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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