Stochastic calculus and its applications
Stochastic calculus and its applicationsAndrey A. DorogovtsevDepartment of Random ProcessesInstitute of Mathematics NAS [email protected], 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.
Andrey Dorogovtsev White noise analysis and coalescing stochastic ows
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 .
Andrey Dorogovtsev White noise analysis and coalescing stochastic ows
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.
Andrey Dorogovtsev White noise analysis and coalescing stochastic ows
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
Andrey Dorogovtsev White noise analysis and coalescing stochastic ows
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
Andrey Dorogovtsev White noise analysis and coalescing stochastic ows
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
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