Stochastic calculus and its applications

Stochastic calculus and its applications Andrey A. Dorogovtsev Department of Random Processes Institute of Mathematics NAS Ukraine andrey.dorogovtsev@gmail.com Sarajevo, 2019

Lecture 1. Ito-Wiener expansion

De�nition

A random variable ξ is Gaussian or normally distributed with parameters a and σ2if its density has a form

p(u) = 1√ 2πσ

exp−1 2

(u−a)2.

The case a = 1, σ = 1 is called standard.

Fact

Eξ = a, V ξ = σ2. Characteristic function E exp itξ = exp ita− σ2t2 2

.

Theorem

Let ξ and η be independent Gaussian random variabless with parameters aξ , σξ , aη , ση correspondingly. Then for arbitrary α and β random variable αξ + βη is Gaussian with parameters αaξ + βaη , α2σ2ξ + β

2σ2η . Andrey Dorogovtsev White noise analysis and coalescing stochastic �ows

Lecture 1. Ito-Wiener expansion

Corollary

Let ξ1, ...,ξn be independent standard Gaussian variables. Then for arbitrary α1, ...,αn the sum

n

∑ k=1

αkξk

has a Gaussian distribution with the parameters 0 and ∑nk=1α2k .

This gives us possibility to de�ne a linear map from Rn to space of

random variables

Rn 3 α 7→ Uα = n

∑ k=1

αkξk

with the properties

1) for every α ∈ Rn, Uα is a Gaussian random variable, 2) EUα = 0, VUα = ||α||2, where ||α||, is a Euqlidean norm of α

Andrey Dorogovtsev White noise analysis and coalescing stochastic �ows

Lecture 1. Ito-Wiener expansion

Fact

U saves the norm.

Corollary

For orthogal α,β ∈ Rn the random variables Uα and Uβ are independent.

Proof.

E exp i(λ1Uα + λ2Uβ ) =

= E exp

( i

n

∑ k=1

(λ1αk + λ2βk)ξk

) =

= exp−1 2

n

∑ k=1

(λ1αk + λ2βk)2 =

Andrey Dorogovtsev White noise analysis and coalescing stochastic �ows

Lecture 1. Ito-Wiener expansion

Proof.

exp

{ −λ

2

1

2

n

∑ k=1

α2k − λ 2 2

2

n

∑ k=1

β 2k −λ1λ2 n

∑ k=1

αkβk

} =

= E exp iλ1UαE exp iλ2Uβ

due to the equality

(α,β ) = n

∑ k=1

αkβk = 0.

Andrey Dorogovtsev White noise analysis and coalescing stochastic �ows

Lecture 1. Ito-Wiener expansion

De�nition

White noise in the Hilbert space H is a linear map from H to space

of random variables

H 3 α 7→ (α,ξ )

1) for every α ∈ H, (α,ξ ) is a Gaussian random variable, 2) E (α,ξ ) = 0, V (α,ξ ) = ||α||2, where ||α||, is a norm of α .

Example

H = l2, {ξn;n ≥ 1} be the sequence of the independent standard Gaussian random variables. De�ne

(h,ξ ) := ∞

∑ n=1

hnξn

Now ξ is not a random element in H.

Andrey Dorogovtsev White noise analysis and coalescing stochastic �ows

Lecture 1. Ito-Wiener expansion

How about H = L2([0; 1])? Let ξ be a white noise in L2([0; 1]). Recall that for f ,g ∈ L2([0; 1]) their product is de�ned as

(f , g) = ∫

1

0

f (t)g(t)dt.

Consequently, one can denote

(f , ξ ) = ∫

1

0

f (t)ξ (t)dt.

But since dimL2([0; 1]) = ∞ , then ξ is not a random function (some people say that ξ is a generalized function)!

Andrey Dorogovtsev White noise analysis and coalescing stochastic �ows

Lecture 1. Ito-Wiener expansion

The integral from ξ has a right meaning.

De�nition

w(t) = ∫ t 0

ξ (s)ds = (1[0; t], ξ ), t ∈ [0; 1]. w is called by the Wiener process or the process of Brownian

motion.

Fact

Properties of the Wiener process

1. w is a Gaussian process, which means that for arbitrary

t1, ..., tn ∈ [0; 1], α1, ...,αn ∈ R the sum

n

∑ k=1

αkw(tk)

is a Gaussian random variable.

2. w has independent and stationary increments.

Andrey Dorogovtsev White noise analysis and coalescing stochastic �ows

Lecture 1. Ito-Wiener expansion

Note that one can write

(f , ξ ) = ∫

1

0

f (t)ξ (t)dt = ∫

1

0

f (t)dw(t).

It follows from the de�nition of white noise, that

E

∫ 1

0

f (t)dw(t) = 0, V ∫

1

0

f (t)dw(t) = ∫

1

0

f (t)2dt.

In particular,

Ew(t) = 0, Vw(t) = t.

Andrey Dorogovtsev White noise analysis and coalescing stochastic �ows

Lecture 1. Ito-Wiener expansion

De�nition

n-th degree Hermite polynomial is

Hn(x) = (−1)ne x2

2

( d

dx

)n e−

x2

2 .

The �rst Hermite polynomials are

H0(x) = 1, H1(x) = x , H2(x) = x 2−1.

Fact

For arbitrary n ≥ 0 the following statement holds. 1) Hn is odd if n is odd and even if n is even,

2) Hn+2(x) = xHn+1(x)− (n+1)Hn(x), 3) H ′n+1(x) = (n+1)Hn(x).

Andrey Dorogovtsev White noise analysis and coalescing stochastic �ows

Lecture 1. Ito-Wiener expansion

Fact

For every n,m ≥ 0

1√ 2π

∫ R Hn(x)Hm(x)e

− x22 dx = δmnn!

The sequence {

1√ n! Hn;n ≥ 1

} is an orthonormal basis in

L2

( R, 1√

2π e − x22 dx

) .

Translation of this fact on the �probability language� will be useful.

Andrey Dorogovtsev White noise analysis and coalescing stochastic �ows

Lecture 1. Ito-Wiener expansion

Theorem

Every square-integrable random variable α which is measurable with respect to the standard Gaussian variable ξ can be uniquely represented by the series

α = ∞

∑ n=0

anHn(ξ ).

The series converges in the square mean and

Eα = a0, Eα2 = ∞

∑ n=0

n!a2n.

Andrey Dorogovtsev White noise analysis and coalescing stochastic �ows

Lecture 1. Ito-Wiener expansion

The properties of Hermite polynomials can be derived from their

generating function. Let us note that for arbitrary x ,y ∈ R

exy− y2

2 = ∞

∑ n=0

Hn(x)y n

n! .

From this formula one can get

Hn(ξ ) = ∂ n

∂yn eξy−

y2

2 |y=0.

Andrey Dorogovtsev White noise analysis and coalescing stochastic �ows

Lecture 1. Ito-Wiener expansion

Let ξ be white noise in H.

De�nition

Stochastic exponent for h ∈ H is de�ned as follows

E (h) = exp((h,ξ )− 1 2 ||h||2)

Andrey Dorogovtsev White noise analysis and coalescing stochastic �ows

Lecture 1. Ito-Wiener expansion

Multi-dimensional Hermite polinomials from white noise ξ . Let An be a symmetric n-multiple linear form on H.

De�nition

An is a Hilbert-Shmidt form if for arbitrary orthonormal basis

{en; n ≥ 1} the following sum

||An||2n = ∞

∑ k1,...,kn=1

An(ek1 , ...,ekn) 2

is �nite. The value of the sum does not depend on choice of basis.

Andrey Dorogovtsev White noise analysis and coalescing stochastic �ows

Lecture 1. Ito-Wiener expansion

De�nition

The value of the form An on the white noise ξ (the in�nite-dimensional Hermite polinomial corresponding to An) is the

following product

An(ξ , ...,ξ ) = (An, ∇nhE )|h=0.

Fact

Properties of An(ξ , ...,ξ ) 1.

EAn(ξ , . . . ,ξ ) = 0, EAn(ξ , . . . ,ξ )Bn(ξ , . . . ,ξ ) = n!(An,Bn)n.

2. Let m 6= n. Then

EAn(ξ , . . . ,ξ )Bm(ξ , . . . ,ξ ) = 0.

Andrey Dorogovtsev White noise analysis and coalescing stochastic �ows

Lecture 1. Ito-Wiener expansion

Fact

3. Ito-Wiener expansion. Every square integrable random variable

α which is measurable with respect to ξ has a unique orthogonal expansion

α = ∞

∑ n=0

An(ξ , . . . ,ξ ),

Eα = A0, Eα2 = ∞

∑ n=0

n!‖An‖2n.

Andrey Dorogovtsev White noise analysis and coalescing stochastic �ows

Lecture 1. Ito-Wiener expansion

Examples

1. Let A = e⊗n, where e ∈ H and ‖e‖= 1. Then

e⊗n(ξ , . . . ,ξ ) = Hn((e,ξ )).

If ‖e‖ 6= 1 then

e⊗n(ξ , . . . ,ξ ) = ‖e‖nHn ((

e

‖e‖ ,ξ ))

.

2. Let {e1, . . . ,en} be an orthonormal system. Then

e1⊗ . . .⊗ en(ξ , . . . ,ξ ) = n

∏ k=1

(ek ,ξ ).

Andrey Dorogovtsev White noise analysis and coalescing stochastic �ows

Lecture 1. Ito-Wiener expansion

Examples

3.

e⊗r1 1 ⊗ . . .e⊗rnn (ξ , . . . ,ξ ) =

n

∏ k=1

Hrk ((ek ,ξ )).

4. For arbitrary f1, f2 ∈ H

f1⊗ f2(ξ ,ξ ) = (f1,ξ )(f2,ξ )− (f1, f2).

Andrey Dorogovtsev White noise analysis and coalescing stochastic �ows

Lecture 1. Ito-Wiener expansion

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