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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
The c