wt Ws which is independent from

Fs = σ fWu : 0 u sg.

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Properties IIBrownian motion

Property 3. 8ω 2 Ω, the path t ! Wt (ω) is nowheredi¤erentiable.Such a property is well illustrated by the construction ofBrownian motion.

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Multidimensional Brownian motionI

DenitionStandard Brownian motion W of dimension n is a family ofrandom vectors

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

t

>: t 0

where W (1), ...,W (n) represent independent Brownian motionsbuilt on the ltered probability space (Ω,F ,F,P) .

BrownianMotion




Multidimensional Brownian motionII

Multidimensional Brownian motion is very commonly usedin continuous-time market models.

For example, when modeling simultaneously several riskyasset prices.

However, the shocks received by such risky assets shouldnot be independent.

That is why we would like to build a multidimensionalBrownian motion, the components of which are correlated.

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Multidimensional Brownian motionIII

Starting from a standard Brownian motion W ofdimension n, it is possible to create a Brownian motion ofdimension n, the components of which are correlated.

BrownianMotion




Multidimensional Brownian motionIV

Theorem

Γ =γiji ,j2f1,2,...,ng is a matrix of constants and

W =W (1), ...,W (n)

>is a vector made up of independent

Brownian motions. For all t, lets set Bt = ΓWt .Then Bt is un random vector of dimension n, the ithcomponent of which is B (i )t = ∑n

k=1 γikW(k )t . Moreover

CovhB (i )t ,B

(j)t

i= t

n

∑k=1

γikγjk

et CorhB (i )t ,B

(j)t

i=

∑nk=1 γikγjkq

∑nk=1 γ2ik

q∑nk=1 γ2jk

.

BrownianMotion




Proof IMultidimensional Brownian motion

CovhB (i )t ,B

(j)t

i= Cov

"n

∑k=1

γikW(k )t ,

n

∑k =1

γjk W(k )t

#

=n

∑k=1

n

∑k =1

γikγjk CovhW (k )t ,W (k )

t

i=

n

∑k=1

γikγjkCovhW (k )t ,W (k )

t

icar Cov

hW (k )t ,W (k )

t

i= 0 if k 6= k

= tn

∑k=1

γikγjk

car CovhW (k )t ,W (k )

t

i= Var

hW (k )t

i= t

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Proof IIMultidimensional Brownian motion

VarhB (i )t

i= Cov

hB (i )t ,B

(i )t

i= t

n

∑k=1

γ2ik

CorhB (i )t ,B

(j)t

i=

CovhB (i )t ,B

(j)t

ir

VarhB (i )t

irVar

hB (j)t

i=

t ∑nk=1 γikγjkq

t ∑nk=1 γ2ik

qt ∑n

k=1 γ2jk

BrownianMotion




Proof IIIMultidimensional Brownian motion

We have just shown that it is possible to build a Brownianmotion B, the components of which are correlated,starting from a standard Brownian motion W (theelements of which are independent). More precisely, ifB =ΓW, then we know how to nd the correlation matrixof B.

Can we follow the other way around, i.e., if we know thecorrelation matrix of B, can we determine the matrix Γallowing us to express the components of B as a linearcombination of independent Brownian motions ?

BrownianMotion




Proof IVMultidimensional Brownian motion

Theorem

Lets now assume that B (1), ...,B (n) represent correlatedBrownian motions, built on the ltered probability space(Ω,F ,F,P) and that

8i , j 2 f1, ..., ng and 8t 0, CorhB (i )t ,B

(j)t

i= ρij .

There exists a matrix A of format n n such that

(i) B = AW(ii) Cor

hB (i )t ,B

(j)t

i= ρij

(iii) W is made of independent Brownian motions.

BrownianMotion




Proof VMultidimensional Brownian motion

Proof. Let VB = thρij

ii ,j=1,...,n

be the variance-covariance

matrix of random vectorB (1)t , ...,B (n)t

. Since B = AW then

VB = AtIA> = tAA>

where I represents the identity matrix of dimension n.Since a variance-covariance matrix is a symmetric positivedenite matrix, there exists an invertible upper triangularmatrix U such that VB = U>U (Cholesky decomposition).(Several software packages, including MATLAB, have afunction to calculate such a matrix).As a consequence, one only has to set

A =1ptU>.

BrownianMotion




Other properties IBrownian motion

Let a > 0. Lets dene

τa (ω) =

8<:inf fs 0 : Ws (ω) = ag if fs 0 : Ws (ω) = ag 6= ?

∞ if fs 0 : Ws (ω) = ag = ?,

the rst time when Brownian motion W reaches point a.

The next two results are intended to show that theBrownian motion will eventually reach, with probability 1,any real number, however large.

BrownianMotion




Other properties IIBrownian motion

TheoremLemma. The random variable τa is a stopping time.

DenitionLet (Ω,F ) be a measurable space equipped with the ltrationF = fFt : t 0g. A stopping time τ is a function of Ω into[0,∞] Fmeasurable such that

fω 2 Ω : τ (ω) tg 2 Ft .

BrownianMotion




Other properties IIIBrownian motion

Proof of the lemma. We must show that for all t 0, theevent fω 2 Ω : τa tg belongs to the sigma-algebra Ft .If Q represents the set of all rational numbers, then

fω 2 Ω : τa tg

=

ω 2 Ω : sup

0stWs (ω) a

=

∞\n=1

ω 2 Ω : sup

0stWs (ω) > a

1n

=

∞\n=1

[r2Q\[0,t ]

ω 2 Ω : Wr (ω) > a

1n

| z

2Fr therefore 2Ft| z 2Ft

2 Ft

BrownianMotion




Other properties IVBrownian motion

where the last equality is obtained from the fact thatsup0stWs (ω) > a 1

n if and only if there exists at least onerational number r smaller than or equal to t for whichWr (ω) > a 1

n .

BrownianMotion




Other properties VBrownian motion

TheoremLemma. The stopping time τa is nite almost surely, i.e.P [τa = ∞] = 0.

Proof of the lemma. We want to use the martingale stoppingtime theorem...

BrownianMotion




Other properties VIBrownian motion

TheoremOptional Stopping Theorem. Let X = fXt : t 0g be aprocess with càdlàg (or RCLL - right continuous with leftlimits) paths, built on the ltered probability space(Ω,F ,F,P), where F is the ltration fFt : t 0g. Letsassume that the stochastic process X is Fadapted and that itis integrable, i.e. EP [jXt j] < ∞. Then X is a martingale if andonly if EP [Xτ] = EP [X0] for all stopping time τ bounded, i.e.for any stopping time τ considered, there exists a constant bsuch that

8ω 2 Ω, 0 τ (ω) b.(ref. Revuz and Yor, proposition 3.5, page 67)

BrownianMotion




Other properties VIIBrownian motion

TheoremTheorem. If the martingale M = fMt : t 0g and thestopping time τ are built on the same ltered probability space(Ω,F ,F,P) then the stopped process Mτ is also a martingaleon that space. (ref. Revuz and Yor, corollaire 3.6, page 67)

Proof of the lemma. We want to use the martingale stoppingtime theorem and, in order to do so, a bounded stopping time isneeded. But the stopping time τa is not bounded; however, forall n 2 N, the stopping time τa ^ n, for its part, is bounded.

Using the martingale stopping time theoremM =

nMt = exp

hσWt σ2

2 ti

: t 0o, we obtain

E [Mτa^n ] = E [M0] = 1.

BrownianMotion




Other properties VIIIBrownian motion

Since

Mτa^n (ω) =

8<: exphσa σ2

2 τa (ω)i

if τa (ω) nexp

hσWn (ω) σ2

2 niif τa (ω) > n,

then8n 2 N, Mτa^n exp [σa] ,

therefore the sequence fMτa^n : n 2 Ng is dominated bythe constant exp [σa] .

BrownianMotion




Other properties IXBrownian motion

Moreover, for all ω 2 fω 2 Ω : τa (ω) < ∞g,

limn!∞

Mτa^n (ω) = Mτa (ω) = exp

σa σ2

2τa (ω)

while for all ω 2 fω 2 Ω : τa (ω) = ∞g and for all t 0,

Mt (ω) = exp

σWt (ω)σ2

2t exp

σa σ2

2t

which yields that, for all ω 2 fω 2 Ω : τa (ω) = ∞g,

limn!∞

Mτa^n (ω) = 0.

BrownianMotion




Other properties XBrownian motion

Lebesgues dominated convergence theorem yiields that

Eexp

σa σ2

2τa

1fτa<∞g

= E

Mτa1fτa<∞g

= E

2664 limn!∞Mτa^n1fτa<∞g + lim

n!∞Mτa^n1fτa=∞g| z

=0

3775= E

hlimn!∞

Mτa^ni

= limn!∞

E [Mτa^n ]

= 1.

BrownianMotion




Other properties XIBrownian motion

As a consequence,

Eexp

σ2

2τa

1fτa<∞g

= exp [σa] .

By letting σ tend to 0, we obtain

P [τa < ∞] = E1fτa<∞g

= E

limσ!0

expσ2

2τa

1fτa<∞g

= lim

σ!0Eexp

σ2

2τa

1fτa<∞g

by dominated convergence theorem

= limσ!0

exp [σa]

= 1.

BrownianMotion




Other properties XIIBrownian motion

We also obtain, incidentally, the moment-generatingfunction E

eλτa

of τa. Indeed, if λ = σ2

2 , then

Eexp [λτa] 1fτa<∞g

= exp

hp2λa

i.

But, since exp [λτa] 1fτa=∞g = 0 almost surely,

exp [λτa] = Eexp [λτa] 1fτa<∞g

almost surely

etE [exp [λτa]] = exp

hp2λa

i.

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Other properties XIIIBrownian motion

We continue the study of Brownian motion surprisingbehavior.

TheoremLemma. The Brownian motion paths on the interval [0,T ] arenot of bounded variationa.

aSee Appendix B.

Intuitively, this latter result means that each of theBrownian motion paths on the interval [0,T ] is of innitelength.

BrownianMotion




Other properties XIVBrownian motion

TheoremLemma. The Brownian motion is recurrent.

It means that the Brownian motion visits an innite number oftimes each of its states, i.e. all real numbers.

BrownianMotion

Brownianmotion

Constructionof theBrownianmotion1stapproximation2ndapproximation3rdapproximationn thapproximationPassage to thelimit


Construction of the Brownianmotion I

Constructing a Brownian motion is to build a probabilityspace (Ω,F ,P) et a stochastic process on that space,satisfying conditions (MB1), (MB2), (MB3) and (MB4).

To simplify our task, we will construct the Brownianmotion on the interval de temps [0, 1] since, if there existsa Brownian motion on that interval, we can construct oneon any bounded time interval. Indeed, if fWt : t 2 [0, 1]gis a Brownian motion on the interval [0, 1] then 8T > 0,

W =nW t = T

12W t

T: t 2 [0,T ]

ois a Brownian motion on the interval [0,T ] .

BrownianMotion

Brownianmotion



Construction of the Brownianmotion II

We will construct the Brownian motion by successiveapproximations. Let

I (n) = fodd integers comprised between 0 and 2ng .

For example,

I (0) = f1g , I (1) = f1g , I (2) = f1, 3g , I (3) = f1, 3, 5, 7g , etc.

BrownianMotion

Brownianmotion



Construction of the Brownianmotion III

Let (Ω,F ,P) be a probability space on which there existsa sequencen

ξ(n)i : i 2 I (n) and n 2 N

o=

nξ(0)1 , ξ

(1)1 , ξ

(2)1 , ξ

(2)3 , ξ

(3)1 , ξ

(3)3 , ξ

(3)5 , ξ

(3)7 , ...

oof independent random variables, all following standardnormal law (N (0, 1)).

Starting from these variables, we will construct a sequenceof stochastic processes that approaches the Brownianmotion.

BrownianMotion

Brownianmotion



First approximation IConstruction of the Brownian motion

The rst approximation is very rough: we set

B (0)0 (ω) = 0 and B (0)1 (ω) = ξ(0)1 (ω)

and all other B (0)t (ω) are linear interpolations betweenthese two points

B (0)t (ω) =

8><>:0 si t = 0

ξ(0)1 (ω) t si 0 < t < 1

ξ(0)1 (ω) si t = 1

.

BrownianMotion

Brownianmotion



First approximation IIConstruction of the Brownian motion

Note that the graph below represents a single path of theprocess B (0). Since it is possible that the random variableξ(0)1 takes negative values, then it is also possible that ourrst approximation has paths with negative slopes.

A path of the rst approximation t ! B (0)t (ω)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

t

B0

BrownianMotion

Brownianmotion



Second approximation IConstruction of the Brownian motion

The second approximation is constructed based on the rstone. Both ends remain xed,

B (1)0 (ω) = B (0)0 (ω) = 0 and B (1)1 (ω) = B (0)1 (ω) = ξ(0)1 (ω) ,

while the midpoint is moved:

B (1)12(ω) =

12

B (0)0 (ω) + B (0)1 (ω)

+12

ξ11 (ω) .

The other points on the path are obtained from linearinterpolation.

B (1)t (ω) =

8>>><>>>:B (0)0 (ω) if t = 0

12

B (0)0 (ω) + B (0)1 (ω)

+ 1

2 ξ(1)1 (ω) if t = 1

2

B (0)1 (ω) if t = 1

BrownianMotion

Brownianmotion



Second approximation IIConstruction of the Brownian motion

B (1)t (ω) =

8>>><>>>:B (0)0 (ω) if t = 0

12

B (0)0 (ω) + B (0)1 (ω)

+ 1

2 ξ(1)1 (ω) if t = 1

2

B (0)1 (ω) if t = 1

A path of the second approximation t ! B (1)t (ω)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

t

B1

BrownianMotion

Brownianmotion



Second approximation IIIConstruction of the Brownian motion

Note that

B (1)0 = B (0)0 = 0

B (1)12

=

12

B (0)0 + B (0)1

+12

ξ(1)1

=

12

0+ ξ

(0)1

+12

ξ(1)1

=

12

ξ(0)1 + ξ

(1)1

N

0,14(1+ 1)

= N

0,12

B (1)1 = B (0)1 = ξ

(0)1 N (0, 1)

BrownianMotion

Brownianmotion



Second approximation IVConstruction of the Brownian motion

implies that

B (1)1 B (1)12

= B (0)1 12

B (0)0 + B (0)1

+12

ξ(1)1

= ξ

(0)1

12

0+ ξ

(0)1

+12

ξ(1)1

=

12

ξ(0)1 ξ

(1)1

N

0,12

B (1)12 B (1)0 =

12

0+ ξ

(0)1

+12

ξ(1)1

B (0)0

=12

ξ(0)1 + ξ

(1)1

N

0,12

BrownianMotion

Brownianmotion



Second approximation VConstruction of the Brownian motion

and both these random variables are independent since they areGaussian, and

CovhB (1)1 B (1)1

2,B (1)1

2 B (1)0

i= Cov

12

ξ(0)1 ξ

(1)1

,12

ξ(0)1 + ξ

(1)1

=14

0@ Covhξ(0)1 , ξ

(0)1

i+Cov

hξ(0)1 , ξ

(1)1

iCov

hξ(1)1 , ξ

(0)1

iCov

hξ(1)1 , ξ

(1)1

i 1A=

14

Var

hξ(0)1

i+ 0 0Var

hξ(1)1

i=

14(1+ 0 0 1) = 0.

BrownianMotion

Brownianmotion



Third approximation IConstruction of the Brownian motion

The third approximation is obtained from the second one :

B (2)t (ω) =

8>>>>>>>>>><>>>>>>>>>>:

B (1)0 (ω) if t = 012

B (1)0 (ω) + B (1)1

2(ω)

+ 1

232

ξ(2)1 (ω) if t = 1

4

B (1)12(ω) if t = 1

2

12

B (1)12(ω) + B (1)1 (ω)

+ 1

232

ξ(2)3 (ω) if t = 3

4

B (1)1 (ω) if t = 1

BrownianMotion

Brownianmotion



Third approximation IIConstruction of the Brownian motion

A path of the third approximation t ! B (2)t (ω)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

t

B2

BrownianMotion

Brownianmotion



Third approximation IIIConstruction of the Brownian motion

Note that

B (2)0 = B (1)0 (ω) = 0

B (2)14

=12

B (1)0 + B (1)1

2

+1

232

ξ(2)1

=12

0+

12

ξ(0)1 + ξ

(1)1

+1

232

ξ(2)1

=14

ξ(0)1 +

14

ξ(1)1 +

1

232

ξ(2)1

N0,116+116+18

= N

0,14

BrownianMotion

Brownianmotion



Third approximation IVConstruction of the Brownian motion

B (2)12

= B (1)12=12

ξ(0)1 + ξ

(1)1

N

0,12

B (2)34

=12

B (1)12+ B (1)1

+1

232

ξ(2)3

=12

12

ξ(0)1 + ξ

(1)1

+ ξ

(0)1

+1

232

ξ(2)3

=34

ξ(0)1 +

14

ξ(1)1 +

1

232

ξ(2)3

N0,916+116+18

= N

0,34

B (2)1 = B (1)1 = ξ

(0)1 N (0, 1)

BrownianMotion

Brownianmotion



Third approximation VConstruction of the Brownian motion

implies

B (2)1 B (2)34

= ξ(0)1

34

ξ(0)1 +

14

ξ(1)1 +

1

232

ξ(2)3

=

14

ξ(0)1 1

4ξ(1)1 1

232

ξ(2)3

N0,116+116+18

= N

0,14

B (2)34 B (2)1

2=

34

ξ(0)1 +

14

ξ(1)1 +

1

232

ξ(2)3

12

ξ(0)1 + ξ

(1)1

=

14

ξ(0)1 1

4ξ(1)1 +

1

232

ξ(2)3

N0,116+116+18

= N

0,14

BrownianMotion

Brownianmotion



Third approximation VIConstruction of the Brownian motion

B (2)12 B (2)1

4=

12

ξ(0)1 + ξ

(1)1

14

ξ(0)1 +

14

ξ(1)1 +

1

232

ξ(2)1

=

14

ξ(0)1 +

14

ξ(1)1 1

232

ξ(2)1

N0,116+116+18

= N

0,14

B (2)14 B (2)0 = B (2)1

4 N

0,14

BrownianMotion

Brownianmotion



Third approximation VIIConstruction of the Brownian motion

and these four random variables are mutually independent since

CovhB (2)1 B (2)3

4,B (2)3

4 B (2)1

2

i= Cov

"ξ(0)1

4 ξ

(1)1

4 ξ

(2)3

232,

ξ(0)1

4 ξ

(1)1

4+

ξ(2)3

232

#=

116

Varhξ(0)1

i+116

Varhξ(1)1

i 18

Varhξ(2)3

i= 0

CovhB (2)1 B (2)3

4,B (2)1

2 B (2)1

4

i= Cov

"ξ(0)1

4 ξ

(1)1

4 ξ

(2)3

232,

ξ(0)1

4+

ξ(1)1

4 ξ

(2)1

232

#=

116

Varhξ(0)1

i 116

Varhξ(1)1

i= 0

BrownianMotion

Brownianmotion



Third approximation VIIIConstruction of the Brownian motion

CovhB (2)1 B (2)3

4,B (2)1

4 B (2)0

i= Cov

"ξ(0)1

4 ξ

(1)1

4 ξ

(2)3

232,

ξ(0)1

4+

ξ(1)1

4+

ξ(2)1

232

#=

116

Varhξ(0)1

i 116

Varhξ(1)1

i= 0

CovhB (2)34 B (2)1

2,B (2)1

2 B (2)1

4

i= Cov

"ξ(0)1

4 ξ

(1)1

4+

ξ(2)3

232,

ξ(0)1

4+

ξ(1)1

4 ξ

(2)1

232

#=

116

Varhξ(0)1

i 116

Varhξ(1)1

i= 0

BrownianMotion

Brownianmotion



Third approximation IXConstruction of the Brownian motion

CovhB (2)34 B (2)1

2,B (2)1

4 B (2)0

i= Cov

"ξ(0)1

4 ξ

(1)1

4+

ξ(2)3

232,

ξ(0)1

4+

ξ(1)1

4+

ξ(2)1

232

#=

116

Varhξ(0)1

i 116

Varhξ(1)1

i= 0

CovhB (2)12 B (2)1

4,B (2)1

4 B (2)0

i= Cov

"ξ(0)1

4+

ξ(1)1

4 ξ

(2)1

232,

ξ(0)1

4+

ξ(1)1

4+

ξ(2)1

232

#=

116

Varhξ(0)1

i+116

Varhξ(1)1

i 18

Varhξ(2)1

i= 0.

BrownianMotion

Brownianmotion



n th approximationConstruction of the Brownian motion

In order to be able to express the n th approximation, weintroduce, for all natural integer n and for all k 2 I (n) theHaar function, Hnk : [0, 1]! R, and the Schauder function,Snk : [0, 1]! R :

BrownianMotion

Brownianmotion



Haar functionn th approximation

8t 2 [0, 1] ,

H (0)1 (t) = 1

et

8n 2 N, 8k 2 I (n) , 8t 2 [0, 1] ,

H (n)k (t) =

8><>:2n12 if k12n t < k

2n

2 n12 if k2n t < k+1

2n

0 otherwise

BrownianMotion

Brownianmotion



Schauder function In th approximation

8n 2 N[ f0g , 8k 2 I (n) , 8t 2 [0, 1] ,

S (n)k (t) =Z t

0H (n)k (s) ds.

Schauder functionS (0)1

0.0 0.5 1.00

1

Schauder function S (1)1

0.0 0.5 1.00.0

0.5

BrownianMotion

Brownianmotion



Schauder function IIn th approximation

Schauder function S (2)1 Schauder function S (2)3

0.0 0.5 1.00.0

0.3

0.0 0.5 1.00.0

0.3

BrownianMotion

Brownianmotion



Schauder function IIIn th approximation


0.0 0.5 1.00.0

0.2

0.0 0.5 1.00.0

0.2


0.0 0.5 1.00.0

0.2

0.0 0.5 1.00.0

0.2

Note that such functions are deterministic, they are notrandom.

BrownianMotion

Brownianmotion



n th approximation

The n+ 1 th stochastic process in the approximation sequenceapproaching the Brownian motion can be written as

B (n)t (ω) =n

∑k=0

∑j2I (k )

S (k )j (t) ξ(k )j (ω) .

If B (n)t (ω) is to converge, it will converge to

Bt (ω) =∞

∑k=0

∑j2I (k )

S (k )j (t) ξ(k )j (ω) .

We claim that for most ω that limit exists2 and that theprocess fBt : 0 t 1g thus obtained is a Brownian motion.

2Which is to say that there exists a subset A Ω such that P (A) = 1and that 8ω 2 A, the limit exists. We then say that the limit existsPalmost surely.

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Brownianmotion



Passage to the limit I

Such a construction allows us to grasp the erratic nature of theBrownian motion paths. We will not complete the constructionin detail since some steps require a knowledge of measuretheory that is beyond this course objectives. We will simplyoutline each of the steps in the proof :

(i) By construction, we have that 8ω 2 Ω,B0 (ω) = limn!∞ B

(n)0 (ω) = 0, so condition (MB1) is

satised.(ii) First, it must be shown that, for most ω, the series

n

∑k=0

∑j2I (k )

S (k )j (t) ξ(k )j (ω)

converges uniformly on the interval [0, 1] when n tends toinnity (see Karatzas and Shreve, 1988, Lemma 3.1, page 57).

BrownianMotion

Brownianmotion



Convergence uniforme IPassage to the limit

Recall that a sequence of functions ffn : R ! R jn 2 Ngconverges to another function f : R ! R at the pointx 2 R if

limn!∞

jfn (x) f (x)j = 0

while that very sequence converges uniformly on theinterval [a, b] to that function f if

limn!∞

supaxb

jfn (x) f (x)j = 0.

BrownianMotion

Brownianmotion



Convergence uniforme IIPassage to the limit

The reason why we need the B (n)t (ω) to convergeuniformly to Bt (ω) is that we want to preserve pathcontinuity. Indeed, we have constructed our approximationsequence in such a way that, for each ω, the patht ! B (n)t (ω) is continue. But it is possible that asequence of continuous functions converges, pointwise, toa function that is not continuous.

BrownianMotion

Brownianmotion



Convergence uniforme IIIPassage to the limit

Example. For all natural integer n, the function fn : [0, 1]! R

dened as

fn (t) =

8<:0 if 0 t 1

2 12n

nt n2 +

12 if 12

12n < t <

12 +

12n

1 if 12 +12n t 1

is continuous.But the limit of the sequence of fn is not a continuous function:

limn!∞

fn (t) = f (t) =

8<:0 si 0 t < 1

212 si t = 1

21 si 12 < t 1

.

Such a sequence does not converge uniformly to f . Indeed,

limn!∞

sup0x1

jfn (x) f (x)j = limn!∞

12=126= 0.

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Brownianmotion



Convergence uniforme IVPassage to the limit

By contrast, if a sequence of continuous functions convergesuniformly, the limit will also be a continuous function. Thus,with this result, we will obtain that the limit exists and such alimit process satises condition (MB4) about path continuity.

BrownianMotion

Brownianmotion



Passage to the limit I

(iii) It is possible to show by recurrence on n that

8n 2 N[ f0g , 8k 2 I (n) , B k2n B k1

2n N

0,12n

and that 8n 2 N, the setnB k2n B k1

2njk 2 I (n)

ois made of

independent random variables.Now, for all real numbers 0 r < s < t < u 1, we canconstruct decreasing sequences of real numbers frn jn 2 Ng,fsn jn 2 Ng, ftn jn 2 Ng and fun jn 2 Ng such that

limn!∞

rn = r , limn!∞

sn = s, limn!∞

tn = t, limn!∞

un = u

BrownianMotion

Brownianmotion



Passage to the limit IIet rn, sn, tn, un 2

0, 12n , ...,

2n12n , 1

. Bt path continuity then

implies that

limn!∞

Brn (ω) = Br (ω) , limn!∞

Bsn (ω) = Bs (ω) ,

limn!∞

Btn (ω) = Bt (ω) and limn!∞

Bun (ω) = Bu (ω)

hence

Bs (ω) Br (ω) = limn!∞

[Bsn (ω) Brn (ω)] ,Bt (ω) Bs (ω) = lim

n!∞[Btn (ω) Bsn (ω)] ,

Bu (ω) Bt (ω) = limn!∞

[Bun (ω) Btn (ω)] .

We can then use the results established for Bsn (ω) Brn (ω),Btn (ω) Bsn (ω) and Bun (ω) Btn (ω) in order to verifyconditions (MB2) and (MB3).

BrownianMotion

Brownianmotion



Random walks I

Lets construct, for all m 2 N, a sequence of independentand identically distributed random variablesξ(m) =

nξ(m)k : k 2 N

osuch that

ξ(m)k =

8><>:m 1

2 with probability 12

m12 with probability 1

2

, k 2 N.

Note that

Ehξ(m)k

i= 0 and Var

hξ(m)k

i=1m.

BrownianMotion

Brownianmotion



Random walks II

For all m 2 N, lets set

X (m)0 = 0 and 8n 20,1m,2m, ...

, X (m)n =

mn

∑k=1

ξ(m)k .

The process X (m) =nX (m)n : n 2

0, 1m ,

2m , ...

ois a

random walk. As m increases, we take shorter and shortersteps (of length m

12 ), as well as faster and faster (every

1/m time unit). Note that

EhX (m)n

i=

mn

∑k=1

Ehξ(m)k

i= 0

et VarhX (m)n

i=

mn

∑k=1

Varhξ(m)k

i=mnm= n.

BrownianMotion

Brownianmotion



Random walks III

Moreover, the central limit theorem implies that X (m)n

converges in law to a N (0, n) distribution as m increasesto innity.

Lets set

Y (m)0 = 0 and Y (m)t = X (m)nm

pour tout t 2nm,n+ 1m

.

The process Y (m) is dened for all t 0. It is possible toshow that this sequence

nY (m) : m 2 N

oof stochastic

processes obtained from random walks converges in law toa Brownian motion process as m tends to innity.

BrownianMotion

Brownianmotion



References I

ANDERSON, T.W. (1981). An Introduction toMultivariate Statistical Analysis, second edition, Wiley,New-York.

BAXTER, Martin and RENNIE, Andrew (1996). FinancialCalculus : An Introduction to Derivative Pricing,Cambridge University Press, New York.

DURRETT, Richard (1996). Stochastic Calculus, APractical Introduction, CRC Press, New York.

KARATZAS, Ioannis and SHREVE, Steven E. (1988).Brownian Motion and Stochastic Calculus,Springer-Verlag, New York.

KARLIN, Samuel and TAYLOR, Howard M. (1975). AFirst Course in Stochastic Processes, second edition,Academic Press, New York.

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Brownianmotion



References II

LAMBERTON, Damien and LAPEYRE, Bernard (1991).Introduction au calcul stochastique appliqué à la nance,Éllipses, Paris.

REVUZ, Daniel and YOR, Marc (1991). ContinuousMartingale and Brownian Motion, Springer-Verlag, NewYork.

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