80-646-08StochasticcalculusGeneviŁveGauthier TheBrownianmotion

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Motion

Brownian motion Construction of the Brownian motion An alternative construction

The Brownian motion

80-646-08 Stochastic calculus

Geneviève Gauthier

HEC Montréal

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Motion

Brownian motion

De…nition Refresher:

normal law Properties Other properties Construction of the Brownian motion An alternative construction

Introduction

Brownian motion

In 1827, Robert Brown observed that small particles suspended in a drop of water exhibit ceaseless irregular motions. Historically, Brownian motion was at …rst an attempt to model such a phenomenon. Today, the Brownian motion process occurs in diverse areas such as economics, communication theory, biology, management science and mathematics. (Adapted from the introductory paragraph about Brownian motion, S. Karlin and H. M.

Taylor (1975).)

The mathematician Norbert Wiener is credited with

rigorously analyzing the mathematics related to Brownian

motion, and that is why such a process is also known as a

Wiener process.

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Motion

Brownian motion

De…nition I

Brownian motion

Let ( Ω , F ^, ^F ^, ^P ) be a …ltered probability space.

Technical condition. Since we will work with almost-sure equalities ¹ , we require that the set of events which have a zero-probability to occur be included in the sigma-algebra F ⁰ , which is to say that the set

N = f ^A 2 F ^: ^P ( A ) = 0 g F ⁰ ^. Thus, if X is F ^t measurable and Y = X

P almost-surely, then we know that Y is F ^t measurable.

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Motion

Brownian motion

De…nition II

Brownian motion

De…nition

A standard Brownian motion f ^W ^t ^: ^t ⁰ g is an adapted stochastic process, built on a …ltered probability space ( Ω , F ^, F , P ) such that:

(MB1) 8 ^ω 2 ^Ω ^, ^W ⁰ ( ω ) = 0,

(MB2) 8 ⁰ ^t ⁰ ^t ¹ ^... ^t ^k ^, the random variables W t

1

W t

0

, W t

2

W t

1

, ..., W t

_k

W t

_k ₁

are independent, (MB3) 8 ^s ^, ^t 0 such that s < t, the random variable W _t W _s is normally distributed with expectation 0 and variance t s i.e. W t W s N ( 0, t s ) ,

(MB4) 8 ^ω 2 ^Ω ^, ^{the path} ^t ! ^W ^t ( ω ) is continuous.

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Motion

Brownian motion

De…nition III

Brownian motion

In general, the …ltration we use is F = fF ^t ^: ^t ⁰ g ^where F ^t = σ ff ^W ^s ^: ⁰ ^s ^t g [ N g

is the smallest sigma-algebra for which the random

variables W _s : 0 s t are measurable and that contains the zero-measure sets.

1

X = Y P almost-surely if the set of ω for which X di¤ers from Y has a zero probability, i.e.

P f ^ω 2 ^Ω ^: ^X ( ω ) 6 = Y ( ω ) g = 0.

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Motion

Brownian motion

Reminder about normal law I

If X is a random variable following a normal law with expectation µ and standard deviation σ > 0, then its probability density function is

f _X ( x ) = ¹ σ p

2π exp

( ( x µ ) ² 2σ ²

) ,

which allows us to determine 8 ^a, ^b 2 R , a < b, P [ a < X b ] =

Z b

a

f _X ( x ) dx.

Unfortunately, the integral above has no primitive. So we need to value it numerically. The cumulative distribution function of X is

F _X ( x ) =

Z x

∞

f _X ( y ) dy .

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Motion

Brownian motion

Reminder about normal law II

In general, if two random variables X and Y , built on the same probability space, are independent, then their covariance

Cov [ X , Y ] = E [ XY ] E [ X ] E [ Y ] is nil.

However, it is possible that two variables have zero

covariance, but that they are not independent. Here is an

illustrative example:

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Motion

Brownian motion

Reminder about normal law III

Example

ω X ( ω ) Y ( ω ) X ( ω ) Y ( ω ) _P ( ω )

ω 1 0 1 0 ¹ ₄

ω 2 0 1 0 ¹ ₄

ω 3 1 0 0 ¹ ₄

ω 4 1 0 0 ¹ ₄

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Motion

Brownian motion

Reminder about normal law IV

The covariance between these two variables is zero since:

E

^P

[ X ] = 0, E

^P

[ Y ] = 0, E

^P

[ XY ] = 0 ) ^Cov

^P

[ X , Y ] = E

^P

[ XY ] E

^P

[ X ] E

^P

[ Y ] = 0 but they are dependent since

P [ X = 0 and Y = 0 ] = 0 6 = ¹

4 = _P [ X = 0 ] _P [ Y = 0 ] . However, when variables are normally distributed (not necessarily with the same expectation and standard

deviation), there is a result that allows us to verify whether

such variables are independent by using covariance :

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Motion

Brownian motion

Reminder about normal law V

Theorem

Proposition. If X and Y are two random variables following a

multivariate normale distribution, both built on the same

probability space, then X and Y are independent if and only if

their covariance is zero (T. W. Anderson, 1984, Theorem 2.4.4,

page 28).

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Motion

Brownian motion

Properties I

The Brownian motion

Theorem

Lemma 1. Let f ^W ^t ^: ^t ⁰ g be a standard Brownian motion.

Then

(i) For all s > 0, f ^W ^t ⁺ ^s ^W ^s ^: ^t ⁰ g (time homogeneity) (ii) f ^W ^t ^: ^t ⁰ g ^(symmetry)

(iii) n cW

^t

c2

: t 0 o

(time rescaling) (iv) n

W _t = tW

1

t

1 _t > 0 : t 0 o

(time inversion) are also standard Brownian motions.

Exercise. Verify it.

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Motion

Brownian motion

Properties II

The Brownian motion

Theorem

Lemma 2. Brownian motion is a martingale.

De…nition

De…nition. On the …ltered probability space ( _Ω , F ^, F , P ) , where F is the …ltration fF ^t ^: ^t ⁰ g , the stochastic process M = f ^M ^t ^: ^t ⁰ g ^{is a} martingale in continuous time if

( M 1 ) 8 ^t ^0, ^E

^P

[ j ^M ^t j ] < _∞ ; ( M 2 ) 8 ^t ^0, ^M ^t ^is F ^t measurable;

( M 3 ) 8 ^s ^, ^t ^{0 tel que} ^s < t, E

^P

[ M _t jF ^s ] = M _s .

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Motion

Brownian motion

Proof of Lemma 2 I

Brownian motion properties

From the very de…nition of the …ltration, it is obvious that W is an adapted stochastic process.

For all time t, the random variable W _t is integrable since E

^P

[ j ^W ^t j ] =

Z

_∞

∞

j ^z j

p 2πt exp z ² 2t dz

= 2 Z

_∞

0 p z

2πt exp z ² 2t dz

=

r 2t

π exp z ² 2t

∞

0 = r 2t

π < _∞ .

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Motion

Brownian motion

Proof of Lemma 2 II

Brownian motion properties

The only thing left to verify is that 8 ^s, ^t 0 such that s < t, E

^P

[ W t jF ^s ] = W s .

E

^P

[ W _t jF ^s ] = E

^P

[ W _t W _s + W _s jF ^s ]

= E

^P

[ W t W s jF ^s ] + E

^P

[ W s jF ^s ]

= E

^P

[ W t W s ] + W s from ( MB2 )

= W _s from ( MB 3 )

The proof is complete.

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Motion

Brownian motion

Properties I

Brownian motion

Theorem

Lemma 3. The Brownian motion is a Markov process.

Idea of the proof of Lemma 3 : For all u 2 [ 0, s ] , the random variables W _t W _s and W _u are independent since

Cov [ W t W s , W u ]

= Cov [ W t W u + W u W s , W u ]

= Cov [ W _t W _u , W _u ] Cov [ W _s W _u , W _u ]

= 0 + 0 par ( MB 2 ) .

As a consequence, W

t

= ( W

t

W

s

) + W

s

can be written as the sum of

two random variables: W

s

which depends on information available at time

s, F

^s

, only through σ ( W

s

) and W

t

W

s

which is independent from

F

^s

= σ f ^W

^u

^: ⁰ ^u ^s g ^.

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Motion

Brownian motion

Properties II

Brownian motion

Property 3. 8 ^ω 2 ^Ω ^, ^{the path t} ! ^W ^t ( ω ) is nowhere di¤erentiable.

Such a property is well illustrated by the construction of

Brownian motion.

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Motion

Brownian motion

Multidimensional Brownian motion I

De…nition

Standard Brownian motion W of dimension n is a family of random vectors

W _t = W _t ⁽ ¹ ⁾ , ..., W _t ⁽ ⁿ ⁾ ^> : t 0

where W ⁽ ¹ ⁾ , ..., W ⁽ ⁿ ⁾ represent independent Brownian motions

built on the …ltered probability space ( Ω , F ^, ^F ^, ^P ) .

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Motion

Brownian motion

Multidimensional Brownian motion II

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

For example, when modeling simultaneously several risky asset prices.

However, the shocks received by such risky assets should not be independent.

That is why we would like to build a multidimensional

Brownian motion, the components of which are correlated.

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Motion

Brownian motion

Multidimensional Brownian motion III

Starting from a standard Brownian motion W of

dimension n, it is possible to create a Brownian motion of

dimension n, the components of which are correlated.

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Motion

Brownian motion

Multidimensional Brownian motion IV

Theorem Γ = γ _{ij i}

_,j

2f ^1,2,...,n g is a matrix of constants and

W = W ⁽ ¹ ⁾ , ..., W ⁽ ⁿ ⁾ ^> is a vector made up of independent Brownian motions.

For all t, let’s set B t = _Γ W t .

Then B _t is un random vector of dimension n, the i th component of which is B _t ⁽ ⁱ ⁾ = _∑ ⁿ _k ₌ ₁ γ _ik W _t ⁽ ^k ⁾ . Moreover

Cov h

B _t ⁽ ⁱ ⁾ , B _t ⁽ ^j ⁾ i

= t

∑ n k = 1

γ _ik γ _jk

et Cor h

B _t ⁽ ⁱ ⁾ , B _t ⁽ ^j ⁾ i

= ^∑

n

k = 1 γ _ik γ _jk q ∑ ⁿ k = 1 γ ² _ik

q ∑ ⁿ k = 1 γ ² _jk

.

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Motion

Brownian motion

Proof I

Multidimensional Brownian motion

Cov h

B _t ⁽ ⁱ ⁾ , B _t ⁽ ^j ⁾ i

= Cov

" _n

k ∑ = 1

γ _ik W _t ⁽ ^k ⁾ ,

∑ n k = 1

γ _jk W _t ⁽ ^k ⁾

#

=

∑ n k = 1

γ _ik γ _jk Cov h

W _t ⁽ ^k ⁾ , W _t ⁽ ^k ⁾ i

=

∑ n k = 1

γ _ik γ _jk Cov h

W _t ⁽ ^k ⁾ , W _t ⁽ ^k ⁾ i

car Cov h

W _t ⁽ ^k ⁾ , W _t ⁽ ^k ⁾ i

= 0 if k 6 = k

= t

∑ n k = 1

γ _ik γ _jk

car Cov h

W _t ⁽ ^k ⁾ , W _t ⁽ ^k ⁾ i

= Var h W _t ⁽ ^k ⁾ i

= t

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Motion

Brownian motion

Proof II

Multidimensional Brownian motion

Var h B _t ⁽ ⁱ ⁾ i

= Cov h

B _t ⁽ ⁱ ⁾ , B _t ⁽ ⁱ ⁾ i

= t

∑ n k = 1

γ ² _ik

Cor h

B _t ⁽ ⁱ ⁾ , B _t ⁽ ^j ⁾ i

=

Cov h

B _t ⁽ ⁱ ⁾ , B _t ⁽ ^j ⁾ i r

Var h

B _t ⁽ ⁱ ⁾ ir Var h

B _t ⁽ ^j ⁾ i

= ^t ^∑

n

k = 1 γ _ik γ _jk q

t ∑ ⁿ k = 1 γ ² _ik q

t ∑ ⁿ k = 1 γ ² _jk

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Motion

Brownian motion

Proof III

Multidimensional Brownian motion

We have just shown that it is possible to build a Brownian motion B, the components of which are correlated,

starting from a standard Brownian motion W (the elements of which are independent). More precisely, if B = Γ W, then we know how to …nd the correlation matrix of B.

Can we follow the other way around, i.e., if we know the

correlation matrix of B, can we determine the matrix Γ

allowing us to express the components of B as a linear

combination of independent Brownian motions ?

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Motion

Brownian motion

Proof IV

Multidimensional Brownian motion

Theorem

Let’s now assume that B ⁽ ¹ ⁾ , ..., B ⁽ ⁿ ⁾ represent correlated Brownian motions, built on the …ltered probability space ( _Ω , F ^, F , P ) and that

8 ^i, ^j 2 f ^{1, ...,} ⁿ g ^and 8 ^t ^0, ^Cor ^h ^B t ⁽ ⁱ ⁾ , B _t ⁽ ^j ⁾ i

= ρ _ij .

There exists a matrix A of format n n such that

( i ) B = AW

( ii ) Cor h

B _t ⁽ ⁱ ⁾ , B _t ⁽ ^j ⁾ i

= ρ _ij

( iii ) W is made of independent Brownian motions.

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Motion

Brownian motion

Proof V

Multidimensional Brownian motion

Proof. Let V

_B

= t h ρ _ij

i

i,j = 1,...,n be the variance-covariance matrix of random vector B _t ⁽ ¹ ⁾ , ..., B _t ⁽ ⁿ ⁾ . Since B = AW then

V

_B

= AtIA ^> = tAA ^>

where I represents the identity matrix of dimension n.

Since a variance-covariance matrix is a symmetric positive de…nite matrix, there exists an invertible upper triangular matrix U such that V

B

= U ^> U (Cholesky decomposition).

(Several software packages, including MATLAB, have a function to calculate such a matrix).

As a consequence, one only has to set A = p ¹

t U ^> .

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Motion

Brownian motion

Other properties I

Brownian motion

Let a > 0. Let’s de…ne

τ

a

( ω ) = 8 <

:

inf f ^s ⁰ ^: ^W

^s

( ω ) = a g ^if f ^s ⁰ ^: ^W

^s

( ω ) = a g 6 = ?

∞ if f ^s ⁰ ^: ^W

^s

( ω ) = a g = ? ,

the …rst time when Brownian motion W reaches point a.

The next two results are intended to show that the

Brownian motion will eventually reach, with probability 1,

any real number, however large.

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Motion

Brownian motion

Other properties II

Brownian motion

Theorem

Lemma. The random variable τ a is a stopping time.

De…nition

Let ( Ω , F ) be a measurable space equipped with the …ltration F = fF ^t ^: ^t ⁰ g ^{. A} stopping time τ is a function of Ω into [ 0, ∞ ] F measurable such that

f ^ω 2 Ω : τ ( ω ) t g 2 F ^t ^.

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Motion

Brownian motion

Other properties III

Brownian motion

Proof of the lemma. We must show that for all t 0, the event f ^ω 2 Ω : τ a t g belongs to the sigma-algebra F ^t ^. If Q represents the set of all rational numbers, then

f ^ω 2 ^Ω ^: ^τ ^a ^t g

= ω 2 ^Ω ^: ^sup

0 s t

W _s ( ω ) a

=

\

∞

n = 1

ω 2 Ω : sup

0 s t

W s ( ω ) > a 1 n

=

\

∞

n = 1

[

r 2

Q

\ [ 0,t ]

ω 2 ^Ω ^: ^W ^r ( ω ) > a 1

| {z n }

2F

^r

^therefore 2F

^t

| {z }

2F

^t

2 F ^t

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Motion

Brownian motion

Other properties IV

Brownian motion

where the last equality is obtained from the fact that

sup ₀ _s _t W _s ( ω ) > a ¹ _n if and only if there exists at least one rational number r smaller than or equal to t for which

W r ( ω ) > a ¹ _n .

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Motion

Brownian motion

Other properties V

Brownian motion

Theorem

Lemma . The stopping time τ a is …nite almost surely, i.e.

P [ τ a = _∞ ] = 0.

Proof of the lemma. We want to use the martingale stopping

time theorem...

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Motion

Brownian motion

Other properties VI

Brownian motion

Theorem

Optional Stopping Theorem . Let X = f ^X ^t ^: ^t ⁰ g ^{be a} process with càdlàg (or RCLL - right continuous with left limits) paths, built on the …ltered probability space

( Ω , F ^, ^F ^, ^P ) , where F is the …ltration fF ^t ^: ^t ⁰ g ^{. Let’s} assume that the stochastic process X is F adapted and that it is integrable, i.e. E

^P

[ j ^X ^t j ] < _∞ . Then X is a martingale if and only if E

^P

[ X

_τ

] = E

^P

[ X ₀ ] for all stopping time τ bounded, i.e.

for any stopping time τ considered, there exists a constant b such that

8 ^ω 2 ^Ω ^, ⁰ ^τ ( ω ) b.

(ref. Revuz and Yor, proposition 3.5, page 67)

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Motion

Brownian motion

Other properties VII

Brownian motion

Theorem

Theorem. If the martingale M = f ^M ^t ^: ^t ⁰ g ^{and the} stopping time τ are built on the same …ltered probability space ( _Ω , F ^, F , P ) then the stopped process M

^τ

is also a martingale on that space. (ref. Revuz and Yor, corollaire 3.6, page 67)

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

Using the martingale stopping time theorem M = ⁿ M _t = exp h

σW _t

^σ

₂

²

t i

: t 0 o

, we obtain

E [ M

_τ_a

_^ n ] = E [ M 0 ] = 1.

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Motion

Brownian motion

Other properties VIII

Brownian motion

Since

M

_τ_a

_^ _n ( ω ) = 8 <

: exp h

σa

^σ

₂

²

τ a ( ω ) ⁱ if τ a ( ω ) n exp h

σW _n ( ω )

^σ

₂

²

n i

if τ a ( ω ) > n, then

8 ⁿ 2 ^N ^, ^M

τa

^ ⁿ exp [ σa ] ,

therefore the sequence f ^M

^τa

^ ⁿ : n 2 ^N g is dominated by

the constant exp [ σa ] .

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Motion

Brownian motion

Other properties IX

Brownian motion

Moreover, for all ω 2 f ^ω 2 Ω : τ a ( ω ) < _∞ g ^,

n lim !

∞

M

_τ_a

_^ n ( ω ) = M

_τ_a

( ω ) = exp σa σ ² 2 τ a ( ω ) while for all ω 2 f ^ω 2 Ω : τ a ( ω ) = ∞ g and for all t 0,

M t ( ω ) = exp σW t ( ω ) ^σ

2 2 t exp σa σ ² 2 t which yields that, for all ω 2 f ^ω 2 Ω : τ a ( ω ) = ∞ g ^,

n lim !

∞

M

_τ_a

_^ _n ( ω ) = 0.

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Motion

Brownian motion

Other properties X

Brownian motion

Lebesgue’s dominated convergence theorem yiields that E exp σa σ ²

2 τ a 1 _f

_τ_a

_<

_∞

_g

= E M

_τ_a

1 _f

_τ_a

_<

_∞

_g

= E 2 6 6 4 lim

n !

∞

M

_τ_a

_^ _n 1 _f

_τ_a

_<

_∞

_g + lim

n !

∞

M

_τ_a

_^ _n 1 _f

_τ_a

₌

_∞

_g

| {z }

= 0

3 7 7 5

= E h

n lim !

∞

M

_τ_a

_^ _n i

= lim

n !

∞

E [ M

_τ_a

_^ _n ]

= 1.

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Motion

Brownian motion

Other properties XI

Brownian motion

As a consequence, E exp σ ²

2 τ a 1 _f

_τ_a

_<

_∞

_g = exp [ σa ] . By letting σ tend to 0, we obtain

P [ τ a < ∞ ] = E 1 _f

_τ_a

_<

_∞

_g

= E lim

σ

! ⁰ exp σ ²

2 τ a 1 _f

_τ_a

_<

_∞

_g

= lim

σ

! ⁰ E exp σ ²

2 τ a 1 _f

_τ_a

_<

_∞

_g by dominated convergence theorem

= lim

σ

! ⁰ exp [ σa ]

= 1.

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Motion

Brownian motion

Other properties XII

Brownian motion

We also obtain, incidentally, the moment-generating function E e

^λτ^a

of τ a . Indeed, if λ =

^σ

₂

²

, then

E exp [ λτ a ] 1 _f

_τ_a

_<

_∞

_g = exp h p 2λa i

.

But, since exp [ λτ a ] 1 _f

_τ_a

₌

_∞

_g = 0 almost surely, exp [ λτ a ] = E exp [ λτ a ] 1 _f

_τ_a

_<

_∞

_g almost surely et

E [ exp [ λτ a ]] = exp h p 2λa i

.

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Motion

Brownian motion

Other properties XIII

Brownian motion

We continue the study of Brownian motion surprising behavior.

Theorem

Lemma . The Brownian motion paths on the interval [ 0, T ] are not of bounded variation ^a .

a

See Appendix B.

Intuitively, this latter result means that each of the

Brownian motion paths on the interval [ 0, T ] is of in…nite

length.

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Motion

Brownian motion

Other properties XIV

Brownian motion

Theorem

Lemma. The Brownian motion is recurrent.

It means that the Brownian motion visits an in…nite number of

times each of its states, i.e. all real numbers.

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Motion

Brownian motion Construction of the Brownian motion

1st approximation 2nd approximation 3rd approximation n th approximation Passage to the limit An alternative construction

Construction of the Brownian motion I

Constructing a Brownian motion is to build a probability space ( Ω , F ^, ^P ) et a stochastic process on that space, satisfying conditions ( MB 1 ) , ( MB 2 ) , ( MB 3 ) and ( MB4 ) . To simplify our task, we will construct the Brownian motion on the interval de temps [ 0, 1 ] since, if there exists a Brownian motion on that interval, we can construct one on any bounded time interval. Indeed, if f ^W ^t ^: ^t 2 [ 0, 1 ] g is a Brownian motion on the interval [ 0, 1 ] then 8 ^T > 0,

W = ⁿ W _t = T

¹²

W

^t

T

: t 2 [ 0, T ] ^o

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

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Motion

Construction of the Brownian motion II

We will construct the Brownian motion by successive approximations. Let

I ( n ) = f odd integers comprised between 0 and 2 ⁿ g ^. For example,

I ( 0 ) = f ¹ g ^, Î ( 1 ) = f ¹ g ^, Î ( 2 ) = f ^1, ³ g ^, Î ( 3 ) = f ^1, ^3, ^5, ⁷ g ^{, etc.}

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Motion

Construction of the Brownian motion III

Let ( Ω , F ^, ^P ) be a probability space on which there exists a sequence

n

ξ ⁽ _i ⁿ ⁾ : i 2 ^I ( n ) and n 2 N o

= ⁿ ξ ⁽ ₁ ⁰ ⁾ , ξ ⁽ ₁ ¹ ⁾ , ξ ⁽ ₁ ² ⁾ , ξ ₃ ⁽ ² ⁾ , ξ ⁽ ₁ ³ ⁾ , ξ ⁽ ₃ ³ ⁾ , ξ ⁽ ₅ ³ ⁾ , ξ ⁽ ₇ ³ ⁾ , ... o of independent random variables, all following standard normal law ( N ( 0, 1 )) .

Starting from these variables, we will construct a sequence

of stochastic processes that approaches the Brownian

motion.

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Motion

First approximation I

Construction of the Brownian motion

The …rst approximation is very rough: we set B ₀ ⁽ ⁰ ⁾ ( ω ) = 0 and B ₁ ⁽ ⁰ ⁾ ( ω ) = ξ ⁽ ₁ ⁰ ⁾ ( ω ) and all other B _t ⁽ ⁰ ⁾ ( ω ) are linear interpolations between these two points

B _t ⁽ ⁰ ⁾ ( ω ) = 8 >

<

> :

0 si t = 0

ξ ₁ ⁽ ⁰ ⁾ ( ω ) t si 0 < t < 1 ξ ⁽ ₁ ⁰ ⁾ ( ω ) si t = 1

.

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Motion

First approximation II

Construction of the Brownian motion

Note that the graph below represents a single path of the process B ⁽ ⁰ ⁾ . Since it is possible that the random variable ξ ⁽ ₁ ⁰ ⁾ takes negative values, then it is also possible that our

…rst approximation has paths with negative slopes.

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

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4

t

B0

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Motion

Second approximation I

Construction of the Brownian motion

The second approximation is constructed based on the …rst one. Both ends remain …xed,

B ₀ ⁽ ¹ ⁾ ( ω ) = B ₀ ⁽ ⁰ ⁾ ( ω ) = 0 and B ₁ ⁽ ¹ ⁾ ( ω ) = B ₁ ⁽ ⁰ ⁾ ( ω ) = ξ ⁽ ₁ ⁰ ⁾ ( ω ) , while the midpoint is moved:

B ⁽

1

¹ ⁾

2

( ω ) = ¹

2 B ₀ ⁽ ⁰ ⁾ ( ω ) + B ₁ ⁽ ⁰ ⁾ ( ω ) + ¹

2 ξ ¹ ₁ ( ω ) . The other points on the path are obtained from linear interpolation.

B _t ⁽ ¹ ⁾ ( ω ) = 8 >

> >

<

> >

> :

B ₀ ⁽ ⁰ ⁾ ( ω ) if t = 0

1 2 B ₀ ⁽ ⁰ ⁾ ( ω ) + B ₁ ⁽ ⁰ ⁾ ( ω ) + ¹ ₂ ξ ⁽ ₁ ¹ ⁾ ( ω ) if t = ¹ ₂

B ₁ ⁽ ⁰ ⁾ ( ω ) if t = 1

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Motion

Second approximation II

Construction of the Brownian motion

B _t ⁽ ¹ ⁾ ( ω ) = 8 >

> >

<

> >

> :

B ₀ ⁽ ⁰ ⁾ ( ω ) if t = 0

1 2 B ₀ ⁽ ⁰ ⁾ ( ω ) + B ₁ ⁽ ⁰ ⁾ ( ω ) + ¹ ₂ ξ ⁽ ₁ ¹ ⁾ ( ω ) if t = ¹ ₂ B ₁ ⁽ ⁰ ⁾ ( ω ) if t = 1 A path of the second approximation t ! ^B t ⁽ ¹ ⁾ ( ω )

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6

t

B1

(47)

Motion

Second approximation III

Construction of the Brownian motion

Note that

B ₀ ⁽ ¹ ⁾ = B ₀ ⁽ ⁰ ⁾ = 0

B ⁽

1

¹ ⁾

2

= ¹

2 B ₀ ⁽ ⁰ ⁾ + B ₁ ⁽ ⁰ ⁾ + ¹ 2 ξ ⁽ ₁ ¹ ⁾

= ¹

2 0 + ξ ⁽ ₁ ⁰ ⁾ + ¹ 2 ξ ⁽ ₁ ¹ ⁾

= ¹

2 ξ ₁ ⁽ ⁰ ⁾ + ξ ⁽ ₁ ¹ ⁾ N 0, 1

4 ( 1 + 1 ) = N 0, 1

2 B ₁ ⁽ ¹ ⁾ = B ₁ ⁽ ⁰ ⁾ = ξ ⁽ ₁ ⁰ ⁾ N ( 0, 1 )

(48)

Motion

Second approximation IV

Construction of the Brownian motion

implies that B ₁ ⁽ ¹ ⁾ B ⁽

1

¹ ⁾

2

= B ₁ ⁽ ⁰ ⁾ 1

2 B ₀ ⁽ ⁰ ⁾ + B ₁ ⁽ ⁰ ⁾ + ¹ 2 ξ ⁽ ₁ ¹ ⁾

= ξ ⁽ ₁ ⁰ ⁾ 1

2 0 + ξ ⁽ ₁ ⁰ ⁾ + ¹ 2 ξ ⁽ ₁ ¹ ⁾

= ¹

2 ξ ⁽ ₁ ⁰ ⁾ ξ ⁽ ₁ ¹ ⁾ N 0, 1 2 B ⁽

1

¹ ⁾

2

B ₀ ⁽ ¹ ⁾ = ¹

2 0 + ξ ₁ ⁽ ⁰ ⁾ + ¹

2 ξ ⁽ ₁ ¹ ⁾ B ₀ ⁽ ⁰ ⁾

= ¹

2 ξ ⁽ ₁ ⁰ ⁾ + ξ ⁽ ₁ ¹ ⁾ N 0, 1

2

(49)

Motion

Second approximation V

Construction of the Brownian motion

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

Cov h

B ₁ ⁽ ¹ ⁾ B ⁽

₁

¹ ⁾

2

, B ⁽

₁

¹ ⁾

2

B ₀ ⁽ ¹ ⁾ i

= Cov 1

2 ξ ⁽ ₁ ⁰ ⁾ ξ ⁽ ₁ ¹ ⁾ , 1

2 ξ ₁ ⁽ ⁰ ⁾ + ξ ⁽ ₁ ¹ ⁾

= ¹ 4

0 @ Cov h

ξ ⁽ ₁ ⁰ ⁾ , ξ ⁽ ₁ ⁰ ⁾ i

+ Cov h

ξ ⁽ ₁ ⁰ ⁾ , ξ ⁽ ₁ ¹ ⁾ i

Cov h

ξ ⁽ ₁ ¹ ⁾ , ξ ⁽ ₁ ⁰ ⁾ i

Cov h

ξ ₁ ⁽ ¹ ⁾ , ξ ⁽ ₁ ¹ ⁾ i

1 A

= ¹

4 Var h ξ ⁽ ₁ ⁰ ⁾

i

+ 0 0 Var h ξ ⁽ ₁ ¹ ⁾

i

= ¹

4 ( 1 + 0 0 1 ) = 0.

(50)

Motion

Third approximation I

Construction of the Brownian motion

The third approximation is obtained from the second one :

B _t ⁽ ² ⁾ ( ω ) = 8 >

> >

> <

> >

:

B ₀ ⁽ ¹ ⁾ ( ω ) if t = 0

1 2 B ₀ ⁽ ¹ ⁾ ( ω ) + B ⁽

1

¹ ⁾

2

( ω ) + ¹

2

³²

ξ ⁽ ₁ ² ⁾ ( ω ) if t = ¹ ₄ B ⁽

1

¹ ⁾

2

( ω ) if t = ¹ ₂

1 2 B ⁽

1

¹ ⁾

2

( ω ) + B ₁ ⁽ ¹ ⁾ ( ω ) + ¹

2

³²

ξ ⁽ ₃ ² ⁾ ( ω ) if t = ³ ₄

B ₁ ⁽ ¹ ⁾ ( ω ) if t = 1

(51)

Motion

Third approximation II

Construction of the Brownian motion

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

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6

t

B2

(52)

Motion

Third approximation III

Construction of the Brownian motion

Note that

B ₀ ⁽ ² ⁾ = B ₀ ⁽ ¹ ⁾ ( ω ) = 0

B ⁽

1

² ⁾

4

= ¹

2 B ₀ ⁽ ¹ ⁾ + B ⁽

1

¹ ⁾

2

+ ¹ 2

³²

ξ ₁ ⁽ ² ⁾

= ¹

2 0 + ¹

2 ξ ⁽ ₁ ⁰ ⁾ + ξ ⁽ ₁ ¹ ⁾ + ¹ 2

³²

ξ ₁ ⁽ ² ⁾

= ¹

4 ξ ⁽ ₁ ⁰ ⁾ + ¹

4 ξ ⁽ ₁ ¹ ⁾ + ¹ 2

³²

ξ ⁽ ₁ ² ⁾ N 0, 1

16 + ¹ 16 + ¹

8 = N 0, 1

4

(53)

Motion

Third approximation IV

Construction of the Brownian motion

B ⁽

1

² ⁾

2

= B ⁽

1

¹ ⁾

2

= ¹

2 ξ ⁽ ₁ ⁰ ⁾ + ξ ₁ ⁽ ¹ ⁾ N 0, 1 2 B ⁽

3

² ⁾

4

= ¹ 2 B ⁽

1

¹ ⁾

2

+ B ₁ ⁽ ¹ ⁾ + ¹ 2

³²

ξ ⁽ ₃ ² ⁾

= ¹ 2

1 2 ξ ⁽ ₁ ⁰ ⁾ + ξ ⁽ ₁ ¹ ⁾ + ξ ⁽ ₁ ⁰ ⁾ + ¹ 2

³²

ξ ⁽ ₃ ² ⁾

= ³

4 ξ ⁽ ₁ ⁰ ⁾ + ¹

4 ξ ⁽ ₁ ¹ ⁾ + ¹ 2

³²

ξ ⁽ ₃ ² ⁾ N 0, 9

16 + ¹ 16 + ¹

8 = N 0, 3

4 B ₁ ⁽ ² ⁾ = B ₁ ⁽ ¹ ⁾ = ξ ⁽ ₁ ⁰ ⁾ N ( 0, 1 )

(54)

Motion

Third approximation V

Construction of the Brownian motion

implies B ₁ ⁽ ² ⁾ B ⁽

3

² ⁾

4

= ξ ⁽ ₁ ⁰ ⁾ 3

4 ξ ⁽ ₁ ⁰ ⁾ + ¹

4 ξ ⁽ ₁ ¹ ⁾ + ¹ 2

³²

ξ ⁽ ₃ ² ⁾

= ¹

4 ξ ⁽ ₁ ⁰ ⁾ 1

4 ξ ⁽ ₁ ¹ ⁾ 1 2

³²

ξ ⁽ ₃ ² ⁾ N 0, 1

16 + ¹ 16 + ¹

8 = N 0, 1 4 B ⁽

₃

² ⁾

4

B ⁽

₁

² ⁾

2

= ³

4 ξ ⁽ ₁ ⁰ ⁾ + ¹

4 ξ ⁽ ₁ ¹ ⁾ + ¹

2

³²

ξ ₃ ⁽ ² ⁾ 1

2 ξ ₁ ⁽ ⁰ ⁾ + ξ ⁽ ₁ ¹ ⁾

= ¹

4 ξ ⁽ ₁ ⁰ ⁾ 1

4 ξ ⁽ ₁ ¹ ⁾ + ¹ 2

³²

ξ ⁽ ₃ ² ⁾ N 0, 1

16 + ¹ 16 + ¹

8 = N 0, 1

4

(55)

Motion

Third approximation VI

Construction of the Brownian motion

B ⁽

₁

² ⁾

2

B ⁽

₁

² ⁾

4

= ¹

2 ξ ₁ ⁽ ⁰ ⁾ + ξ ₁ ⁽ ¹ ⁾ 1

4 ξ ⁽ ₁ ⁰ ⁾ + ¹

4 ξ ⁽ ₁ ¹ ⁾ + ¹ 2

³²

ξ ⁽ ₁ ² ⁾

= ¹

4 ξ ⁽ ₁ ⁰ ⁾ + ¹

4 ξ ⁽ ₁ ¹ ⁾ 1 2

³²

ξ ⁽ ₁ ² ⁾ N 0, 1

16 + ¹ 16 + ¹

8 = N 0, 1 4 B ⁽

1

² ⁾

4

B ₀ ⁽ ² ⁾ = B ⁽

1

² ⁾

4

N 0, 1

4

(56)

Motion

Third approximation VII

Construction of the Brownian motion

and these four random variables are mutually independent since Cov h

B ₁ ⁽ ² ⁾ B ⁽

3

² ⁾

4

, B ⁽

3

² ⁾

4

B ⁽

1

² ⁾

2

i

= Cov

"

ξ ⁽ ₁ ⁰ ⁾ 4

ξ ⁽ ₁ ¹ ⁾ 4

ξ ⁽ ₃ ² ⁾ 2

³²

, ξ ⁽ ₁ ⁰ ⁾

4 ξ ⁽ ₁ ¹ ⁾ 4 + ^ξ

( 2 ) 3

2

³²

#

= ¹ 16 Var h

ξ ⁽ ₁ ⁰ ⁾ i

+ ¹ 16 Var h

ξ ⁽ ₁ ¹ ⁾

i 1

8 Var h ξ ⁽ ₃ ² ⁾

i

= 0

Cov h

B ₁ ⁽ ² ⁾ B ⁽

3

² ⁾

4

, B ⁽

1

² ⁾

2

B ⁽

1

² ⁾

4

i

= Cov

"

ξ ⁽ ₁ ⁰ ⁾ 4

ξ ⁽ ₁ ¹ ⁾ 4

ξ ⁽ ₃ ² ⁾ 2

³²

, ξ ⁽ ₁ ⁰ ⁾

4 + ^ξ

( 1 ) 1

4 ξ ⁽ ₁ ² ⁾ 2

³²

#

= ¹ 16 Var h

ξ ⁽ ₁ ⁰ ⁾

i 1

16 Var h ξ ⁽ ₁ ¹ ⁾

i

= 0

(57)

Motion

Third approximation VIII

Construction of the Brownian motion

Cov h

B ₁ ⁽ ² ⁾ B ⁽

3

² ⁾

4

, B ⁽

1

² ⁾

4

B ₀ ⁽ ² ⁾ i

= Cov

"

ξ ⁽ ₁ ⁰ ⁾ 4

ξ ⁽ ₁ ¹ ⁾ 4

ξ ⁽ ₃ ² ⁾ 2

³²

, ξ ⁽ ₁ ⁰ ⁾

4 + ^ξ

( 1 ) 1

4 + ^ξ

( 2 ) 1

2

³²

#

= ¹ 16 Var

h ξ ⁽ ₁ ⁰ ⁾

i 1

16 Var h

ξ ⁽ ₁ ¹ ⁾ i

= 0

Cov h B ⁽

3

² ⁾

4

B ⁽

1

² ⁾

2

, B ⁽

1

² ⁾

2

B ⁽

1

² ⁾

4

i

= Cov

"

ξ ⁽ ₁ ⁰ ⁾ 4

ξ ⁽ ₁ ¹ ⁾ 4 + ^ξ

( 2 ) 3

2

³²

, ξ ⁽ ₁ ⁰ ⁾ 4 + ^ξ

( 1 ) 1

4 ξ ⁽ ₁ ² ⁾ 2

³²

80-646-08StochasticcalculusGeneviŁveGauthier TheBrownianmotion

The Brownian motion

80-646-08 Stochastic calculus

Geneviève Gauthier

HEC Montréal

Introduction

Brownian motion

Taylor (1975).)

The mathematician Norbert Wiener is credited with

rigorously analyzing the mathematics related to Brownian

motion, and that is why such a process is also known as a

Wiener process.

De…nition I

Brownian motion

Let ( Ω , F , F , P ) be a …ltered probability space.

Technical condition. Since we will work with almost-sure equalities 1 , we require that the set of events which have a zero-probability to occur be included in the sigma-algebra F 0 , which is to say that the set

N = f A 2 F : P ( A ) = 0 g F 0 . Thus, if X is F t measurable and Y = X

P almost-surely, then we know that Y is F t measurable.

De…nition II

Brownian motion

De…nition

A standard Brownian motion f W t : t 0 g is an adapted stochastic process, built on a …ltered probability space ( Ω , F , F , P ) such that:

(MB1) 8 ω 2 Ω , W 0 ( ω ) = 0,

(MB2) 8 0 t 0 t 1 ... t k , the random variables W t

W t

, W t

W t

, ..., W t

W t

are independent, (MB3) 8 s , t 0 such that s < t, the random variable W t W s is normally distributed with expectation 0 and variance t s i.e. W t W s N ( 0, t s ) ,

(MB4) 8 ω 2 Ω , the path t ! W t ( ω ) is continuous.

De…nition III

Brownian motion

In general, the …ltration we use is F = fF t : t 0 g where F t = σ ff W s : 0 s t g [ N g

is the smallest sigma-algebra for which the random

variables W s : 0 s t are measurable and that contains the zero-measure sets.

X = Y P almost-surely if the set of ω for which X di¤ers from Y has a zero probability, i.e.

P f ω 2 Ω : X ( ω ) 6 = Y ( ω ) g = 0.

Reminder about normal law I

If X is a random variable following a normal law with expectation µ and standard deviation σ > 0, then its probability density function is

f X ( x ) = 1 σ p

2π exp

( ( x µ ) 2 2σ 2

) ,

which allows us to determine 8 a, b 2 R , a < b, P [ a < X b ] =

Z b

a

f X ( x ) dx.

Unfortunately, the integral above has no primitive. So we need to value it numerically. The cumulative distribution function of X is

F X ( x ) =

Z x

f X ( y ) dy .

Reminder about normal law II

In general, if two random variables X and Y , built on the same probability space, are independent, then their covariance

Cov [ X , Y ] = E [ XY ] E [ X ] E [ Y ] is nil.

However, it is possible that two variables have zero

covariance, but that they are not independent. Here is an

illustrative example:

Reminder about normal law III

Example

ω X ( ω ) Y ( ω ) X ( ω ) Y ( ω ) P ( ω )

ω 1 0 1 0 1 4

ω 2 0 1 0 1 4

ω 3 1 0 0 1 4

ω 4 1 0 0 1 4

Reminder about normal law IV

The covariance between these two variables is zero since:

E

[ X ] = 0, E

[ Y ] = 0, E

[ XY ] = 0 ) Cov

[ X , Y ] = E

[ XY ] E

[ X ] E

[ Y ] = 0 but they are dependent since

P [ X = 0 and Y = 0 ] = 0 6 = 1

4 = P [ X = 0 ] P [ Y = 0 ] . However, when variables are normally distributed (not necessarily with the same expectation and standard

deviation), there is a result that allows us to verify whether

such variables are independent by using covariance :

Reminder about normal law V

Let ( Ω , F ^, ^F ^, ^P ) be a …ltered probability space.

Technical condition. Since we will work with almost-sure equalities ¹ , we require that the set of events which have a zero-probability to occur be included in the sigma-algebra F ⁰ , which is to say that the set

N = f ^A 2 F ^: ^P ( A ) = 0 g F ⁰ ^. Thus, if X is F ^t measurable and Y = X

P almost-surely, then we know that Y is F ^t measurable.

A standard Brownian motion f ^W ^t ^: ^t ⁰ g is an adapted stochastic process, built on a …ltered probability space ( Ω , F ^, F , P ) such that:

(MB1) 8 ^ω 2 ^Ω ^, ^W ⁰ ( ω ) = 0,

(MB2) 8 ⁰ ^t ⁰ ^t ¹ ^... ^t ^k ^, the random variables W t

are independent, (MB3) 8 ^s ^, ^t 0 such that s < t, the random variable W _t W _s is normally distributed with expectation 0 and variance t s i.e. W t W s N ( 0, t s ) ,

(MB4) 8 ^ω 2 ^Ω ^, ^{the path} ^t ! ^W ^t ( ω ) is continuous.

In general, the …ltration we use is F = fF ^t ^: ^t ⁰ g ^where F ^t = σ ff ^W ^s ^: ⁰ ^s ^t g [ N g

variables W _s : 0 s t are measurable and that contains the zero-measure sets.

P f ^ω 2 ^Ω ^: ^X ( ω ) 6 = Y ( ω ) g = 0.

f _X ( x ) = ¹ σ p

( ( x µ ) ² 2σ ²

which allows us to determine 8 ^a, ^b 2 R , a < b, P [ a < X b ] =

f _X ( x ) dx.

F _X ( x ) =

f _X ( y ) dy .

ω X ( ω ) Y ( ω ) X ( ω ) Y ( ω ) _P ( ω )

ω 1 0 1 0 ¹ ₄

ω 2 0 1 0 ¹ ₄

ω 3 1 0 0 ¹ ₄

ω 4 1 0 0 ¹ ₄

[ XY ] = 0 ) ^Cov

P [ X = 0 and Y = 0 ] = 0 6 = ¹

4 = _P [ X = 0 ] _P [ Y = 0 ] . However, when variables are normally distributed (not necessarily with the same expectation and standard

Lemma 1. Let f ^W ^t ^: ^t ⁰ g be a standard Brownian motion.

(i) For all s > 0, f ^W ^t ⁺ ^s ^W ^s ^: ^t ⁰ g (time homogeneity) (ii) f ^W ^t ^: ^t ⁰ g ^(symmetry)

W _t = tW

1 _t > 0 : t 0 o

De…nition. On the …ltered probability space ( _Ω , F ^, F , P ) , where F is the …ltration fF ^t ^: ^t ⁰ g , the stochastic process M = f ^M ^t ^: ^t ⁰ g ^{is a} martingale in continuous time if

( M 1 ) 8 ^t ^0, ^E

[ j ^M ^t j ] < _∞ ; ( M 2 ) 8 ^t ^0, ^M ^t ^is F ^t measurable;

( M 3 ) 8 ^s ^, ^t ^{0 tel que} ^s < t, E

[ M _t jF ^s ] = M _s .

For all time t, the random variable W _t is integrable since E

[ j ^W ^t j ] =

j ^z j

p 2πt exp z ² 2t dz

2πt exp z ² 2t dz

π exp z ² 2t

π < _∞ .

The only thing left to verify is that 8 ^s, ^t 0 such that s < t, E

[ W t jF ^s ] = W s .

[ W _t jF ^s ] = E

[ W _t W _s + W _s jF ^s ]

[ W t W s jF ^s ] + E

[ W s jF ^s ]

= W _s from ( MB 3 )

Idea of the proof of Lemma 3 : For all u 2 [ 0, s ] , the random variables W _t W _s and W _u are independent since

= Cov [ W _t W _u , W _u ] Cov [ W _s W _u , W _u ]