Stochastic calculus and its applications

Stochastic calculus and its applicationsAndrey A. DorogovtsevDepartment of Random ProcessesInstitute of Mathematics NAS Ukraineandrey.dorogovtsev@gmail.comSarajevo, 2019

Lecture 1. Ito-Wiener expansion

Denition

A random variable ξ is Gaussian or normally distributed with

parameters a and σ2if its density has a form

p(u) =1√2πσ

exp−12

(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 , α2σ2

ξ+ β 2σ2

η .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=1

α2

k .

This gives us possibility to dene 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−12

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

α2

k −λ 2

2

2

n

∑k=1

β2

k −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

Denition

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. Dene

(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 dened 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.

Denition

w(t) =∫ t0

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 denition 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

Denition

n-th degree Hermite polynomial is

Hn(x) = (−1)nex2

2

(d

dx

)n

e−x2

2 .

The rst Hermite polynomials are

H0(x) = 1, H1(x) = x , H2(x) = x2−1.

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

∫RHn(x)Hm(x)e−

x2

2 dx = δmnn!

The sequence

1√n!Hn;n ≥ 1

is an orthonormal basis in

L2

(R, 1√

2πe−

x2

2 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)yn

n!.

From this formula one can get

Hn(ξ ) =∂ n

∂yneξy− 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.

Denition

Stochastic exponent for h ∈ H is dened 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.

Denition

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

Denition

The value of the form An on the white noise ξ (the

innite-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⊗r11⊗ . . .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 case H = L2([0; 1]). Now the n−tiple Hibert-Shmidt form An

can be associated with the square-integrable kernel a ∈ L2([0; 1]n)as follows

An(f1, ..., fn) =∫

...∫

1

0

a(t1, ..., tn)f1(t1)...fn(tn)dt1...dtn.

Since ξ can be viewed as the fromal derivative of the Wiener

process w then An(ξ , ...,ξ ) is n−tiple integral with respect to w∫...∫

1

0

a(t1, ..., tn)dw(t1)...w(dtn) = An(ξ , ...,ξ ).

Remark This is denition of the multiple integral!

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 1. Ito-Wiener expansion

Example ∫0≤t1≤...≤tn≤1

dw(t1)...w(dtn) =

=1

n!

∫...∫

1

0

dw(t1)...w(dtn) =1

n!Hn(w(1))

Remark This is the dierence between the usual and stochastic

calculus ∫1

0

w(t)dw(t) =1

2H2(w(1)) =

1

2(w(1)2−1)!!!

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 1. Ito-Wiener expansion

Fact

Ito-Wiener expansion. Every square integrable random variable α

which is measurable with respect to w has a unique orthogonal

expansion

α = A0 +∞

∑n=1

∫0≤t1≤...≤tn≤1

an(t1, ..., tn)dw(t1)...w(dtn),

Eα = A0, Eα2 = A2

0 +∞

∑n=1

∫0≤t1≤...≤tn≤1

an(t1, ..., tn)2dt1...dtn.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 1. Ito-Wiener expansion

Corollary

Clark representation. Let the square-integrable random variable α

be measurable with respect to a Wiener process w(t); t ∈ [0;1].Then there exists non-anticipating random function

x(t); t ∈ [0;1] such that

E

∫1

0

x(t)2dt < +∞

and

α = Eα +∫

1

0

x(t)dw(t).

The random function x is unique up to the stochastic equivalence.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 1. Ito-Wiener expansion

How one can nd Ito-Wiener expansion or Clark representation?

Let us consider this expansion for the stochastic exponent.

Theorem

E (f ) = exp∫

1

0

f (t)dw(t)− 1

2

∫1

0

f (t)2dt=

= 1+∞

∑n=1

∫0≤t1≤...≤tn≤1

f (t1)...f (tn)dw(t1)...w(dtn).

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 1. Ito-Wiener expansion

Corollary

For

α = A0 +∞

∑n=1

∫0≤t1≤...≤tn≤1

an(t1, ..., tn)dw(t1)...w(dtn)

the following relation holds

EαE (f ) = A0 +∞

∑n=1

∫0≤t1≤...≤tn≤1

an(t1, ..., tn)f (t1)...f (tn)dt1...dtn.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 1. Ito-Wiener expansion

It follows from the relation

Eα2 = A2

0 +∞

∑n=1

∫0≤t1≤...≤tn≤1

an(t1, ..., tn)2dt1...dtn

that the function T (α)(f ) = EαE (f ), f ∈ L2([0; 1]) is analytic

with respect to f .

Denition

The function T (α)(f ) = EαE (f ), f ∈ L2([0; 1]) is called by theFourier-Wiener transform of the random variable α.

Fact

Fourier-Wiener transform uniquely defines random variable.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 1. Ito-Wiener expansion

Calculation of the Fourier-Wiener transform is related to the shift

of the white noise functionals.

Note, that in the case when dimH < +∞ one can consider the

random variables measurable with respect to ξ as a functions from

ξ (ξ now is a random element in H). So its shift along the vectors

from H can be dened trivially. The situation change when

dimH = ∞. As it was mentioned before ξ is not a random element

in the innitely dimensional space H. Consequently, the random

variable measurable with respect to ξ can not be represented as a

function from ξ . Nevertheless the shift of the random variable still

can be dened.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 1. Ito-Wiener expansion

Let α be a random variable which is measurable with respect to ξ .

Denition

The shift of the random variable α on the vector h ∈ H is the such

random variable β , which is measurable with respect to ξ and for

every bounded Borel function f : R → R and arbitrary l ∈ H the

following equality holds

Ef (α)E (h+ l) = Ef (β )E (l).

Denote the shift of α on the vector h ∈ H as Thα.

Theorem

The shift Thα exists, is uniquely dened and coincides with the

usual shift when dimH < +∞.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 1. Ito-Wiener expansion

Example

1. Let w be a Wiener process on [0;1]. Consider the random

variable α = w(1). Then for arbitrary h ∈ L2([0;1])

Thw(1) = w(1) +∫

1

0

h(t)dt.

It follows from the denition, that the following fundamental

relation holds

Theorem

For the square-integrable random variable α

T (α)(h) = EThα.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 1. Ito-Wiener expansion

Theorem

Let z(t), t ∈ [0;1] be adapted to the Wiener ltration random

process with

E

∫1

0

z(t)4dt < +∞.

Then the random process Thz is adapted and

E

∫1

0

(Thz(t))2dt < +∞.

Moreover

Th

∫1

0

z(t)dw(t) =∫

1

0

Thz(t)dw(t) +∫

1

0

Thz(t)h(t)dt.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 1. Ito-Wiener expansion

Theorem

Let x be a solution to the following Cauchy problemdx(t) = a(x(t))dt +b(x(t))dw(t),

x(0) = x0,

where a,b are the bounded continuous functions have the

continuous bounded rst derivatives and a nonrandom x0 ∈ R. Takeh ∈ C ([0;1]). Then the random process y(t) = Thx(t), t ∈ [0;1] isthe solution to the following Cauchy problem

dy(t) = a(y(t))dt +b(y(t))dw(t) +b(y(t))h(t)dt,

y(0) = x0.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 1. Ito-Wiener expansion

Having the shift of the random variable α one can try to dene the

derivative of this variable as

Dhα = limε→0

Tεhα−α

ε

In the terms of Ito-Wiener expansion the stochastic derivative can

be dened as follows. For square integrable random variable α

which is measurable with respect to ξ consider its ItoWiener

expansion

α =∞

∑n=0

An(ξ , . . . ,ξ ).

For every n ≥ 1 An is a symmetric n-linear HilbertShmidt form on

H. Consequently, for arbitrary h ∈ H An(h, . . . , ·) is a symmetric

n−1-linear HilbertShmidt form on H and one can consider the

series∞

∑n=0

nAn(h,ξ , . . . ,ξ ).

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 1. Ito-Wiener expansion

Theorem

The stochastic derivative of α is a such square integrable random

element Dα in H that for every h ∈ H

(Dα,h) =∞

∑n=0

nAn(h,ξ , . . . ,ξ ).

Example

Suppose that H = R. In this case Gaussian white noise ξ is a

standard Gaussian variable. Every random variable α measurable

with respect to ξ can be represented as a function from ξ

α = ϕ(ξ ). Suppose that ϕ is bounded and has a continuous

bounded derivative. The stochastic derivative of ϕ(ξ ) exists and

Dϕ(ξ ) = ϕ′(ξ ).

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 1. Ito-Wiener expansion

Theorem

Let the random variables α1, . . . ,αn be stochastically dierentiable.

Suppose that the function F ∈ C 1(Rn) has a bounded derivative.

Then the random variable F (α1, . . . ,αn) has a stochastic derivative

and

DF (α1, . . . ,αn) =n

∑k=1

F ′k(α1, . . . ,αn)Dαk .

Example

DF (w(t1), . . . ,w(tn)) =n

∑k=1

F ′k(w(t1), . . . ,w(tn))1[0;tk ].

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 1. Ito-Wiener expansion

The stochastic derivative D can be considered as an operator

acting from the space of square integrable random variables

L2(Ω, F , P) to the space of square integrable random H−valuedelements L2(Ω, P, H).

Denition

Adjoint operator I = DFis called by the extended (or Skorokhod)

stochastic integral.

Example

Consider the case H = Rn. It can be shown that for function~f : Rn→ Rn which has a continuous derivative of polynomial growth

I (~f (~ξ )) =n

∑k=1

ξk fk(~ξ )− ∂

∂ξkfk(~ξ ).

Here ~f = (f1, . . . , fn), ~ξ = (ξ1, . . . ,ξn).

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 1. Ito-Wiener expansion

The previous example gives us an experience that the extended

stochastic integral is a special dierential operator in the space of

random elements. Nevertheless the extended stochastic integral has

a properties of the usual integration operator and coincides with

the Itô integral in adapted case.

Example

Suppose that H = L2([0;1]) and ξ is generated by the Wiener

process w . Let x be a random process adapted to the ltration of

w and such that

E

∫1

0

x(s)2ds < +∞.

Then x can be considered as a square integrable random element in

H. It turns out that I (x) is properly dened and coincides with the

Itô integral ∫1

0

x(s)dw(s).

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 1. Ito-Wiener expansion

Denition

The operator N = ID in the space L2(Ω, F , P) is called by the

Ornstain-Uhlenbeck operator (or number of particles operator).

In terms of elements of the Ito-Wiener expansion the operator N

can be expressed very simply.

Fact

If the random variable α has the representation

α =∞

∑n=0

An(ξ , . . . ,ξ ),

then

Nα =∞

∑n=0

nAn(ξ , . . . ,ξ ).

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 1. Ito-Wiener expansion

The operators Tt = exp−Nt, t ≥ 0 are the contraction semi-group

in L2(Ω, F , P).What is interesting and surprising that Tt are

nonnegative in a sence, that for α ≥ 0 Ttα ≥ 0. What is the

reason? Let us note that

Ttα =∞

∑n=0

e−ntAn(ξ , . . . ,ξ ) =∞

∑n=0

An(e−tξ , . . . ,e−tξ ).

More over, for every continuous linear operator C in H with the

operator norm ||C || ≤ 1 the following operator is properly dened

on the random variables

Γ(C )α =∞

∑n=0

An(Cξ , . . . ,Cξ ).

It is called by the second quantization operator based on C and

also is nonnegative!

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 1. Ito-Wiener expansion

The explanation is very simple!

Dene the operator

C =√I −CFC

and consider the white noise η independent from ξ . Then

ξ = Cξ + Cη is again white noise in H. The following theorem

takes place.

Theorem

For any square-integrable α = ∑∞n=0An(ξ , . . . ,ξ )

Γ(C )α = E (∞

∑n=0

An(η , . . . ,η)/ξ ).

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

This lecture deals with the special class of Gaussian process which

admit the integration theory almost the same as a Wiener process.

There are several reasons for consideration the processes dierent

from Wiener processes. Some of such processes like a fractional

Brownian motion are important for applications. Also a such

processes arise naturally in the ltration problems as it will be

shown in the future. Moreover the theory of integration for such

processes can be developed straightforwardly and in a simple way.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Denition

Gaussian process γ(t); t ∈ [0;1] is an integrator if there exists

such positive constant C that for every partition

0 = t0 < t1 < .. . < tn = 1 and arbitrary real numbers a0, . . . ,an−1the following inequality holds

E (n−1

∑k=0

ak(γ(tk+1)− γ(tk)))2 ≤ Cn−1

∑k=0

a2k

(tk+1− tk).

Example

The processes w and w(t)− tw(1), t ∈ [0;1] are integrators. Fora Wiener process holds with C = 1. Consider γ(t) = w(t)− tw(1).

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Example

In this case

E (n−1

∑k=0

ak(γ(tk+1)− γ(tk)))2 ≤

≤ 2E (n−1

∑k=0

ak(w(tk+1)−w(tk)))2 +2E (n−1

∑k=0

ak(tk+1− tk)w(1))2 =

= 2(n−1

∑k=0

a2k

(tk+1− tk) +2(n−1

∑k=0

ak(tk+1− tk))2 ≤

≤ 3n−1

∑k=0

a2k

(tk+1− tk).

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Theorem

Suppose that Gaussian process γ is C 1 with probability one. Then γ

is an integrator.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Theorem

Let the process γ is obtained from w as follows

γ(t) = Γ(C )w(t)

Then γ is an integrator.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

The integrators have simple description as a stochastic integrals

with respect to Wiener process.

Theorem

Let w be a Wiener process on [0;1] and γ be a measurable with

respect to w and jointly Gaussian with it process, γ(0) = 0. Then γ

is an integrator if and only if there exists such continuous linear

operator A in L2([0;1]) that

γ(t) =∫

1

0

A(1[0;t])(s)dw(s).

Remark Despite such simple description integrators can have

unexpectivly complicated behaviour. For example integrator can

have not a quadratic variation in L2-sense and, correspondingly, can

have not semimartingale structure.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Theorem

If A is a HilbertShmidt operator, then a quadratic variation of γ in

the mean is equal to zero.

Idea Denote by ξ the white noise generated by w . Then the

process γ can be represented as

γ(t) =∫

1

0

A(1[0;t])(s)dw(s) = (A(1[0;t]), ξ ) = (1[0;t], A∗ξ ) =

=∫

t

0

(A∗ξ )(s)ds.

The last expression is well-dened because A∗ is a HilbertShmidt

operator and A∗ξ is a usual random Gaussian element in L2([0;1]).

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Theorem

Let the restriction of the operator A onto the space C ([0;1]) be a

continuous linear operator that maps C ([0;1]) into itself. Then the

corresponding process γ has a quadratic variation in the square

mean.

Proof.

It follows from the conditions of the theorem that the operator A

can be associated with the function µ : [0;1]×B([0;1])→ R such

that

(i) for any t ∈ [0;1],µ(t, ·) is a nite signed measure on B([0;1]);(ii) for any ∆ ∈B([0;1]), µ(·,∆) is a Borel function on [0;1];(iii) for every f ∈ C ([0;1]), t ∈ [0;1] :

(Af )(t) =∫

1

0

f (s)µ(t,ds).

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Proof.

Now

A(1[0;t])(s) = µ(s, [0; t]).

Therefore, for any partition 0 = t0 < t1 < .. . < tn = 1

En−1

∑i=0

(γ(ti+1)− γ(ti ))2 =n−1

∑i=0

‖A1[ti ;ti+1)‖2 =

=n−1

∑i=0

∫1

0

µ(s, [ti ; ti+1))2ds =∫

1

0

n−1

∑i=0

µ(s, [ti ; ti+1))2ds.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Proof.

For xed sn−1

∑i=0

µ(s, [ti ; ti+1))2→∑t

µ(s,t)2,

max(ti+1− ti )→ 0.

Since A is bounded operator in C ([0;1]) then

c = sup[0;1] |µ|(s, [0;1]) < +∞. Consequently,

sup[0;1]

supt0<...<tn

n−1

∑i=0

µ(s, [ti ; ti+1))2 ≤ c2.

It follows from the dominated convergence theorem, that

limmax(ti+1−ti )→0

n−1

∑i=0

‖A1[ti ;ti+1)‖2 =

∫1

0∑t

µ(s,t)2ds.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Let us consider processes satisfying the conditions of Theorem

Example

1. Let γ be equal w . In this case A is identity operator.

Consequently the measures µ(s, ·) now have the form

µ(s, ·) = δs

and the corresponding limit equals∫1

0

1ds = 1.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Example

2. γ(t) = w(t)−2tw(1). For this process operator A has the

following form

(Af )(s) = f (s)−2

∫1

0

f (t)dt.

Consequently A satises the conditions of Theorem with

µ(s, ·) = δs −2λ ,

where λ is Lebesgue measure on [0;1]. Hence the quadraticvariation in the square mean for γ equals

∫1

01ds = 1. Note that γ

has a covariance t ∧ s, i.e. γ is a Wiener process (with respect to

dierent ltration as an initial process w).

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Now we will dene the extended stochastic integral with respect to

integrators. Let γ be an integrator jointly Gaussian and measurable

with respect to the Wiener process w on [0;1]. As it was mentioned

at the previous section, γ has a representation

γ(t) =∫

1

0

(A1[0;t])(s)dw(s)

with a certain continuous operator A in L2([0;1]).

Denition

Square integrable random element x in L2([0;1]) belongs to the

domain of denition of the extended integral with respect to γ if

Ax belongs to the domain of denition of I (extended integral with

respect to w). In this case∫1

0

x(s)dγ(s) := I (Ax).

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Note, that since A is continuous, every nonrandom function from

L2([0;1]) can be integrated with respect to γ. Also, everystochastically dierentiable random element can be integrated with

respect to γ. Sometime for simplicity we will denote the integral∫1

0xdγ as Iγ (x).

Fact

The properties of a stochastic integral with respect to γ .

1) For x from the domain of denition Iγ and stochastically

dierentiable random variable α

EIγ (x)α = E (x ,A∗Dα),

in particular EIγ (x) = 0.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Fact

2) For stochastically dierentiable x

EIγ (x)2 = E‖Ax‖2 +Etr(ADx)2.

3) For stochastically dierentiable x and bounded stochastically

dierentiable α

Iγ (αx) = α Iγ (x)− (x ,A∗Dα).

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Note that in the case of integration with respect to Wiener process

w there was a class of adapted random functions for which the

extended integral coincide with Itô integral and has nice properties.

The same happens with integration with respect to γ. Suppose thatγ is a martingale (with respect its own ltration). Since γ is a

Gaussian process then the characteristics < γ > is nonrandom

increasing function . Moreover < γ > is Lipshitz function.

Consequently, every random function x from L2([0;1]) with the

nite second moment adapted to the ltration of γ is integrable

with respect to γ in the Itô sense.

Theorem

Let x and γ be as it was described above. Then the Itô integral∫1

0x(t)dγ(t) coinside with Iγ (x).

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

As it was mentioned before, there exists one-to-one correspondence

between the integrators and continuous linear operators in the

space L2([0;1]). Let γ be an integrator and A be an operator

corresponding to γ.

Denition

The operator A∗A is called by the characteristic operator for

integrator γ.

Remark. If γ1 and γ2 have the same characteristic operator then

they are equidistributed. This can be easily checked by calculation

a covariance function.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Theorem

Let A∗A be a HilbertShmidt operator. Then γ has a quadratic

variation in the square mean i.e. the following limit exists

limmax(tk+1−tk)→0

En−1

∑k=0

(γ(tk+1)− γ(tk))2 =

= limmax(tk+1−tk)→0

n−1

∑k=0

(A∗A1[tk ;tk+1), 1[tk ;tk+1)) = 0.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Consider an Itô formula for γ.

Theorem

Suppose that A satises one of the following conditions:

(i) A is HilbertShmidt operator,

(ii) ∃ C > 0 ∀ t ∈ [0;1] ∀ f ∈ L2([0;1])∩C ([0; t]) :

A∗A(f ) ∈ C ([0; t]),

max[0;t]|A∗A(f )| ≤ Cmax

[0;t]|f |.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Theorem

Then for any twice dierentiable function F : [0;1]×R → R with

bounded derivatives, the equalities

F (t,γ(t)) = F (0,0) +∫

t

0

F ′1(s,γ(s))ds+

∫t

0

F ′2(s,γ(s))dγ(s) +1

2trAΨtA

∗,

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Theorem

and

F (t,γ(t)) = F (0,0) +∫

t

0

F ′1(s,γ(s))ds+

∫t

0

F ′2(s,γ(s))dγ(s) +1

2

∫t

0

A∗A(F ′′22(s ∨·,γ(s ∨·)))(s)ds

are true in cases (i) or (ii), respectively. Here Ψt is an integral

operator in L2([0;1]) with the kernel

1[0;t](s ∨ r)F ′′22(s ∨ r ,γ(s ∨ r)).

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Theorem

Let Γ(C ) be an operator of the second quantization, x be a random

element in the complete separable metric space X . Then there

exists the random measure µ on X such that for every bounded

measurable function ∫fdµ = Γ(C )f (x).

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

For every s ∈ [0; 1] and u ∈ R denote by x(u,s,T ) the solution at

time T of the following Cauchy problem

dx(t) = a(x(t))dt +b(x(t))dw(t), x(s) = u.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian integrators

Theorem

U(u, t) = Γ(C )f (x(u,0, t))

U(u, t) satises the following SPDE

dU(u, t) = (1

2b(u)2

∂ 2

∂u2U(u, t) +a(u)

∂

∂uU(u, t))dt+

+b(u)∂

∂uU(u, t)dγ(t)

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 2. Gaussian strong random operators and its action on therandom elements

Let H be a separable real Hilbert space with the norm ‖ · ‖ and inner product(·, ·). Suppose that ξ is the generalized Gaussian random element in H with zeromean and identical covariation. In other words ξ is the family of jointly Gaussianrandom variables denoted by (ϕ, ξ), ϕ ∈ H with the properties1) (ϕ, ξ) has the normal distribution with zero mean and variance ‖ϕ‖2 for everyϕ ∈ H,2) (ϕ, ξ) is linear with respect to ϕ.

69

Lecture 2. Gaussian strong random operators and its action on therandom elements

Denition

Th e G a u s s i a n s t r o n g r a n d om l i n e a r o p e r a t o r ( G SRO ) A in His the mapping, which maps every element x of H into the jointly Gaussian with ξrandom element in H and is continuous in the square mean.

As an example of GSRO the integral with respect to Wiener process can beconsidered.

70

Lecture 2. Gaussian strong random operators and its action on therandom elements

Example

Dene GSRO A in the following way

∀ϕ ∈ H : (Aϕ)(t) =

∫ t

0

ϕ(s)dw(s), t ∈ [0;T ].

It can be easily seen that Aϕ now is a Gaussian random element in H, and A iscontinuous in square mean.

To include in this picture the integration with respect to another Gaussianprocesses (for example with respect to the fractional Brownian motion) considermore general GSRO. Suppose, that K be a bounded linear operator, which actsfrom L2([0;T ]) to L2([0;T ]2). Dene

∀ϕ ∈ H : (Aϕ)(t) =

∫ T

0

(Kϕ)(t, s)dw(s).

71

Lecture 2. Gaussian strong random operators and its action on therandom elements

It can be checked, that A is GSRO in H. Making an obvious changes one candene the GSRO acting from the dierent Hilbert space H1 into H. For exampleconsider for α ∈

(12 ; 1)the covariation function of the fractional Brownian motion

with Hurst parameter α

R(s, t) =1

2(t2α + s2α − |t− s|2α).

Dene the space H1 as a completion of the set of step functions on [0;T ] withrespect the inner product under which

(1I[0;s], 1I[0;t]) = R(s, t).

72

Lecture 2. Gaussian strong random operators and its action on therandom elements

Consider the kernel Kα from the integral representation of the fractionalBrownian motion Bα

Bα(t) =

∫ t

0

Kα(t, s)dw(s)

and∂Kα

∂t(t, s) = cα

(α− 1

2

)(t− s)α − 3

2

(st

) 12−α

.

Dene for ϕ ∈ H1

(Kϕ)(t, s) =

∫ t

s

ϕ(r)∂Kα

∂r(r, s)dr1I[0;t](s).

73

Lecture 2. Gaussian strong random operators and its action on therandom elements

Now let

(Aϕ)(t) =

∫ T

0

(Kϕ)(t, s)dw(s) =

∫ t

0

(Kϕ)(t, s)dw(s).

Then

(Aϕ)(t) =

∫ t

0

ϕ(s)dBα(s).

74

Lecture 2. Gaussian strong random operators and its action on therandom elements

We will consider the action of GSRO on the random elements in H. Considerarbitrary GSRO A in H. Then for every ϕ ∈ H the Ito-Wiener expansion of Aϕcontains only two terms

Aϕ = α0ϕ+ α1(ϕ)(ξ).

Here α0 is a continuous linear operator in H and α1 is a continuous linearoperator from H to the space of Hilbert-Shmidt operators in H. Now let x be arandom element in H with the nite second moment. Then α1(x) has a nitesecond moment in the space of Hilbert-Shmidt operators. So for every ϕ ∈ H

α1(x)(ϕ) =

∞∑k=0

Bk(ϕ; ξ, . . . , ξ).

It can be easily veried, that Bk is k + 1-linear H-valued Hilbert-Shmidt form onH. Dene ΛBk as a symmetrization of Bk with respect to all k + 1 variables.

75

Lecture 2. Gaussian strong random operators and its action on therandom elements

Denition (A. A. Dorogovtsev, 1988)

The random element x b e l o n g s t o t h e d oma i n o f d e f i n i t i o n o fG SRO A if the series

∞∑k=0

ΛBk(ξ, . . . , ξ)

converges in H in the square mean and in this case

Ax = α0x+

∞∑k=0

ΛBk(ξ, . . . , ξ).

76

Lecture 2. Gaussian strong random operators and its action on therandom elements

So one can dene GSRO Aγ associated with the integrator γ by the rule

∀ϕ ∈ L2([0;T ]) : (Aγϕ)(t) =

∫ t

0

ϕdγ.

In this situation the denition Aγ(x) is a denition of the extended stochasticintegral of x with respect to γ. Note that in the case γ = w it will be a usualextended integral.

77

Lecture 2. Gaussian strong random operators and its action on therandom elements

For every continuous linear operator C in H with the operator norm ‖C‖ ≤ 1 thefollowing operator is properly dened on the random variables

Γ(C)α =

∞∑n=0

An(Cξ, . . . , Cξ).

It is called by the second quantization operator based on C and also isnonnegative!

78

Lecture 2. Gaussian strong random operators and its action on therandom elements

The explanation is very simple! Dene the operator

C =√I − C?C

and consider the white noise η independent from ξ. Then ξ = Cξ + Cη is againwhite noise in H. The following theorem takes place.

Theorem

For any square-integrable α =∑∞n=0An(ξ, . . . , ξ)

Γ(C)α = E(

∞∑n=0

An(ξ, . . . , ξ/ξ).

79

Lecture 2. Gaussian strong random operators and its action on therandom elements

Theorem (A. A. Dorogovtsev, 1998)

Let A be a GSRO in H and Γ(C) be an operator of the second quantization.Suppose that the random element x lies in the domain of denition of A in thesence of denition. Then Γ(C)x belongs to the domain of denition of GSROΓ(C)A and the following equality holds

Γ(C)(Ax) = Γ(C)A(Γ(C)x).

Here Γ(C)A is the GSRO which acts by the rule

∀ϕ ∈ H : Γ(C)Aϕ = Γ(C)(Aϕ).

80

Lecture 2. Gaussian strong random operators and its action on therandom elements

Example

Consider in the situation of the example 1 GSRO of integration with respect toWiener process w. Suppose that random function x in L2([0;T ]) with the nitesecond moment is adapted to the ow of σ-elds generated by w. It iswell-known, that in this case the extended stochastic integral∫ t

0

x(s)dw(s), t ∈ [0;T ]

exists and coincides with the Ito integral. Now the theorem 1 says us that

Γ(C)

(∫ t

0

x(s)dw(s)

)=

∫ t

0

Γ(C)x(s)dγ(s),

where γ is an integrator of the type γ(t) = Γ(C)w(t) and the integral in the rightpart is an extended stochastic integral.

81

Lecture 2. Equations with the strong Gaussian random operators

Consider equationx = y +A(x).

Here y and x are known and unknown random elements in H respectively.Our aim is to solve this equation for y = βu, where u ∈ H is nonrandom and β isa random variable. To do this suppose rstly, that1. α0 = 0.2. There exists a solution for arbitrary nonrandom y.In this case the solution xu of the equation

xu = u+A(xu)

is unique for every u ∈ H and can be obtained as a sum

xu =

∞∑k=0

Ak(u)

82

Lecture 2. Equations with the strong Gaussian random operators

Theorem (A.A.Dorogovtsev)

Let for arbitrary u ∈ H xu have the stochastic derivatives of all orders. Then forevery β, which has a nite ItoWiener expansion, the equation

x = βu+A(x)

has unique solution

x =

∞∑j=0

(−1)j

j!

∞∑t1...tj=1

Djβ(ϕt1 , . . . , ϕtj )Djxu(ϕt1 , . . . , ϕyj ),

where ϕt; t ≥ 1 is an arbitrary orthonormal basis in H.

83

Lecture 2. Equations with the strong Gaussian random operators

Example

Linear one-dimensional equation. Let w(t); t ∈ [0; 1] be a Wiener process. Theequation

dx(t) = a0(t)x(t)dt+ a(t)x(t)dw(t), x(0) = β,

with the anticipating initial condition β measurable with respect σ(w(t); t ∈ [0; 1]),can be considered as equation with the Gaussian strong random operator.

x(t) = β ·X0(t)+

+

∞∑j=1

(−1)j∫0≤τ1≤···≤τj≤t

Djβ(τ1, . . . , τj)a(τ1) . . . a(τj)dτ1 . . . dτj ·X0(t).

84

Lecture 2. Equations with the strong Gaussian random operators

Example

Consequently, by the Taylor expansion (which is valid in this situation for β, whichcan be considered as a function from w)

x(t) = Tt(β) ·X0(t),

where

Tt(β)(w) = β

(w −

∫ t∧.

0

a(τ)dτ

).

85

Lecture 2. Equations with the strong Gaussian random operators

Consider the equationdU(x, t) = 1

2∂2

∂x2U(x, t)dt+ ∂∂xU(x, t)dw(t),

U(x, 0) = f(x), x ∈ R.

Suppose, that the initial condition f has the form

f(x) = α · ϕ(x),

where ϕ is the deterministic function from L2(R) and α is the random variable

measurable with respect to w. Consider the Fourier transform U of the solution.Then it satises following Cauchy problem

dU(λ, t) = − 12λ

2U(λ, t)dt+ iλU(λ, t)dw(t),

U(λ, 0) = αϕ(λ).

86

Lecture 2. Equations with the strong Gaussian random operators

U(λ, t) = ϕ(λ)eiλw(t)∞∑k=0

ikλk∫

k. . .

∫0≤t1≤...≤tk≤t

Dkα(t1, . . . , tk)dt1 . . . dtk.

Taking the inverse Fourier transform

U(x, t) =

∞∑k=0

ϕ(k)(x+ w(t))

∫k. . .

∫0≤t1≤...≤tk≤t

Dkα(t1, . . . , tk)dt1 . . . dtk.

87

Lecture 3. Anticipating equations and ltration

Let us consider for GSRO A, random element x from the domain of A and ϕ ∈ H

E(Ax)e(ϕ,ξ)−12‖ϕ‖

2

.

This expectation in Hilbert space is a Bochner integral.

Lemma

E(Ax)e(ϕ,ξ)−12‖ϕ‖

2

= α0xϕ + α1(xϕ)(ϕ).

Here α0 and α1 are the terms from expansion of A and

xϕ = Exe(ϕ,ξ)−12‖ϕ‖

2

.

88

Lecture 3. Anticipating equations and ltration

This lemma allows to dene the action of the unbounded Gaussian randomoperator on the random elements in the weak sense. Let H1 be a separable realHilbert space densely and continuously embedded into H. Consider GSRO Aacting from H1 to H (in another word for every u ∈ H1 Au is a Gaussian elementin H and this correspondence is continuous in the square mean with respect tothe convergence in H1). Then A can be treated as an unbounded Gaussianrandom operator in H. As it was mentioned above, A can be described with thehelp of two deterministic linear operators α0 : H1 → H, α1 : H1 → σ2(H) (hereσ2(H) is the space of the Hilbert-Shmidt operators in H with the correspondentnorm). Operators α0 and α1 can be considered as an operators on H. Then thecorrespondence

H ⊃ H1 3 u 7→ α0(u) + α1(u)(ξ)

denes an unbounded Gaussian random operator in H with the domain ofdenition H1. The next denition is closely related to the lemma.

89

Lecture 3. Anticipating equations and ltration

Denition

The random element x in H with the nite second moment b e l o n g s i n t h ew e a k s e n s e t o t h e d oma i n o f d e f i n i t i o n o f t h e u n b o u n d e dG a u s s i a n r a n d om o p e r a t o r A if there exist the dense subset L ⊂ H andthe random element y in H with the nite second moment such, that

∀ϕ ∈ L : xϕ = Exe(ϕ,ξ)−12‖ξ‖

2

∈ H1, α0(xϕ) + α1(xϕ)(ϕ) = yϕ.

Here yϕ is dened in the same way as xϕ.

90

Lecture 3. Anticipating equations and ltration

Example

Case of deterministic operator. Let H ′ = L2(R× [0; 1], e−x2

2 dx× dt). Considerthe nonrandom operator A in H ′ which is dened by the formula

Af(x, t) =

∫ t

0

∂

∂xf(x, s)ds.

As a Hilbert space H1 let us use

H1 = W 12 (R, e−

x2

2 dx)× L2([0; 1]),

91

Lecture 3. Anticipating equations and ltration

Example

i.e. H1 consists of the functions from x ∈ R, t ∈ [0; 1] which have one Sobolevderivative with respect to x. Consider the following random element in H ′

X(x, t) = 1Iw(t)≤x.

It is easy to verify that X has a nite second moment

E

∫R

∫ 1

0

X(x, t)2 · e− x2

2 dxdt ≤√

2π.

92

Lecture 3. Anticipating equations and ltration

Example

Note, that for xed t X(·, t) /∈W 12 (R, e− x2

2 dx). Indeed due to the Sobolev

embedding theorem all functions from W 12 (R, e− x2

2 dx) must be continuous. Letus prove that X belongs to the domain of denition A in the sense of denition 4.Take ϕ ∈ L2([0; 1]) and consider

Xϕ(x, t) = E1Iw(t)≤x exp

∫ 1

0

ϕ(s)dw(s)− 1

2

∫ 1

0

ϕ2(s)ds

.

It follows from Girsanov theorem that

Xϕ(x, t) = E1Iw(t)+∫ t0ϕ(s)ds≤x.

93

Lecture 3. Anticipating equations and ltration

Example

Then Xϕ ∈ H1 and∫ t

0

∂

∂xXϕ(x, s)ds =

∫ t

0

1√2πs

exp− 1

2s

x−

∫ s

0

ϕ(r)dr

2

ds. (∗)

Now nd Y with the nite second moment in H such, that Yϕ equals to (*).Consider the sequence

Y n(x, t) = 2n

∫ t

0

1I[x− 1n ;x+ 1

n ](w(s))ds, n ≥ 1.

Note , that there exists the random eld Y (x, t), x ∈ R, t ∈ [0; 1] such, that

E(Y n(x, t)− Y (x, t))2 → 0, n→∞.

94

Lecture 3. Anticipating equations and ltration

Example

Also note, thatsupn≥1

EY n(x, t)2 ≤ c|x|.

This can be easily checked using Tanaka approach. Hence Y is the randomelement in H with the nite second moment. Moreover

Yϕ(x, t) = EY (x, t) exp

∫ 1

0

ϕ(s)dw(s)− 1

2

∫ t

0

ϕ2(s)ds

=

= limn→∞

EY n(x, t) exp

∫ 1

0

ϕ(s)dw(s)− 1

2

∫ t

0

ϕ2(s)ds

=

= limn→∞

E2n

∫ t

0

1I[x− 1n ;x+ 1

n ](w(s))ds exp

∫ 1

0

ϕ(s)dw(s)− 1

2

∫ t

0

ϕ2(s)ds

=

95

Lecture 3. Anticipating equations and ltration

Example

= limn→∞

E2n

∫ t

0

1I[x− 1n ;x+ 1

n ](w(s) +

∫ s

0

ϕ(r)dr)ds =

=

∫ t

0

1√2πs

exp− 1

2s

x−

∫ s

0

ϕ(r)dr

2

ds.

So,

Yϕ(x, t) =

∫ t

0

∂

∂xXϕ(x, s)ds.

Consequently X belongs to the domain of denition of operator A in the weaksense and AX is the local time of the Wiener process.

96

Lecture 3. Anticipating equations and ltration

Example

Let the spaces H be as in the previous example. Suppose, that the space H1 is

built similarly to the previous example with the substitution of W 12 (R, e− x2

2 dx) by

W 22 (R, e− x2

2 dx). Dene the Gaussian random operator A on H1 in the followingway

Af(x, t) =1

2

∫ t

0

∂2

∂x2f(x, s)ds+

∫ t

0

∂

∂xf(x, s)dw(s).

Let us consider arbitrary bounded and measurable function h on R and dene therandom element X in H by the formula

X(x, t) = h(x+ w(t)).

Prove, that X belongs to the domain of denition of A in the weak sense.

97

Lecture 3. Anticipating equations and ltration

Example

Really, the operators α0 and α1 from the Ito-Wiener representation of A now hasthe form

α0(u)(x, t) =1

2

∫ t

0

∂2

∂x2u(x, s)ds,

α1(u)(ϕ)(x, t) =

∫ t

0

∂

∂xu(x, s)ϕ(s)ds.

Consider

Xϕ(x, t) = Eh(x+ w(t)) exp

∫ 1

0

ϕ(s)dw(s)− 1

2

∫ t

0

ϕ2(s)ds

=

= Eh(x+w(t)+

∫ t

0

ϕ(s)ds) =1√2πt

∫Rh(u) exp− 1

2t

u− x−

∫ t

0

ϕ(s)ds

2

du.

98

Lecture 3. Anticipating equations and ltration

Theorem

Let the process γ is obtained from w as follows

γ(t) = Γ(C)w(t).

Then γ is an integrator.

99

Lecture 3. Anticipating equations and ltration

For every s ∈ [0; 1] and u ∈ R denote by x(u, s, T ) the solution at time T of thefollowing Cauchy problem

dx(t) = a(x(t))dt+ b(x(t))dw(t), x(s) = u.

100

Lecture 3. Anticipating equations and ltration

Theorem

U(u; t) = Γ(C)f(x(u, 0, t))

U(u, t) satises the following SPDE

dU(u, t) = (1

2b(u)2

∂2

∂u2U(u, t) + a(u)

∂

∂uU(u, t))dt+

+b(u)∂

∂uU(u, t)dγ(t)

101

Lecture 3. Anticipating equations and ltration

Suppose that x satises the Skorokhod SDE in the domain Gdx(t) = b(x(t))dt+ σ(x(t))dw(t) + 1I∂G(x(t))γ(x(t))dη(t),

x(0) = x0, η(0) = 0.

Consider for the function f ∈ C2(G) and the certain bounded linear operator C inL2([0;T ],Rd) following random function

U(x0, t) = Γ(G)f(x(t)).

102

Lecture 3. Anticipating equations and ltration

Theorem

Suppose, that σ, b and γ have three bounded continuous derivatives and boundaryof G is the C3-bounded manifold. Let also (γ,∇f)|∂G = 0. Then U satises thefollowing anticipating PSDE

dU(x, t) =

1

2

d∑ij=1

aij(x)∂2

∂xi∂xjU(x, t) +

d∑j=1

bj(x)∂

∂xjU(x, t)

dt−−

d∑ij=1

σij(x)∂

∂xiU(x, t)dγj(t),

103

Lecture 3. Anticipating equations and ltration

Theorem

and the boundary condition(γ,∇U)|∂G = 0.

Here for j = 1, . . . , dγj(t) = Γ(G)wj(t).

The equation is understood in the weak sense.

104

Lecture 3. Anticipating equations and ltration

Denote by τ the hitting time of y on the boundary ∂G. Let g ∈ C(∂G).

Q(x, t) = Γ(C)

(∫ T∧τ

t

f(y(s))ds+ g(y(T ∧ τ))

).

Here g is the unique deterministic function from C2(G)⋂C(G) which satises

the Dirichlet problem12

∑dij=1 aij(x) ∂2

∂xi∂xjg(x) +

∑dj=1 bj(x)∂g(x)∂xj

= 0, x ∈ G,g|∂G = g.

105

Lecture 3. Anticipating equations and ltration

Theorem

The random function Q satises in the weak sense the following anticipatingSPDE with the boundary conditionsdQ(x, t) =

[− 1

2

∑dij=1 aij(x) ∂2

∂xi∂xjQ(x, t)−

∑dj=1 bj(x) ∂

∂xjQ(x, t) + f(x)

]dt−

− 12

∑dij=1 aij(x) ∂

∂xjQ(x, t)dγi(t),

Q|∂G = g, Q(x, T ) = g(x).

Here γi, i = 1, . . . , d are the same as in the previous theorem and the stochasticintegrals related to dγi have the same meaning.

106

Lecture 3. Anticipating equations and ltration

Applications:

1. Generalized ltration problem.Let (w1, w2) be the pair of jointly Gaussian one-dimensional Wiener processes. Letthe processes x1, x2 satisfy the relations

dx1(t) = a1(x1(t))dt+ dw1(t),

dx2(t) = a2(x1(t))dt+ dw2(t),

x1(0) = x2(0) = 0.

107

Lecture 3. Anticipating equations and ltration

Applications:

The problem is to nd the conditional distribution of x1(t) for t ∈ [0; 1] undergiven x2(s); s ∈ [0; 1]. We will try to get the equation for

E(f(x1(t))/x2)

for the appropriate functions f.The distribution of (x1, x2) is absolutely continuous with respect to thedistribution (w1, w2) for suciently small a1, a2. The corresponding density willbe denoted by p.

108

Lecture 3. Anticipating equations and ltration

Dene for u ∈ C([01]) and Borel ∆ ⊂ C([0; 1])

ν(u,∆) = P (x1 ∈ ∆/x2 = u).

For measurable bounded ϕ : C([0; 1])→ R put

Ψ(u) =

∫C([0;1])

ϕ(v)p(v, u)ν(u, dv)·

·(∫

C([0;1])

p(v, u)ν(u, dv))

109

Lecture 3. Anticipating equations and ltration

Lemma

E(ϕ(x1)/x2) = Ψ(x2)

110

Lecture 3. Anticipating equations and ltration

Theorem

The random function

U(r, t) = E(f(r + w1(t))p(w1, w2)/w2)

satises relation

dU(r, t) =1

2

∂2

∂r2U(r, t)dt+

+∂

∂rU(r, t)γ(dt) + Ef ′(r + w1(t))(SDp(w1, w2))1(t)dt.

γ(t) = E(w1(t)/w2).

Here (SDp(w1, w2))1 is a stochastic derivative of p with respect to w1.

111

Lecture 3. Ito_Wiener expansion of the special functionals

Let w(t), t ≥ 0 be the one-dimensional Wiener process starting

from the point x > 0. Denote by τ the hitting time for w on the

level 0. We are interesting in the value Γ(C )1τ≤t .

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 3. Ito-Wiener expansion of the special functionals

Let us nd V (x ,s) = Γ(C )f (x +w(t ∧ τ)−w(s)) where f satises

condition f ′′(0) = 0 and

τ = inft : t ≥ s, w(t)−w(s) = 0

Theorem

V satises the following anticipating SPDE

dV (x ,s) =−12

∂ 2

∂x2V (x ,s)− ∂

∂xV (x ,s)dγ(s),

V (0,s)′′(0) = 0, V (x , t) = f (x).

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 3. Ito-Wiener expansion of the special functionals

Example

C = e⊗ e, ||e||= 1. Γ(C ) now is a conditional expectation with

respect to the random variable

η =∫

1

0

e(s)dw(s).

Now

γ(t) = η

∫ t

0

e(s)ds.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 3. Ito-Wiener expansion of the special functionals

Example

Looking for the solution of the kind

V (x ,s) = exp(−12

η2)

∞

∑k=0

Hk(η)Vk(x ,s)

one can nd

Vk(x ,s) =∫

...∫s≤r1≤...rk≤t

∫...

∫∞

0

qt−rk (x ,y1)∂

∂y1qrk−rk−1(y1,y2)...

∂

∂yk−1qr1−s(yk−1,yk)f (yk)dy1...dykdr1...drk .

Here q is the transition density of the killed at zero Wiener process.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Ito-Wiener expansion of the special functionals

Theorem

(Krylov-Veretennikov expansion) If

dx(u, t) = a(x(u, t))dt +b(x(u, t))dw(t), x(u,0) = u

then

f (x(u, t)) =

=∞

∑n=0

∫0≤s1≤...≤sn≤t

Tt−snBTsn−sn−1B . . .Ts1 f (u)dw(s1) . . .dw(sn).

Here Tt is a transition semigroup and B = b ddu.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 3. Ito-Wiener expansion of the special functionals

KrylovVeretennikov expansion for the Wiener processstopped at zero.

τ = inft : w(t) = 0

w(t) = w(τ ∧ t)

Tt(f )(u) = Euf (w(t)).

Lemma For a measurable bounded function f : R→ R andu ≥ 0

f (w(t)) = Tt f (u) +∞

∑k=1

∫∆k(t)

Tt−rk∂

∂vkTrk−rk−1 . . .

∂

∂v1Tr1f (v1)dw(r1) . . .dw(rk)

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 3. Ito-Wiener expansion of the special functionals

Coalescing Wiener processeswk ; k ≥ 1 are independent Wiener processes, rk ; k = 1, ...,nare real numbers .

x(rk , t) = wk(t), t ≥ 0.

σk+1 = inft :k

∏j=1

(x(rj , t)−wk+1(t)) = 0,

x(rk+1, t) =

wk+1(t), t ≤ σk+1

x(rk∗ , t), t ≥ σk+1.

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Lecture 3. Ito-Wiener expansion of the special functionals

Denition

An arbitrary set of the kind i , i +1, . . . , j, where i , j ∈N,i ≤ jis called a block.A representation of the block 1,2, . . . ,n as a union ofdisjoint blocks is called a partition of the block 1,2, . . . ,n.We say that a partition π2 follows from a partition π1 if itcoincides with π1 or if it is obtained by the union of twosubsequent blocks from π1.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 3. Ito-Wiener expansion of the special functionals

Denition

R is the set of all sequences of partitions π0, . . . ,πl where π0

is a trivial partition, π0 = 1,2, . . . ,n and every πi+1

follows from πi .Rk is the set of all sequences from R that have exactly kmatching pairs: πi = πi+1.

The set of strongly decreasing sequences we denote by R .

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Lecture 3. Ito-Wiener expansion of the special functionals

Let us associate with every partition π a vector ~λπ ∈ Rn withthe next property. For each block s, . . . , t from π thefollowing relation holds

t

∑q=s

λ2πq = 1.

Dene the processes xs , . . . ,xt after the moment τ and up tothe next moment of coalescence in the whole systemx1, . . . ,xn by the rule

xi (t) = xi (τ) +t

∑q=s

λπq(wq(t)−wq(τ)).

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 3. Ito-Wiener expansion of the special functionals

Denition

Operators related to a sequence of partitions π ∈ R .τ0 = 0< τ1 < .. . < τn−1 are the moments of coalescence forxk(t), k = 1, . . . ,n, ν = π0,ν1, . . . ,νn−1 is related randomsequence of partitions. The numbers i and j belong to thesame block in the partition νk if and only if xi (t) = xj(t) forτk ≤ t.

T πt f (u1, . . . ,un) =Ef (x1(t), . . . ,xn(t))1ν1=π1,...,νk=πk , τk≤t<τk+1.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 3. Ito-Wiener expansion of the special functionals

Theorem

f (x1(t), . . . ,xn(t)) = ∑π∈R

T πt f (u1, . . . ,un)+

∞

∑k=1

n

∑i1,...,ik=1

∑π∈Rk

k

∏j=1

λπj ij

∫4k(t)

T π1s1

∂i1Tπ2s2−s1 ...∂ikT

πk+1

t−sk

f (u1, . . . ,un)dwi1(s1)...dwik (sk).

If i ∈ s, ..., t then

∂i f =q=t

∑q=s

f ′q.

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Lecture 4. Arratia ow and web

Equation

dx(t) = a(x(t))dt +b(x(t))dw(t), x(s) = u

generates a ow of dieomorphisms ϕs,t

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Lecture 4. Arratia ow and web

1. X = Rd , for every u the process x(u, t) is a solution to the

Cauchy problem for SDE

dx(u, t) = a(x(u, t))dt +b(x(u, t))dw(t), x(u,0) = u.

2. The Harris ow of Brownian particles

X = R, x(u, t); u ∈ X , t ≥ 0 is a family of Brownian martingales

with respect to the common ltration, x is order-preserving and

d < x(u1,•), x(u2, •) >= ϕ(x(u1, t)− x(u2, t))dt,

where ϕ is a positive denite function.

3. The Arratia ow

ϕ(x) = 1x=0

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Lecture 4. Arratia ow and web

dxε (u, t) =∫R

ψε (xε (u, t)−p)W (dp,dt)∫R

ψ2

ε (u)du = 1, suppψε ⊂ [−ε, ε]

Theorem

The n−point motions of xε converge to the n−point motions of

the Arratia ow when ε → 0.The same statement holds when

ψ2

ε → p1δ−1 +p2δ1

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 4. Arratia ow and web

ξn, n ≥ 1 are the independent stationary Gaussian processes with

zero mean and covariation function Γ

x0(u) = u, xn+1(u) = xn(u) + ξn+1(xn(u)), u ∈ R

The sequences xn(u);n ≥ 0 and xn(u2)−xn(u1);n ≥ 0 have thesame distributions as the sequences yn(u);n ≥ 0, zn(u);n ≥ 0,which are dened by the following rules:

y0 = u, yn+1 = yn + ηn,

z0 = u2−u1, zn+1 = zn +√2Γ(0)−2Γ(zn)ηn,

where ηn;n ≥ 1 is a sequence of independent standard normal

variables.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 4. Arratia ow and web

xn(u, t) = n

(k+1

n− t

)xk(u)+n

(t− k

n

)xk+1(u),

u ∈ R, t ∈[k

n;k+1

n

],k = 0, . . . ,n−1.

Theorem

Let Γ be positive denite function on R such that Γ(0) = 1 and Γhas two continuous bounded derivatives. Suppose that xn is built

upon a sequence ξk ;k ≥ 1 with covariance 1√n

Γ. Then for every

u1, . . . ,ul ∈ R the random processes xn(uj , ·), j = 1, . . . , l weaklyconverge in C ([0;1], Rl ) to the l -point motion of the Harris ow

with the local characteristic Γ.

For the Arratia ow Γ = 1x=0

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 4. Arratia ow and web

Let

Cn = supR

2−2Γn(x)

x2

Theorem

Suppose that the following conditions hold

1)

limn→∞

CneCn

n= 0,

2)

supR\[−δ ;δ ]

|Γm(x)| → 0,m→ ∞,

for every δ > 0. Then for every u1, . . . ,ul ∈ R the random processes

xn(uj , ·), j = 1, . . . , l weakly converge in C ([0;1], Rl ) to the

l -point motion of the Arratia ow.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows

Lecture 4. Arratia ow and web

Take ψ ∈ C∞0

(R), suppψ ⊂ [−1; 1] and dene ψε (u) = 1√εψ(u

ε).

Put

Γε (u) =∫R

ψε (u)ψε (u−p)dp.

Suppose that

Γn =1√n

Γεn .

Theorem

Let the following conditions holds

εn→ 0, ε−2n = o(llnn)

Then the random functions xn converge in generalized

LévyProkhorov distance to the Arratia ow.

Andrey Dorogovtsev White noise analysis and coalescing stochastic ows