Stochastic integral of Hitsuda–Skorokhod type on the extended Fock space
We review some recent results related to stochastic integrals of the Hitsuda–Skorokhod type acting on the extended Fock space and its riggings.
Збережено в:
| Дата: | 2009 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2009
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3056 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509081474695168 |
|---|---|
| author | Kachanovskii, N. A. Tesko, V. A. Качановський, М. О. Теско, В. А. |
| author_facet | Kachanovskii, N. A. Tesko, V. A. Качановський, М. О. Теско, В. А. |
| author_sort | Kachanovskii, N. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:44:24Z |
| description | We review some recent results related to stochastic integrals of the Hitsuda–Skorokhod type acting on the extended Fock space and its riggings. |
| first_indexed | 2026-03-24T02:35:26Z |
| format | Article |
| fulltext |
UDC 517.9
N. A. Kachanovsky, V. A. Tesko (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
STOCHASTIC INTEGRAL OF HITSUDA – SKOROKHOD TYPE
ON THE EXTENDED FOCK SPACE
СТОХАСТИЧНИЙ IНТЕГРАЛ ТИПУ ХIТЦУДИ – СКОРОХОДА
НА РОЗШИРЕНОМУ ПРОСТОРI ФОКА
We review some recent results connected with stochastic integrals of Hitsuda – Skorokhod type acting on the
extended Fock space and its riggings.
Наведено огляд деяких останнiх результатiв, пов’язаних iз стохастичними iнтегралами типу Хiтцуди –
Скорохода, що дiють на розширеному просторi Фока та його оснащеннях.
1. Introduction. The problem of extension for an Itô stochastic integral is a subject of
interest of many researchers. First who proposed such extensions were M. Hitsuda [1],
Yu. L. Daletsky [2, 3], A. V. Skorokhod and Yu. M. Kabanov [4 – 6]. The definiti-
ons of the extended stochastic integral proposed by M. Hitsuda and A. V. Skorokhod
were equivalent and given in terms of the Fock space structure by using the Chaos
Representation Property (CRP) of the Wiener process (this property was derived by Itô
in [7]). Yu. L. Daletsky used another approach: his extension based on the integration by
parts formula. In [6] Yu. M. Kabanov introduced the notion of the Hitsuda – Skorokhod
type stochastic integral in the case of integration with respect to a compensated Poisson
process (this process also possesses the CRP, see, e.g., [8]). Afterwards it became clear
that in the construction of the Hitsuda – Skorokhod type integral, the Gaussian and Poi-
sson character of processes never appears. One uses only the CRP of Wiener or Poisson
processes. Thus in [9] (see also [10, 11]) it was shown that the Hitsuda – Skorokhod
integral as an operator on the Fock space is an extension of the Itô integral not only in
the Wiener and Poisson cases but in the case of any normal martingale with CRP (the
reader can find examples and properties of normal martingales with CRP in, e.g., [9,
12 – 16]).
In the present paper we will explore the Hitsuda – Skorokhod type integral connected
with some normal martingales without CRP. But in order to explain our motivation and
make our considerations clear, first we recall the Gaussian case (see, e.g., [17 – 19] for
more detailed presentation).
Let µG be the Gaussian measure on the Schwartz distributions space D′ = D′(R+)
and L2(D′, µG) be the corresponding L2-space. By definition the space D′ = D′(R+)
is the dual one of the Schwartz space D = D(R+) of infinite differentiable functions
on R+ with compact supports. Denote by 〈x, ϕ〉 the action of x ∈ D′ or ϕ ∈ D and
construct a Wiener process {Wt}t∈R+ by the formula
Wt(x) := 〈x,1[0,t)〉 := lim
n→∞
〈x, ϕn〉 (limit in L2(D′, µG)), (1.1)
where {ϕn}∞n=0 ⊂ D is a sequence converging in L2(R+) = L2(R+, dt) to the indicator
function 1[0,t) of the set [0, t). Note that passing to a limit in (1.1) is possible due to
properties of the Gaussian measure µG, see Section 2 for details.
The CRP of {Wt}t∈R+ implies that for any function F ∈ L2(D′, µG) there exists a
uniquely defined vector f = (fn)∞n=0 from the symmetric Fock space
c© N. A. KACHANOVSKY, V. A. TESKO, 2009
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6 733
734 N. A. KACHANOVSKY, V. A. TESKO
F := C⊕
∞⊕
n=1
L2
C(R+)b⊗nn!
such that
F =
∞∑
n=0
In(fn), In(fn) := n!
∫
∆n
fn(t1, . . . , tn) dWt1 . . . dWtn , (1.2)
where ∆n = {(t1, . . . , tn) ∈ Rn+| t1 < . . . < tn} and In(fn) is a multiple stochastic
integral. More exactly, the mapping (the so-called Wiener – Itô – Segal isomorphism)
IG : F → L2(D′, µG), f = (fn)∞n=0 7→ IGf :=
∞∑
n=0
In(fn), (1.3)
is a well-defined isometrically isomorphic (unitary) operator. Note that Wt(x) =
(
IG(0,
1[0,t), 0, 0, . . .)
)
(x).
It should be noticed that the isomorphism IG has a simple and naturale interpretation
from the spectral point of view. Namely, it is possible to understand the mapping IG
as the Fourier transform of a certain family (the so-called free field) of commuting
selfadjoint operators that act in the Fock space F and have a Jacobi structure. This
result was obtained by V. D. Koshmanenko and Yu. S. Samoilenko in [20]; see also [21].
Taking into account this fact, we can rewrite representation (1.3) in the form
(IGf)(·) =
∞∑
n=0
〈Pn(·), fn〉 ∈ L2(D′, µG), (1.4)
where each 〈Pn(·), fn〉 is a polynomial of the first kind connected with the free field or,
in other terminology, 〈Pn(·), fn〉 is a generalized Hermite polynomial on D′, see, e.g.,
[21, 17, 22].
Now we are ready to pass to the definition of the Hitsuda – Skorokhod integral. Let
F ∈ L2(R+;L2(D′, µG)) ∼= L2(D′, µG)⊗L2(R+). Then, for almost all t ∈ R+, we can
apply Wiener – Itô – Segal expansion (1.2) to the function F (t) = F (·, t) ∈ L2(D′, µG)
and write
F (t) =
∞∑
n=0
n!
∫
∆n
fn(t1, . . . , tn; t) dWt1 . . . dWtn . (1.5)
If F is integrable by Itô with respect to W then using term by term integration we obtain∫
R+
F (t) dWt =
∞∑
n=0
(n+ 1)!
∫
∆n+1
f̂n(t1, . . . , tn, t) dWt1 . . . dWtndWt =
=
∞∑
n=0
In+1(f̂n) ∈ L2(D′, µG),
where f̂n is the symmetrization of fn(t1, . . . , tn; t) with respect to n+ 1 variables. This
representation of the Itô integral suggests us to define its extension by∫
R+
F (t) d̂Wt :=
∞∑
n=0
In+1(f̂n) (1.6)
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
STOCHASTIC INTEGRAL OF HITSUDA – SKOROKHOD TYPE ON THE EXTENDED FOCK SPACE 735
for all F ∈ L2(R+;L2(D′, µG)) such that
∞∑
n=0
In+1(f̂n) ∈ L2(D′, µG) or, equivalently, (0, f̂0, f̂1, . . . ) ∈ F .
Note that exactly in such a way the extended stochastic integral was defined by Hitsuda
and Skorokhod.
Clearly, one can identify
∫
R+
F (t) d̂Wt with the vector (0, f̂0, f̂1, . . . ) from the Fock
space F and consider this integral as an unbounded operator
Iext : L2(R+;F)→ F , f(·) = (fn(·))∞n=0 7→ Iext(f) := (0, f̂0, f̂1, . . . ), (1.7)
with the domain
Dom (Iext) :=
{
f(·) = (fn(·))∞n=0 ∈ L2(R+;F)
∣∣ (0, f̂0, f̂1, . . . ) ∈ F
}
.
Further, one can interpret the Malliavin’s gradient (the stochastic derivative, see, e.g.,
[17, 19]) as an operator acting from F to L2(R+;F); to formulate “on this language”
some properties of the stochastic integral and the stochastic derivative (for example, the
stochastic integral and the stochastic derivative are adjoint one to another operators) etc.
If we apply a Wiener – Itô – Segal type isomorphism to the integral Iext we obtain
a naturale extension of the Itô integral not only in the Wiener case but in the case of
any normal martingale with CRP. Moreover, the properties of the extended stochastic
integral and the stochastic derivative that can be formulated “on the language of Fock
spaces”, i.e., with using of the coefficients from (1.2) – (1.6) only, coincide (up to the
corresponding Wiener – Itô – Segal type isomorphisms) for all these martingales. Thus
this point of view enables us to treat the stochastic analysis of all these processes in the
one framework (as the analysis on the Fock space F).
In view of this it is natural to ask: “is it possible to construct an analog of the
Hitsuda – Skorokhod integral for processes without the CRP? ”. Recently it became clear
(see [23 – 25]) that this is possible at least for the cases of stochastic integration with
respect to Gamma, Pascal and Meixner processes (the processes of Meixner type). In
spite of the fact that these processes are normal martingales without the CRP, they are
connected with Jacobi fields (generalizations of the free field), which act in the so-called
extended Fock space
Fext := C⊕
∞⊕
n=1
Fn,extn!. (1.8)
Here each Fn,ext consists of symmetric square integrable with respect to some measure
ρn functions (see, e.g., [26, 27]). It should be noticed that the theory of Jacobi fields in
the Fock space was created by Yu. M. Berezansky in [28] and carried out by him and
his collaborators, see survey [29] for a more complete bibliography.
So, it follows from results of [30] (see also [31 – 38]) that for each process of
Meixner type there exists a Jacobi field in the space Fext that is a certain family
A = {A(ϕ)}ϕ∈D of commuting selfadjoint operators. These operators have a Jacobi
structure and are connected with the orthogonal decomposition in (1.8). Applying the
projection spectral theorem to this field we can construct a Fourier transform I with
respect to the generalized joint eigenvectors of the family A. This transform has a form
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
736 N. A. KACHANOVSKY, V. A. TESKO
similar to (1.4), but the operator I is a unitary between the spaces Fext and L2(D′, µ),
I : Fext → L2(D′, µ), f = (fn)∞n=0 7→ (If)(·) :=
∞∑
n=0
〈Pn(·), fn〉, (1.9)
where µ is a measure of Meixner type and {Pn(x)}∞n=0 = P (x) is a generalized
joint eigenvector of the operator A(ϕ) corresponding to the eigenvalue 〈P1(x), ϕ〉 ≡
≡ 〈x, ϕ〉. The sequence {Pn(x)}∞n=0 is called the sequence of polynomials of the first
kind connected with the family A.
In this case, the process of Meixner type {Mt}t∈R+ is defined by the formula
Mt(x) := 〈x,1[0,t)〉 :=
(
I(0,1[0,t), 0, 0, . . .)
)
(x).
Note also that 〈Pn(·), fn〉, n ∈ Z+, as well as the generalized Hermite polynomials,
are Schefer polynomials, that is orthogonal polynomials with a generating function of
exponential type, see [36, 37, 30] for more details.
In the present paper, using Fourier transform (1.9), we introduce and study extended
stochastic integrals connected with processes of Meixner type. We define these integrals
by analogy with (1.7), but using instead of the Fock space F the extended Fock space
Fext. Note that related results to this topic have been established in [39 – 41].
The paper is organized in the following manner. In the forthcoming section we give
a brief introduction in the Gaussian white noise analysis and recall the construction of
stochastic integrals on a Fock space and its riggings in the framework of this analysis,
this section serves as a model example. In Section 3 we give a necessary information
about the generalized Meixner measure and the extended Fock space. Section 4 is
devoted to the construction and study of the Itô stochastic integral on the extended Fock
space. Finally, in Section 5 we give definitions and establish main properties of extended
stochastic integrals on the extended Fock space and its riggings.
2. Stochastic integrals in the Gaussian white noise analysis. In this section we
recall some basic concepts of the Gaussian white noise analysis (see, e.g., [17, 21, 18] for
more details) and describe a general approach to construction of the extended stochastic
integral.
2.1. Elements of the Gaussian white noise analysis. Denote by D := C∞0 (R+)
the set of all real-valued infinite differentiable functions on R+ with compact supports.
This set can be naturally endowed with a projective limit topology
D = pr lim
τ∈T
Dτ ,
where T denotes the set of all pairs τ = (τ1, τ2) such that τ1 ∈ N and τ2 is an infinite
differentiable function on R+ such that τ2(t) ≥ 1 for all t ∈ R+; Dτ is the closure of
C∞0 (R+) in the norm | · |Dτ generated by the scalar product
(ϕ,ψ)Dτ =
∫
R+
(
τ1∑
k=0
ϕ(k)(t)ψ(k)(t)
)
τ2(t) dt,
i.e., Dτ denotes the Sobolev space of order τ1 weighted by the function τ2. Henceforth
we will regard D as the corresponding topological space.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
STOCHASTIC INTEGRAL OF HITSUDA – SKOROKHOD TYPE ON THE EXTENDED FOCK SPACE 737
As is known (see, e.g., [21, 42]), Dτ are densely and continuously embedded into
the space L2(R+) of square integrable with respect to the Lebesgue measure real-valued
functions on R+. Therefore one can consider the chain (the rigging of L2(R+))
D′ ⊃ D−τ ⊃ L2(R+) ⊃ Dτ ⊃ D, (2.1)
where D−τ , D′ = ind limτ∈T D−τ are the dual of Dτ , D with respect to L2(R+)
spaces respectively. We denote by 〈· , ·〉 the dual pairing between elements of D′ and D
(and also D−τ and Dτ ) inducted by the scalar product (· , ·)L2(R+) in L2(R+), i.e., we
set
〈f, ϕ〉 := (f, ϕ)L2(R+), f ∈ L2(R+), ϕ ∈ D,
and then extend this definition by continuity. We preserve the notation 〈· , ·〉 for the dual
pairings in tensor powers and complexifications of chain (2.1).
We denote by C(D′) the generated by cylinder sets σ-algebra on D′. Let µG be the
Gaussian measure on C(D′), i.e., a probability measure with the Fourier transform∫
D′
ei〈x,ϕ〉µG(dx) = e
− 1
2 |ϕ|
2
L2(R+) , ϕ ∈ D. (2.2)
Denote by L2(D′, µG) the space of square integrable with respect to µG complex-valued
functions on D′. It follows from (2.2) that∫
D′
〈x, ϕ〉2µG(dx) = |ϕ|2L2(R+), ϕ ∈ D.
Therefore, extending the mapping
L2(R+) ⊃ D 3 ϕ 7→ 〈· , ϕ〉 ∈ L2(D′, µG)
by continuity, we obtain a random variable 〈· , f〉 ∈ L2(D′, µG) for each f ∈ L2(R+).
Thus we can define a random process {Wt}t∈R+ as
Wt(·) := 〈· ,1[0,t]〉
(here and below 1A denotes the indicator of a set A). It is easy to see that finite-
dimensional distributions of a random process W· coincide with those of a Wiener one.
Namely, for all N ∈ N, u1, . . . , uN ∈ R and t1, . . . , tN ∈ R+∫
D′
exp
(
i
N∑
k=1
ukWtk(x)
)
µG(dx) =
∫
D′
exp
(
i
〈
x,
N∑
k=1
uk1[0,tk]
〉)
µG(dx) =
= exp
−1
2
∣∣∣∣∣
N∑
k=1
uk1[0,tk]
∣∣∣∣∣
2
L2(R+)
= exp
−1
2
N∑
k,j=1
ukuj min{tk, tj}
.
Hence {Wt}t∈R+ can be interpreted as a Wiener process.
An important technical tool in the Gaussian white noise analysis is the Wiener –
Itô – Segal isomorphism
IG : F → L2(D′, µG)
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
738 N. A. KACHANOVSKY, V. A. TESKO
between the symmetric Fock space F := F(L2(R+)) over L2(R+) and the complex
Hilbert space L2(D′, µG). Let us recall that the symmetric Fock space F is defined as
F =
∞⊕
n=0
L2
C(R+)b⊗nn!, L2
C(R+)b⊗0 := C,
i.e., F is a complex Hilbert space of sequences f = (fn)∞n=0, fn ∈ L2
C(R+)b⊗n, such
that
‖f‖2F =
∞∑
n=0
|fn|2L2
C(R+) b⊗nn! <∞.
Here and below ⊗̂ denotes a symmetric tensor product (⊗ denotes an ordinary tensor
product), the subindex C denotes the complexification of a real space.
In what follows, we always identify in the natural way the space L2
C(R+)b⊗n with the
space L2
C,sym(Rn+) of all complex-valued symmetric functions from L2
C(Rn+). Namely,
we identify each element g1⊗̂ . . . ⊗̂gn ∈ L2
C(R+)b⊗n with the symmetric function
1
n!
∑
σ
g1(tσ(1)) . . . gn(tσ(n)) ∈ L2
C,sym(Rn+)
(σ running over all permutations of {1, . . . , n}) and extend this procedure to all elements
of L2
C(R+)b⊗n. This is possible because the mapping
g1⊗̂ . . . ⊗̂gn 7→
1
n!
∑
σ
g1(tσ(1)) . . . gn(tσ(n))
after being extended by linearity and continuity to the whole space L2
C(R+)b⊗n is a
unitary operator acting from L2
C(R+)b⊗n to L2
C,sym(Rn+). Naturally
|fn|2L2
C(R+) b⊗n =
∫
Rn+
|fn(t1, . . . , tn)|2 dt1 . . . dtn =
= n!
∫
∆n
|fn(t1, . . . , tn)|2 dt1 . . . dtn
for all fn ∈ L2
C(R+)b⊗n ∼= L2
C,sym(Rn+), where ∆n = {(t1, . . . , tn) ∈ Rn+ | t1 < . . .
. . . < tn}.
Return to the Wiener – Itô – Segal isomorphism IG. There are several equivalent ways
of construction of IG: using multiple stochastic integrals either the Jacobi fields approach
or the system of infinite-dimensional Hermite polynomials. We do not discuss this in
details (see, e.g., [17, 19 – 22] and references therein), but we note that IG is completely
characterized by the following properties:
(i) IG : F → L2(D′, µG) is a unitary operator (an isometrical isomorphism);
(ii) IG(f0, 0, 0, . . .) = f0 for all f0 ∈ C;
(iii) for each n ∈ N and any disjoint Borel sets α1, . . . , αn of finite Lebesgue measure(
IG
(
0, . . . , 0︸ ︷︷ ︸
n times
,1α1⊗̂ . . . ⊗̂1αn , 0, 0, . . .
))
(·) = Wα1(·) . . .Wαn(·),
where Wαk(·) := 〈· ,1αk〉 for all k ∈ {1, . . . , n}.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
STOCHASTIC INTEGRAL OF HITSUDA – SKOROKHOD TYPE ON THE EXTENDED FOCK SPACE 739
It should be noted that properties (i) – (iii) of IG and the fact that the set
C
⊕
span
{(
0, . . . , 0︸ ︷︷ ︸
n times
,1α1⊗̂ . . . ⊗̂1αn , 0, 0, . . .
) ∣∣∣∣∣
n ∈ N; αi ∈ B(R+); αi ∩ αj = ∅, i 6= j
}
is dense in the Fock space F play a fundamental role in the construction of the Itô
integral and its extensions in terms of the Fock space structure, see below for more
details.
Let us construct a convenient for our considerations rigging of F . For τ ∈ T and
q ∈ N we set
F(τ, q) :=
∞⊕
n=0
D
b⊗n
τ,C(n!)22qn, F+ := pr lim
τ∈T,q∈N
F(τ, q),
where F(τ, q) denotes a complex Hilbert space of sequences f = (fn)∞n=0 such that
fn ∈ D
b⊗n
τ,C (D
b⊗0
τ,C := C) and
‖f‖2F(τ,q) :=
∞∑
n=0
|fn|2
D
b⊗n
τ,C
(n!)22qn <∞.
It can be shown that for all q ∈ N and τ ∈ T the dense and continuous embedding
F(τ, q) ↪→ F takes place. Thus one can construct a rigging of the Fock space F
F− ⊃ F(−τ,−q) ⊃ F ⊃ F(τ, q) ⊃ F+, (2.3)
where the spaces
F(−τ,−q) =
∞⊕
n=0
D
b⊗n
−τ,C2−qn, F− = ind lim
τ∈T,q∈N
F(−τ,−q) (2.4)
are dual ones of F(τ, q) and F+ with respect to the zero space F respectively. The
(generated by the scalar product in F) pairing between elements of F− and F+ (and
also F(−τ,−q) and F(τ, q)) will be denoted by 〈〈· , ·〉〉F .
Using rigging (2.3) and the isomorphism IG one can construct the rigging
F− ⊃ F ⊃ F+
↓ IG ↓ IG
(D′)− ⊃ L2(D′, µG) ⊃ (D)+,
where the space of test functions (D)+ := IGF+ is the IG-image of the Fock space
F+ with the topology that is inducted by the topology of F+, the space of generalized
functions (D′)− is the dual one of (D)+ with respect to L2(D′, µG). Note that IG can
be extended to an isomorphism between F− and (D′)−. We keep the same notation IG
for the corresponding extension.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
740 N. A. KACHANOVSKY, V. A. TESKO
Now we recall definitions of some important operators on Fock spaces.
For each ξ = (ξn)∞n=0 ∈ F− we define the S-transform by the formula
(Sξ)(λ) :=
∞∑
n=0
〈ξn, λ⊗n〉, λ ∈ DC,
where the series converges absolutely in a (depending on ξ) neighborhood of 0 ∈ DC.
Each vector ξ from F− is uniquely determined by its S-transform. More exactly, let
Hol 0(DC) be the set of all (germs of) functions that are holomorphic at 0 ∈ DC. It
follows from [43] that the S-transform is a one-to-one map between F− and Hol 0(DC).
Taking into account that Hol 0(DC) is an algebra with ordinary algebraic operations
we can define a Wick product ξ♦ζ of ξ, ζ ∈ F− by the formula
ξ♦ζ := S−1(Sξ · Sζ) ∈ F−.
It is easy to calculate that for all ξ = (ξn)∞n=0, ζ = (ζn)∞n=0 ∈ F−
ξ♦ζ =
(
n∑
m=0
ξm⊗̂ζn−m
)∞
n=0
. (2.5)
Furthermore, if ξ = (ξn)∞n=0 ∈ F− and
h(·) =
∞∑
n=0
hn(· − ξ0)n : C→ C
is a holomorphic at (Sξ)(0) = ξ0 function then one defines the Wick version of h by
h♦(ξ) := S−1h(Sξ) ∈ F−.
As is easy to see,
h♦(ξ) =
∞∑
n=0
hn(0, ξ1, ξ2, . . . )♦n, (2.6)
where ξ♦n := ξ♦ . . .♦ξ (n times) and ξ♦0 := 1.
Using the isomorphism IG, all the above definitions and results can be reformulated
in terms of the generalized functions space (D′)−. In particular, a Wick product and
Wick versions of holomorphic functions can be defined on (D′)− and used in order to
study so-called stochastic equations with Wick-type nonlinearities (see, e.g., [44, 17]).
For each t ∈ R+ we define the annihilation operator a−(δt) on F+ and the creati-
on operator a+(δt) on F− (here δt denotes the delta function at t) by setting “on
coordinates”
(a−(δt)ϕn)(t1, . . . , tn−1) := nϕn(t1, . . . , tn−1, t), ϕn ∈ D
b⊗n
C ;
(a+(δt)ξn) := δt⊗̂ξn, ξn ∈ D′C
b⊗n
.
(2.7)
It is easy to show (see, e.g., [22]) that the operators a−(δt) and a+(δt) can be extended
to linear continuous operators on F(τ, q) and F(−τ,−q) respectively, and a+(δt) is the
dual operator of a−(δt) in the sense that for all ξ ∈ F(−τ,−q) and ϕ ∈ F(τ, q)
〈〈a+(δt)ξ, ϕ〉〉F = 〈〈ξ, a−(δt)ϕ〉〉F .
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
STOCHASTIC INTEGRAL OF HITSUDA – SKOROKHOD TYPE ON THE EXTENDED FOCK SPACE 741
It is obvious that
a+(δt)ξ = ξ♦(0, δt, 0, 0, . . .), ξ ∈ F(−τ,−q). (2.8)
Also we note that
∂(δt) := IGa−(δt)I−1
G : (D)+ → (D)+
is the Gateaux derivative in the direction δt:
(∂(δt)F )(x) = lim
ε→0
F (x+ εδt)− F (x)
ε
, F ∈ (D)+, x ∈ D′,
see, e.g., [17]. The operator ∂(δt) is called the Hida derivative.
2.2. Stochastic integrals on a Fock space and its riggings. In this subsection we
recall the construction of stochastic integrals on a Fock space and its riggings in the
framework of the Gaussian white noise analysis. Namely, starting from the classical Itô
integral with respect to a Wiener process, we define the Itô integral on the Fock space
F and construct its generalization — the extended (Hitsuda – Skorokhod type) stochastic
integral on F and its riggings. We stress that this approach enables us to define the
extended stochastic integral not only with respect to a Wiener process but also with
respect to any normal martingale with the Chaos Representation Property (CRP).
We start from a definition of the classical Itô integral (we refer, e.g., to the books
[45, 46] for details). Let {At}t∈R+ be a natural filtration of σ-algebras At = σ{Ws | s ≤
≤ t} generated by a Wiener process {Wt}t∈R+ (this filtration is made complete and
right continuous). We denote by L2
a(R+ × D′) the set of all adapted with respect to
{At}t∈R+ functions from the space
L2(D′ × R+) := L2(D′ × R+, C(D′)× B(R+), µG × dt) ∼= L2(D′, µG)⊗ L2(R+),
where B denotes the Borel σ-algebra. It can be shown that L2
a(R+ ×D′) is a subspace
of L2(R+×D′) (a linear closed subset of L2(R+×D′)). We will refer to L2
a(R+×D′)
as to the space of Itô integrable functions.
Let us recall that a function F ∈ L2(D′ × R+) is adapted (or nonanticipative) with
respect to the filtration {At}t∈R+ if for almost all t ∈ R+ the function F (·, t) is At-
measurable. In other words, F ∈ L2(D′ × R+) is adapted with respect to {At}t∈R+ if
F (·, t) = E[F (·, t) |At] for almost all t ∈ R+, where E[ · |At] denotes the conditional
expectation with respect to the σ-algebra At. We note that E[ · |At] is the orthogonal
projection in L2(D′, µG) onto the subspace of all At-measurable functions.
Let F (t) = F (x, t) be a simple Itô integrable function. That is, F ∈ L2
a(R+ × D′)
can be written as
F (·) =
n−1∑
k=0
F(k)1(tk,tk+1](·) ∈ L2
a(D′ × R+),
where 0 ≤ t0 < t1 < . . . < tn < ∞ (evidently, every F(k) is an Atk -measurable
function from the space L2(D′, µG)). The Itô integral of F with respect to a Wiener
process W· is defined by the formula∫
R+
F (t) dWt :=
n−1∑
k=0
F(k)(Wtk+1 −Wtk) ∈ L2(D′, µG) (2.9)
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
742 N. A. KACHANOVSKY, V. A. TESKO
and has the isometry property∥∥∥∥∥
∫
R+
F (t) dWt
∥∥∥∥∥
2
L2(D′,µG)
= ‖F‖2L2(D′×R+) ≡
∫
R+
‖F (t)‖2L2(D′,µG) dt. (2.10)
Since the set L2
a,s(D′ × R+) of all simple Itô integrable functions is dense in
L2
a(D′ × R+) (with respect to the topology of L2(D′ × R+)), isometry property (2.10)
allows us to extend the Itô integral to the set L2
a(R+ × D′) of Itô integrable functions,
and (2.10) still holds on this set. Namely, extending the mapping
L2
a,s(D′ × R+) 3 F 7→
∫
R+
F (t) dWt ∈ L2(D′, µG)
by continuity we obtain a definition of the Itô integral on L2
a(D′ × R+).
Let us now turn from the Itô integral on the space L2(D′, µG) to one on the Fock
space F . This integral will be defined in the simplest possible way as the I−1
G -image of
the Itô integral
∫
R+
F (t) dWt. To be precise, denote by L2(R+;F) the Hilbert space of
F-valued functions (more exactly, of equivalence classes)
R+ 3 t 7→ f(t) ∈ F , ‖f‖2L2(R+;F) :=
∫
R+
‖f(t)‖2F dt <∞
with the corresponding scalar product. It is clear that any function f from the space
L2(R+;F) has a form f(t) = (fn(t))∞n=0, where each fn(t1, . . . , tn; t) belongs to
the space L2
C(R+)b⊗n ⊗ L2(R+). This means that fn belongs to L2
C(Rn+1
+ ) and fn is
symmetric with respect to first n variables.
Since the spaces L2(D′ ×R+) and L2(R+;F) can be interpreted as tensor products
L2(D′, µG)⊗ L2(R+) and F ⊗ L2(R+) respectively, we conclude that
IG ⊗ 1: L2(R+;F)→ L2(D′ × R+)
is a well-definite unitary operator.
Definition 2.1. We say that a function f ∈ L2(R+;F) is Itô integrable if (IG⊗1)f
belongs to L2
a(D′ × R+), i.e., if
f ∈ L2
a(R+;F) := (IG ⊗ 1)−1L2
a(D′ × R+).
The Itô integral of f ∈ L2
a(R+;F), denoted by I(f), is defined by
I(f) := I−1
G
∫
R+
IG(f(t)) dWt
∈ F .
Remark 2.1. It follows from definition of I and equality (2.10) that, for all f ∈
∈ L2
a(R+;F),
‖I(f)‖2F =
∫
R+
‖f(t)‖2F dt.
As a consequence, the operator I acts isometrically from the subspace L2
a(R+;F) of
L2(R+;F) into the Fock space F .
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
STOCHASTIC INTEGRAL OF HITSUDA – SKOROKHOD TYPE ON THE EXTENDED FOCK SPACE 743
It is natural now to ask: “How to verify that a function f(·) = (fn(·))∞n=0 ∈
∈ L2(R+;F) is Itô integrable and how to express the corresponding Itô integral in
terms of the Fock space structure?” The answer is following.
Theorem 2.1. The following statements are fulfilled:
(I) A function R+ 3 t 7→ f(t) = (fn(t))∞n=0 ∈ F is Itô integrable (i.e., f belongs
to L2
a(R+;F)) if and only if f ∈ L2(R+;F) and for almost all t ∈ R+
f(t) = (f0(t), f1(t)1[0,t), f2(t)1[0,t)2 , . . .).
(II) For each f(·) = (fn(·))∞n=0 ∈ L2
a(R+;F)
I(f) = (0, f̂0, f̂1, . . .) ∈ F , (2.11)
where f̂n ∈ L2
C(R+)b⊗n+1 denotes the symmetrization of fn(t1, . . . , tn; t) with respect
to all variables, or, equivalently, f̂n is the projection of fn ∈ L2
C(R+)b⊗n⊗L2(R+) onto
L2
C(R+)b⊗n+1. Since the function fn(t1, . . . , tn; t) is symmetric with respect to first n
variables, its symmetrization f̂n is given by
f̂n(t1, . . . , tn+1) :=
1
n+ 1
n+1∑
k=1
fn(t1, . . . , tk� , . . . , tn+1; tk).
Although this theorem easily follows from the results of, e.g., [19, 9, 11], for the
reader’s convenience we present here a proof.
Proof. In order to prove (I), it is sufficient to show that
E[IGfn|At] = IG(1[0,t)nfn) (2.12)
for any fn ∈ L2
C(R+)b⊗n (here and below in this proof we identify fn with (0, . . . , 0, fn, 0,
0, . . .) ∈ F , where fn standing at the n-th position). Moreover, since functions
fn = 1α1⊗̂ . . . ⊗̂1αn , αi ∈ B(R+), αi ∩ αj = ∅, i 6= j,
form a total set in L2
C(R+)b⊗n, it is sufficient to check (2.12) for these functions. Using
property (iii) of IG, properties of a conditional expectation and the fact that
E[Ws|At] = Wt, t ≤ s,
since the Wiener process W is a martingale with respect to {At}t∈R+ , we get
E[IGfn|At] = E[IG(1α1⊗̂ . . . ⊗̂1αn)|At] = E[Wα1 . . .Wαn |At] =
= E
[
n∏
i=1
(
Wαi∩[0,t) +Wαi∩[t,∞)
) ∣∣At] =
= Wα1∩[0,t) . . .Wαn∩[0,t) =
= IG(1α1∩[0,t)⊗̂ . . . ⊗̂1αn∩[0,t)) = IG(1[0,t)nfn).
The first part of the theorem is proved.
Let us establish the second part of the theorem. First of all we note that according to
Remark 2.1 and Theorem 3.2 from [11] we have
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
744 N. A. KACHANOVSKY, V. A. TESKO
‖I(f)‖2F =
∫
R+
‖f(t)‖2F dt and ‖(0, f̂0, f̂1, . . .)‖2F =
∫
R+
‖f(t)‖2F dt
for all f ∈ L2
a(R+;F). Hence, the linear mappings
f 7→ I(f) and f 7→ (0, f̂0, f̂1, . . .)
act continuously (more exactly, isometrically) from L2
a(R+;F) to F . Therefore, it is
sufficient to check (2.11) for simple functions f ∈ L2
a(R+;F) of form
f(t) = g1(s1,s2](t), g = 1α1⊗̂ . . . ⊗̂1αn , (2.13)
where n ∈ N, Borel sets αi ∈ B(R+), i ∈ {1, . . . , n}, are disjoint and (s1, s2] ⊂ R+
(these functions form a total set in L2
a(R+;F)). We note that if f(·) = g1(s1,s2](·) ∈
∈ L2
a(R+;F) has form (2.13) then by assertion (I)
g = 1α1⊗̂ . . . ⊗̂1αn = (1α1⊗̂ . . . ⊗̂1αn)1[0,s1]n .
So in this case αi ⊂ [0, s1]. In particular αi ∩ (s1, s2] = ∅ for all i ∈ {1, . . . , n}.
Let f(·) = g1(s1,s2](·) ∈ L2
a(R+;F) be of form (2.13). Evidently, in this case
f(t) =
(
0, . . . , 0︸ ︷︷ ︸
n times
, fn(t), 0, 0, . . .
)
, fn(t) := (1α1⊗̂ . . . ⊗̂1αn)1(s1,s2](t),
and
F := (IG ⊗ 1)f = IG(g)1(s1,s2]
is a simple Itô integrable function with respect toW, i.e., F ∈ L2
a,s(D′×R+). Therefore,
using Definition 2.1, equality (2.9), property (iii) of the isomorphism IG and taking into
account that αi ∈ B(R+), i ∈ {1, . . . , n}, are disjoint and αi ∩ (s1, s2] = ∅, we get
I(f) = I−1
G
∫
R+
IG(f(t)) dWt
= I−1
G
∫
R+
IG(g)1(s1,s2](t) dWt
=
= I−1
G
(
IG(g)W(s1,s2]
)
= I−1
G
(
IG(1α1⊗̂ . . . ⊗̂1αn)W(s1,s2]
)
=
= I−1
G
(
(Wα1 · . . . ·Wαn)W(s1,s2]
)
= 1α1⊗̂ . . . ⊗̂1αn⊗̂1(s1,s2] =
=
(
0, . . . , 0︸ ︷︷ ︸
n+1 times
, f̂n, 0, 0, . . .
)
.
The theorem is proved.
Remark 2.2. This theorem is one of the most useful results for our purpose.
Analyzing the proof, we see that it does not depend upon the Gaussian character of the
Wiener – Itô – Segal isomorphism IG. One only makes use the properties (i) – (iii) of the
isomorphism IG and the fact that the Wiener processW is a martingale. This observation
plays a crucial role in the construction of the extended stochastic integral with respect
to any normal martingale with the CRP (see Remark 2.7).
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
STOCHASTIC INTEGRAL OF HITSUDA – SKOROKHOD TYPE ON THE EXTENDED FOCK SPACE 745
Remark 2.3. Let (s1, s2] ⊂ R+ be fixed. Choose a vector g = (gn)∞n=0 ∈ F such
that g = (g0, g11[0,s1), g21[0,s1)2 , . . .) and define a simple function f ∈ L2
a(R+;F) by
f(t) := g1(s1,s2](t) = (fn(t))∞n=0, fn(t) := gn1(s1,s2](t).
Then, according to (2.11) and (2.5) we obtain
I(g1(s1,s2]) = I(f) = (0, f̂0, f̂1, . . .) =
= (0, g0⊗̂1(s1,s2], g1⊗̂1(s1,s2], . . .) = g♦(0,1(s1,s2], 0, 0, . . .). (2.14)
If we compare (2.14) with (2.9) we will see the relationship between the Wick multipli-
cation ♦ on F and the ordinary multiplication on L2(D′, µG). Namely, suppose t ∈ R+
and F ∈ L2(D′, µG) is an At-adapted function. Then for each interval (s1, s2] ⊂ (t,∞)
the function F (Ws2−Ws1) belongs to L2(D′, µG) and the I−1
G -image of F (Ws2−Ws1)
has the form
I−1
G (F (Ws2 −Ws1)) = I−1
G (F )♦I−1
G (Ws2 −Ws1) = I−1
G (F )♦(0,1(s1,s2], 0, 0, . . .).
However it can be shown that in general case the IG-image of the Wick multiplication
♦ distinguishes from the ordinary multiplication.
We next turn our attention to generalizations of the Itô integral I. The most naive
and natural idea is to define a generalization of I by formula (2.11) for all functions
f(·) = (fn(·))∞n=0 ∈ L2(R+;F) such that (0, f̂0, f̂1, . . .) ∈ F . Namely, we accept the
following definition.
Definition 2.2. For a function f(·) = (fn(·))∞n=0 ∈ L2(R+;F) such that
(0, f̂0, f̂1, . . . ) ∈ F or, equivalently,
∞∑
n=0
|f̂n|2L2
C(R+) b⊗n+1(n+ 1)! <∞ (2.15)
we define its extended stochastic integral by the formula
Iext(f) := (0, f̂0, f̂1, . . .).
Applying the Wiener – Itô – Segal isomorphism to Iext, we obtain the extended
stochastic integral introduced by Hitsuda and Skorokhod.
Properties of Iext can be easily obtained from the corresponding properties of the
extended stochastic integral on L2(D′×R+). In particular, let us consider the annihilation
operator a−(δt) (see (2.7)) as an unbounded one
a−(δ·) : F → L2(R+;F), g = (gn)∞n=0 7→ a−(δ·)g = ((n+ 1)gn+1(·))∞n=0
(2.16)
with the dense in F domain
Dom(a−(δ·)) :=
{
g = (gn)∞n=0 ∈ F
∣∣∣∣ ∞∑
n=0
|gn|2L2
C(R+) b⊗nn!n <∞
}
.
Note that the IG-image of a−(δ·) is the so-called Malliavin’s gradient, see, e.g., [19].
The following statement follows from, e.g., [47] (see also [17, 19]).
Theorem 2.2. The extended stochastic integral Iext : L2(R+;F) → F and the
annihilation operator a−(δ·) : F → L2(R+;F) are adjoint one to another. In particular,
these operators are closed.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
746 N. A. KACHANOVSKY, V. A. TESKO
Remark 2.4. The fact that in the Gaussian case the Skorokhod integral is adjoint
to the stochastic derivative (the Malliavin gradient) was proved for the first time in
[47]. This result is a starting point in developing of a stochastic calculus for nonadapted
processes (the so-called anticipating stochastic calculus). We refer here to the book
[19] and reference therein for an exhaustive presentation of results, techniques and
applications of the anticipating stochastic calculus.
Remark 2.5. Note that one can get rid of restriction (2.15) and introduce elements
of a Wick calculus considering stochastic integrals on the IG-pre-image of a so-called
regular rigging of L2(D′, µG), see, e.g., [25] for details.
We will now show that the extended stochastic integral Iext can be regarded as an
ordinary Bochner one. Before establishing the corresponding result, let as first look at
the following heuristic argumentation.
According to Remark 2.3 for a simple Itô integrable function
f(·) =
n−1∑
k=0
f(k)1(tk,tk+1](·) ∈ L2
a(R+;F), f(k) ∈ F ,
we have
I(f) =
n−1∑
k=0
f(k)♦(0,1(tk,tk+1], 0, 0, . . .).
Using this equality, (2.8) and the formal representation
(0,1(tk,tk+1], 0, 0, . . .) =
∫
(tk,tk+1]
(0, δt, 0, 0, . . .) dt
we obtain (at least formally)
I(f) =
n−1∑
k=0
f(k)♦(0,1(tk,tk+1], 0, 0, . . .) =
=
n−1∑
k=0
f(k)♦
∫
(tk,tk+1]
(0, δt, 0, 0, . . .) dt =
=
n−1∑
k=0
∫
(tk,tk+1]
f(k)♦(0, δt, 0, 0, . . .) dt =
=
∫
R+
(
n−1∑
k=0
f(k)1(tk,tk+1](t)
)
♦(0, δt, 0, 0, . . .) dt =
=
∫
R+
f(t)♦(0, δt, 0, 0, . . .) dt =
∫
R+
a+(δt)f(t) dt.
Since the delta-function δt is not a square integrable one, the last formula can not be
accepted as a definition of the extended stochastic integral on L2(R+;F). However from
results of [17, 11] the correctness of the following definition follows.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
STOCHASTIC INTEGRAL OF HITSUDA – SKOROKHOD TYPE ON THE EXTENDED FOCK SPACE 747
Definition 2.3. The extended stochastic integral of a function
ξ(·) = (ξn(·))∞n=0 ∈ L2(R+;F(−τ,−q))
is defined as a Bochner one in the space F(−τ,−q) (see (2.4)), τ is such that∫
R+
|δt|2D−τ dt <∞, by the formula
Îext(ξ) :=
∫
R+
a+(δt)ξ(t) dt =
∫
R+
ξ(t)♦(0, δt, 0, 0, . . .) dt ∈ F(−τ,−q). (2.17)
Not complicated direct calculation shows that
Îext(ξ) = (0, ξ̂0, ξ̂1, . . .),
where each ξ̂n ∈ D
b⊗n+1
−τ,C is the projection of ξn(·) ∈ Db⊗n+1
−τ,C ⊗ L2(R+) onto D
b⊗n+1
−τ,C .
This property means in particular that Îext is an extension of Iext, i.e.,
Iext(f) = Îext(f) =
∫
R+
a+(δt)f(t) dt, f ∈ Dom (Iext).
This result explains the same name for the integrals Iext and Îext.
It can be easily shown that the analog of Theorem 2.2 holds true for operators
Îext : L2(R+;F(−τ,−q))→ F(−τ,−q), a−(δ·) : F(τ, q)→ L2(R+;F(τ, q)),
where a−(δ·) is the restriction of operator (2.16) on F(τ, q). Moreover, now Îext and
a−(δ·) are continuous operators.
Remark 2.6. The IG-image of integral Îext has the form
IG
(
Îext(ξ)
)
=
∫
R+
∂+(δt)Ψ(t) dt =
∫
R+
Ψ(t)♦DẆt dt,
where Ψ(t) := IGξ(t), ∂+(δt) := IGa+(δt)I−1
G : (D′)− → (D′)− is an adjoint operator
to the Hida derivative ∂(δt), Ẇt := 〈·, δt〉 = IG(0, δt, 0, 0, . . .) ∈ (D′)− is the so-called
Gaussian white noise and ♦D denotes the Wick product in (D′)−, i.e.,
Ψ♦DΦ := IG(I−1
G Ψ♦I−1
G Φ), Ψ, Φ ∈ (D′)−.
Thus in such a way we obtain the well-known presentation∫
R+
Ψ(t)d̂Wt =
∫
R+
∂+(δt)Ψ(t) dt =
∫
R+
Ψ(t)♦DẆt dt,
where
∫
R+
◦(t)d̂Wt denotes the extended (Hitsuda – Skorokhod) integral with respect to
W (see, e.g., [17, 48, 44] and reference therein for more details).
Remark 2.7. Let (Ω,A, P ) be a complete probability space with a right continuous
filtration {At}t∈R+ , i.e., As ⊂ At if s ≤ t and At =
⋂
s>tAs for all t ∈ R+. Suppose
that A coincides with the smallest σ-algebra generated by
⋃
t∈R+
At and A0 contains
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
748 N. A. KACHANOVSKY, V. A. TESKO
all P -zero sets of A. In addition, suppose that A0 is trivial, that is for each α ∈ A0 we
have P (α) = 0 or P (α) = 1.
By definition a process N = {Nt}t∈R+ , N0 = 0, is a normal martingale on
(Ω,A, P ) with respect to {At}t∈R+ if {Nt}t∈R+ and {N2
t − t}t∈R+ are martingales
with respect to {At}t∈R+ . This means that for all s, t ∈ R+ such that s ≤ t
E[Nt −Ns|As] = 0, E[(Nt −Ns)2|As] = t− s,
where as before E[ · |As] denotes the conditional expectation with respect to As.
It is known (see, for example, [13, 15]) that a mapping
IN : F → L2(Ω,A, P ), f = (fn)∞n=0 7→ INf :=
∞∑
n=0
IN,n(fn),
is a well-defined isometry. Here IN,0(f0) := f0 and, for each n ∈ N,
IN,n(fn) := n!
∫
∆n
fn(t1, . . . , tn) dNt1 . . . dNtn ,
∆n =
{
(t1, . . . , tn) ∈ Rn+
∣∣ t1 < . . . < tn
}
,
is an iterated stochastic integral with respect to N. The integrals IN,n(fn) have the
isometry property∥∥IN,n(fn)
∥∥2
L2(Ω,A,P )
= (n!)2
∫
∆n
|fn(t1, . . . , tn)|2 dt1 . . . dtn = |fn|2L2
C(R+) b⊗nn!,
and, moreover, the orthogonality property
(IN,n(fn), IN,m(fm))L2(Ω,A,P ) =
0, n 6= m,
|fn|2L2
C(R+) b⊗nn!, n = m.
When IN : F → L2(Ω,A, P ) is a unitary operator (i.e., IN isometrically maps
the whole space F onto whole L2(Ω,A, P )) one says that N possesses the Chaotic
Representation Property (CRP). The unique decomposition of F ∈ L2(Ω,A, P ) as
F =
∑∞
n=0
IN,n(fn) is called the chaotic expansion of F. We observe that the standard
Wiener process W, the compensated Poisson process and some Azéma martingales are
examples of normal martingales, which possess the CRP. We refer to [13, 15, 9, 16, 49]
for more information about normal martingales and their properties.
Let N be a normal martingale with CRP. Then as in the Gaussian case the mapping
IN is completely characterized by the following properties:
(i) IN : F → L2(Ω,A, P ) is a unitary operator;
(ii) IN,0(f0) = f0 for all f0 ∈ C;
(iii) for each n ∈ N and any disjoint Borel sets α1, . . . , αn of finite Lebesgue
measure,
IN,n(1α1⊗̂ . . . ⊗̂1αn) = N(α1) · . . . ·N(αn),
where B(R+) 3 α 7→ N(α) ∈ L2(Ω,A, P ) is a vector-valued measure generated by the
normal martingale N, i.e., we set
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
STOCHASTIC INTEGRAL OF HITSUDA – SKOROKHOD TYPE ON THE EXTENDED FOCK SPACE 749
N((s1, s2]) = Ns2 −Ns1 , N({0}) := N0 = 0, N(∅) := 0,
and extend this definition to all Borel subsets of R+.
Since IN has properties (i) – (iii) and the proof of Theorem 2.1 is based on the
corresponding properties of the Wiener – Itô – Segal isomorphism only, we can conclude
that the IN -image of I is the Itô integral with respect to the normal martingale N and
as a consequence the IN -image of Iext gives an extension of this Itô integral. We refer
to [9] for properties and applications of extended stochastic integrals connected with
normal martingales.
3. The generalized Meixner measure and the extended Fock space. Recall the
definition of the generalized Meixner measure on D′ , see [30].
Let us fix arbitrary functions
α : R+ → C, β : R+ → C
that are smooth and satisfy the conditions
θ(s) := −α(s)− β(s) ∈ R, η(s) := α(s)β(s) ∈ R+
for each s ∈ R+. We also assume that the functions θ and η are bounded on R+. Note
that in a certain sense η is a “key parameter”, which will be used often below.
For each s ∈ R+ denote by να(s),β(s) a probability measure on R that is defined by
its Fourier transform∫
R
eiλt να(s),β(s)(dt) = exp
(
−iλ
(
α(s) + β(s)
)
+
+ 2
∞∑
m=1
(α(s)β(s))m
m
[ ∞∑
n=2
(−iλ)n
n!
(
β(s)n−2 + β(s)n−3α(s) + . . .+ α(s)n−2
)]m)
.
Definition 3.1. We say that a probability measure µ on the measurable space
(D′, C(D′)) with the Fourier transform
∫
D′
ei〈x,ϕ〉 µ(dx) = exp
∫
R+
∫
R
(
eitϕ(s) − 1− itϕ(s)
) 1
t2
να(s),β(s)(dt) ds
, ϕ ∈ D,
is called the generalized Meixner measure.
Theorem 3.1 [30]. The measure µ is a generalized stochastic process with inde-
pendent values in the sense of [50]. The Laplace transform of µ is a holomorphic at
0 ∈ DC function.
Let α and β be constants. Accordingly to the classical classification [51] (see also
[36, 37, 30]) µ is the Gaussian measure for α = β = 0; µ is the centered Poissonian
measure for α 6= 0, β = 0; µ is the centered Gamma measure [26, 31] for α = β 6= 0;
µ is the centered Pascal measure [33] for α 6= β, αβ 6= 0, α, β ∈ R; µ is the centered
Meixner measure for α = β, Im(α) 6= 0. Thus the “key parameter” η = 0 if and only
if µ is the Gaussian or Poissonian measure.
Denote by (L2) := L2(D′, µ) the space of complex-valued square integrable with
respect to µ functions on D′. A function
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
750 N. A. KACHANOVSKY, V. A. TESKO
D′ 3 x 7→ F (x) =
n∑
k=0
〈x⊗k, ϕk〉 ∈ C, ϕk ∈ D
b⊗k
C , ϕn 6= 0,
is called a continuous polynomial on D′ of order n. Since the measure µ has a
holomorphic at 0 ∈ DC Laplace transform (Theorem 3.1), the set of all continuous
polynomials on D′ is dense in (L2) [52]. Due to this fact, using the procedure of
orthogonalization of polynomials (see, e.g., [21] for details) one can construct an
orthogonal decomposition of the space (L2). Namely, for n ∈ Z+ let Pn be the set
of all continuous polynomials on D′ of order ≤ n, P̃n be the closure of Pn in (L2) and
(L2
n) := P̃n P̃n−1, where denotes the orthogonal difference in (L2), (L2
0) := C.
Thus we can regard (L2) as the orthogonal direct sum of subspaces (L2
n), i.e.,
(L2) =
∞⊕
n=0
(L2
n).
We pass now to the construction of the extended Fock space. To this end, for each
ϕn ∈ D
b⊗n
C we define :〈x⊗n, ϕn〉 : as the orthogonal projection of 〈x⊗n, ϕn〉 onto (L2
n).
It follows from results of [30] that :〈x⊗n, ϕn〉 : = 〈Pn(x), ϕn〉, where Pn(x) ∈ D′ b⊗n
and for µ-almost all x ∈ D′
P0(x) = 1, P1(x) = x,
and for all ϕn ∈ D
b⊗n
C , ψ ∈ DC
〈Pn+1(x), ϕn⊗̂ψ〉 = 〈Pn(x), ϕn〉〈P1(x), ψ〉−
−n〈Pn(x),Pr [θ(·)ψ(·)ϕn(·, ·2, . . . , ·n)]〉 −
− n〈Pn−1(x), ϕψn〉 − n(n− 1)〈Pn−1(x),Pr [η(·)ψ(·)ϕn(·, ·, ·3 . . . , ·n)]〉. (3.1)
Here Pr denotes the symmetrization operator and
ϕψn(·1, . . . , ·n−1) :=
∫
R+
ϕn(·1, . . . , ·n−1, t)ψ(t) dt ∈ Db⊗n−1
C .
It should be noticed that 〈Pn(·), ϕn〉, n ∈ Z+, are Schefer polynomials, i.e., orthogonal
polynomials with a generating function of exponential type, see [37, 36, 30].
Let Fn,ext be a Hilbert space that is obtained as the closure of Db⊗n
C with respect to
the norm | · |Fn,ext generated by the scalar product
(ϕn, ψn)Fn,ext :=
1
n!
∫
D′
〈Pn(x), ϕn〉〈Pn(x), ψn〉µ(dx), ϕn, ψn ∈ D
b⊗n
C
(
note that since this scalar product is real, | · |Fn,ext =
√
(· , ·)Fn,ext ). Note that Fn,ext
depends on a parameter η; but we omit this parameter for simplification of notation. In
the situations when the dependence on η is significant we specify this.
It is possible to give an inner description of the scalar product in the space Fn,ext.
Namely, according to [25] (see also [30]) we have
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
STOCHASTIC INTEGRAL OF HITSUDA – SKOROKHOD TYPE ON THE EXTENDED FOCK SPACE 751
(ϕn, ψn)Fn,ext =
∑
k,lj ,sj∈N: j=1,...,k, l1>l2>...>lk,
l1s1+...+lksk=n
n!
ls11 . . . lskk s1! . . . sk!
×
×
∫
Rs1+...+sk
+
ϕn
(
t1, . . . , t1︸ ︷︷ ︸
l1
, . . . , ts1 , . . . , ts1︸ ︷︷ ︸
l1
, . . . , ts1+...+sk , . . . , ts1+...+sk︸ ︷︷ ︸
lk
)
×
×ψn
(
t1, . . . , t1︸ ︷︷ ︸
l1
, . . . , ts1 , . . . , ts1︸ ︷︷ ︸
l1
, . . . , ts1+...+sk , . . . , ts1+...+sk︸ ︷︷ ︸
lk
)
×
×η(t1)l1−1. . . η(ts1)l1−1η(ts1+1)l2−1 . . . η(ts1+s2)l2−1 . . . η(ts1+...+sk−1+1)lk−1 . . .
. . . η(ts1+...+sk)lk−1 dt1 . . . dts1+...+sk . (3.2)
It follows from (3.2) that actually Fn,ext is not connected directly with the measure
µ and depends on the function η only. Moreover, it can be shown that for all n ∈ N
Fn,ext ⊆ L̂2(Rn+, ρn) :=
{
fn ∈ L2(Rn+, ρn) : fn is symmetric in all variables
}
, (3.3)
where the Borel measure ρn is constructed by using (3.2). In particular, ρ1 is the Lebesgue
measure on R+. If µ is the Gaussian or Poissonian measure then η = 0 and therefore
ρn is the Lebesgue measure on B(Rn+). We refer a reader to [32] for a more detailed
discussion of spaces like Fn,ext.
Definition 3.2. We define the extended Fock space Fext by the formula
Fext :=
∞⊕
n=0
Fn,extn!, F0,ext := C.
Thus Fext is a complex Hilbert space of sequences f = (fn)∞n=0, fn ∈ Fn,ext such that
‖f‖2Fext
=
∞∑
n=0
|fn|2Fn,ext
n! <∞.
Remark 3.1. Let us explain the term the extended Fock space. It is not difficult to
show by analogy with [32] that the space Fn,ext is, generally speaking, the orthogonal
sum of L2
C(R+)b⊗n and some another Hilbert spaces. In this sense Fn,ext is an extension
of L2
C(R+)b⊗n and therefore Fext is an extension of F .
One can give another explanation of the fact that Fn,ext is a more wide space than
L2
C(R+)b⊗n. Namely, let fn ∈ L2
C(R+)b⊗n (fn is an equivalence class in L2
C(R+)b⊗n). We
select a representative (a function) ḟn ∈ fn with a “zero diagonal”, i.e., ḟn(t1, . . . , tn) =
= 0 if there exist i, j ∈ {1, . . . , n}, i 6= j such that ti = tj . This function generates the
equivalence class f̃n in Fn,ext that can be identified with fn (see [25] for details).
Let Ffin denote the set of all finite sequences (ϕn)∞n=0, ϕn ∈ D
b⊗n
C . It is clear that
Ffin is a dense subset of Fext and the mapping
Fext ⊃ Ffin 3 ϕ = (ϕn)∞n=0 7→ (Iϕ)(·) :=
∞∑
n=0
〈Pn(·), ϕn〉 ∈ (L2)
(the series, in fact, finite) is isometric. Extending this mapping by continuity to the whole
space Fext we obtain a unitary operator acting between Fext and (L2). We keep the
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
752 N. A. KACHANOVSKY, V. A. TESKO
notation I for the extension, and we will refer to the operator I : Fext → (L2) as to the
generalized Wiener – Itô – Segal isomorphism. Note that I is the Fourier transform of
a Jacobi field that act in the extended Fock space Fext, see [30] and also [31 – 38].
In what follows, for each fn ∈ Fn,ext we set
〈Pn, fn〉 := I
(
0, . . . , 0︸ ︷︷ ︸
n times
, fn, 0, . . .
)
.
Then for each f = (fn)∞n=0 ∈ Fext we have
If =
∞∑
n=0
〈Pn, fn〉 ∈ (L2). (3.4)
4. The Itô integral on the extended Fock space. We consider the Meixner random
process {
Mt(·) := 〈· ,1[0,t)〉
}
t∈R+
on the probability space
(
D′, C(D′), µ
)
. It follows from Theorem 3.1 that this process
has orthogonal independent increments. Since in additionM· is locally square integrable,
this process is a normal martingale with respect to the natural filtration of σ-algebras
At := σ{Ms | s ≤ t} (the filtration is made complete and right continuous).
Remark 4.1. Note that if the parameter η from the definition of the measure µ
is not a constant then M· is not a Lévy process because in this case M· is not a time
homogeneous one.
We will define the Itô integral on the extended Fock space as the I−1-image of the
classical Itô stochastic integral with respect to the Meixner process. Namely, denote by
(L2)⊗ L2(R+)a the set of all adapted with respect to the filtration {At}t∈R+ functions
from the space L2(D′ × R+, C(D′)× B(R+), µ× dt) ∼= (L2)⊗ L2(R+), i.e.,
(L2)⊗ L2(R+)
a
:=
{
F ∈ (L2)⊗ L2(R+)
∣∣∣F (·, t) = E[F (·, t) |At ] for a.a. t ∈ R+
}
.
(4.1)
Definition 4.1. We say that a function f ∈ L2(R+;Fext) ∼= Fext ⊗L2(R+) is Itô
integrable if (I ⊗ 1)f belongs to (L2)⊗ L2(R+)a, i.e., if
f ∈ L2
a(R+;Fext) ∼= Fext ⊗ L2(R+)
a
:= (I ⊗ 1)−1(L2)⊗ L2(R+)
a
.
On this case the Itô integral of f ∈ L2
a(R+;Fext) is defined as an element of Fext given
by
I(f) := I−1
∫
R+
I(f(t)) dMt
.
Before giving an inner description of the set L2
a(R+;Fext) and express the Itô integral
in terms of the extended Fock space Fext, let us look at the generalized Wiener – Itô –
Segal isomorphism I more carefully. First of all we note that this isomorphism has
analogs of properties (i) – (iii) of the Wiener – Itô – Segal isomorphism IG, i.e.,
(i) I : Fext → (L2) is a unitary operator;
(ii) I(f0, 0, 0, . . .) = f0 for all f0 ∈ C;
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
STOCHASTIC INTEGRAL OF HITSUDA – SKOROKHOD TYPE ON THE EXTENDED FOCK SPACE 753
(iii) for each n ∈ N and any disjoint Borel sets α1, . . . , αn of finite Lebesgue measure(
I(0, . . . , 0︸ ︷︷ ︸
n times
,1α1⊗̂ . . . ⊗̂1αn , 0, 0, . . .)
)
(·) = Mα1(·) . . .Mαn(·),
where Mαk(·) := 〈· ,1αk〉 for all k ∈ {1, . . . , n}.
However in contrast to the Gaussian case if η 6= 0 then the isomorphism I is not
uniquely determined by its properties (i) – (iii) because the set
C
⊕
span
{(
0, . . . , 0︸ ︷︷ ︸
n times
,1α1⊗̂ . . . ⊗̂1αn , 0, 0, . . .
)∣∣∣∣∣
n ∈ N; αi ∈ B(R+); αi ∩ αj = ∅, i 6= j
}
,
is not dense in the extended Fock space Fext. Therefore in order to give a description of
the set L2
a(R+;Fext) and to express the Itô integral in terms of the extended Fock space
structure one can not use the (based on (i) – (iii)) scheme of the proof of Theorem 2.1.
Somehow one must use another properties of the isomorphism I. As it follows from
[25] an appropriate property of I is recurrence relation (3.1), see details below.
We have the following result (cf. Theorem 2.1).
Theorem 4.1. A function f(·) = (fn(·))∞n=0 ∈ L2(R+;Fext) belongs to L2
a(R+;
Fext) if and only if for almost all t ∈ R+
f(t) =
(
f0(t), f1(t)1[0,t), . . . , fn(t)1[0,t)n , . . .
)
.
Taking into account (4.1) and the definition of the space L2
a(R+;Fext), Theorem 4.1
is an immediate consequence of the equality
E
[
〈P0, f0〉|At
]
= 〈P0, f0〉 = f0, t ∈ R+, f0 ∈ C,
and the following statement:
Theorem 4.2. Let fn ∈ Fn,ext, n ∈ N. Then for all t ∈ R+
E
[
〈Pn, fn〉|At
]
= 〈Pn, fn1[0,t)n〉. (4.2)
Proof. Let us fix t ∈ R+. Since a conditional expectation E[ · |At] is an orthogonal
projection in (L2), it is sufficient to prove (4.2) on a total in Fn,ext set. We use the
induction with respect to n. For n = 1 equality (4.2) is fulfilled because
E[〈P1,1[a,b)〉|At] = E[〈P1,1[0,b)〉|At]− E[〈P1,1[0,a)〉|At] =
= E[Mb|At ]− E[Ma|At] = Mmin{b,t} −Mmin{a,t} = 〈P1,1[a,b)1[0,t)〉
and the set of indicators 1[a,b) of intervals [a, b) ⊂ R+ is total in F1,ext = L2
C(R+).
Assume (4.2) is fulfilled for n ∈ {1, 2, . . . ,m} and let us prove this statement for
n = m+ 1. To this end, we need the following technical result.
Lemma 4.1. Let t ∈ R+ and n ∈ N. The set{
ϕ⊗k⊗̂ψ⊗n−k
∣∣ϕ,ψ ∈ DC, suppϕ ⊂ [0, t), suppψ ⊂ [t,∞), k ∈ {0, 1, . . . , n}
}
(4.3)
is total in the space Fn,ext.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
754 N. A. KACHANOVSKY, V. A. TESKO
Proof. Let t ∈ R+ and n ∈ N. For φ ∈ DC we set φ1 := φ1[0,t), φ2 := φ1[t,∞).
Then
φ⊗n = (φ1 + φ2)⊗n =
n∑
k=0
Cknφ
⊗k
1 ⊗̂φ
⊗n−k
2 , (4.4)
where φ⊗k1 ⊗̂φ
⊗n−k
2 denotes the symmetrization with respect to all variables of the
function
φ1(·1) . . . φ1(·k)φ2(·k+1) . . . φ2(·n)(
note that φ⊗k1 ⊗̂φ
⊗n−k
2 is a symmetric function, but not necessary from Db⊗n
C
)
. One
can show by direct calculation that each φ⊗k1 ⊗̂φ
⊗n−k
2 belongs to Fn,ext and can be
approximated in this space by a sequence{
ϕ⊗kl ⊗̂ψ
⊗n−k
l
∣∣∣ϕl, ψl ∈ DC, suppϕl ⊂ [0, t), suppψl ⊂ [t,∞)}
}∞
l=0
(4.5)
(one can select ϕl → φ1, ψl → φ2 pointwisely as l → ∞, then ϕ⊗kl ⊗̂ψ
⊗n−k
l →
→ φ⊗k1 ⊗̂φ
⊗n−k
2 in Fn,ext as l→∞ by the Lebesgue theorem).
Let now fn ∈ Fn,ext be fixed. In order to prove the lemma, it is sufficient to check
that fn can be approximated by linear combinations of elements of set (4.3). Since the
set {φ⊗n |φ ∈ DC} is total in Fn,ext, for arbitrary ε > 0 there exist N ∈ N, constants
c1, . . . , cN and functions φ(1), . . . , φ(N) ∈ DC such that∣∣∣∣∣fn −
N∑
s=1
csφ
⊗n
(s)
∣∣∣∣∣
Fn,ext
<
ε
2
.
Decomposing each φ(s) in the sum φ(s)1 + φ(s)2 as above and using (4.4) we get
N∑
s=1
csφ
⊗n
(s) =
N∑
s=1
n∑
k=0
csC
k
nφ
⊗k
(s)1⊗̂φ
⊗n−k
(s)2 .
Let
{
ϕ⊗k(s),l⊗̂ψ
⊗n−k
(s),l
}∞
l=0
be sequence (4.5) for φ⊗k(s)1⊗̂φ
⊗n−k
(s)2 . Then∣∣∣∣∣fn −
N∑
s=1
n∑
k=0
csC
k
nϕ
⊗k
(s),l⊗̂ψ
⊗n−k
(s),l
∣∣∣∣∣
Fn,ext
≤
≤
∣∣∣∣∣fn −
N∑
s=1
csφ
⊗n
(s)
∣∣∣∣∣
Fn,ext
+
+
∣∣∣∣∣
N∑
s=1
n∑
k=0
csC
k
n
(
φ⊗k(s)1⊗̂φ
⊗n−k
(s)2 − ϕ⊗k(s),l⊗̂ψ
⊗n−k
(s),l
)∣∣∣∣∣
Fn,ext
< ε,
if l is sufficiently large.
Thus the lemma is proved.
We return now to the proof of the theorem. Taking into account the result of this
lemma it is sufficient to prove (4.2) for arbitrary fm+1 of the form
fm+1 = ϕ⊗k⊗̂ψ⊗m+1−k,
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
STOCHASTIC INTEGRAL OF HITSUDA – SKOROKHOD TYPE ON THE EXTENDED FOCK SPACE 755
where ϕ,ψ ∈ DC, suppϕ ⊂ [0, t), suppψ ⊂ [t,∞) and k ∈ {0, 1, . . . ,m + 1}. We
will use recurrent relations (3.1) and the induction hypothesis. The following cases are
possible.
1. Let k = m+ 1, i.e., fm+1 = ϕ⊗m+1. Since suppϕ ⊂ [0, t), we see that〈
Pm+1(·), ϕ⊗m+1
〉
=
〈
Pm+1(·),1[0,t)m+1ϕ⊗m+1
〉
and the function 〈P1(·), ϕ〉 is At-measurable. Hence using (3.1) we get
E
[〈
Pm+1, ϕ
⊗m+1
〉 ∣∣∣At] = E
[
〈P1, ϕ〉〈Pm, ϕ⊗m〉
∣∣∣At]−
−mE
[
〈Pm,Pr [θ(·)ϕ2(·)ϕ⊗m−1(·1, . . . , ·m−1)]〉
∣∣∣At]−
−mE
[
〈Pm−1, 〈ϕ,ϕ〉ϕ⊗m−1〉
∣∣∣At]−
−m(m− 1)E
[〈
Pm−1,Pr
[
η(·)ϕ3(·)ϕ⊗m−2(·1, . . . , ·m−2)
]〉 ∣∣∣At] =
=
〈
P1, ϕ〉〈Pm, ϕ⊗m
〉
−m
〈
Pm,Pr [θ(·)ϕ2(·)ϕ⊗m−1(·1, . . . , ·m−1)]
〉
−
−m
〈
Pm−1, 〈ϕ,ϕ〉ϕ⊗m−1
〉
−m(m− 1)
〈
Pm−1,Pr [η(·)ϕ3(·)ϕ⊗m−2(·1, . . . , ·m−2)]
〉
=
= 〈Pm+1, ϕ
⊗m+1〉 =
〈
Pm+1,1[0,t)m+1ϕ⊗m+1
〉
.
2. Let k = 0, i.e., fm+1 = ψ⊗m+1. Since suppψ ⊂ [t,∞), we conclude that〈
Pm+1(·),1[0,t)m+1ψ⊗m+1
〉
= 0.
Hence
E
[
〈Pm+1, ψ
⊗m+1〉
∣∣At] = E
[
〈P1, ψ〉〈Pm, ψ⊗m〉
∣∣At]−
−mE
[
〈Pm,Pr [θ(·)ψ2(·)ψ⊗m−1(·1, . . . , ·m−1)]〉
∣∣At]−
−mE
[〈
Pm−1, 〈ψ,ψ〉ψ⊗m−1
〉 ∣∣At]−
−m(m− 1)E
[〈
Pm−1,Pr [η(·)ψ3(·)ψ⊗m−2(·1, . . . , ·m−2)]
〉 ∣∣At] =
= E
[
〈P1, ψ〉〈Pm, ψ⊗m〉
]
= 0 =
〈
Pm+1,1[0,t)m+1ψ⊗m+1
〉
if m > 1, and
E
[
〈P2, ψ
⊗2〉 |At
]
=
= E
[
〈P1, ψ〉2
∣∣At]− E[〈P1, θψ
2〉
∣∣At]− E[〈P0, 〈ψ,ψ〉〉
∣∣At] =
= E
[
〈P1, ψ〉2
]
− 〈ψ,ψ〉 = 0 = 〈P2,1[0,t)2ψ
⊗2〉
if m = 1 (here E[·] denotes an expectation).
3. Let k ∈ {1, . . . ,m}, i.e., fm+1 = ϕ⊗k⊗̂ψ⊗m+1−k. Since suppψ ⊂ [t,∞), we
see that 〈
Pm+1(·),1[0,t)m+1ϕ⊗k⊗̂ψ⊗m+1−k
〉
= 0.
In view of the latter and At-measurability of 〈P1(·), ϕ〉 (because suppϕ ⊂ [0, t)) we
obtain
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
756 N. A. KACHANOVSKY, V. A. TESKO
E
[
〈Pm+1, ϕ
⊗k⊗̂ψ⊗m+1−k〉
∣∣At] = E
[
〈P1, ϕ〉〈Pm, ϕ⊗k−1⊗̂ψ⊗m+1−k〉
∣∣At]−
−mE
[
〈Pm,Pr [θ(·)ϕ(·)
(
ϕ⊗k−1⊗̂ψ⊗m+1−k)(·, ·2, . . . , ·m)]〉
∣∣At]−
−mE
[
〈Pm−1, (ϕ⊗k−1⊗̂ψ⊗m+1−k)ϕ〉 |At
]
−
−m(m− 1)E
[
〈Pm−1,Pr [η(·)ϕ(·)
(
ϕ⊗k−1⊗̂ψ⊗m+1−k)(·, ·, ·3, . . . , ·m)]〉 |At
]
=
= 0 = 〈Pm+1,1[0,t)m+1ϕ⊗k⊗̂ψ⊗m+1−k〉.
The theorem is proved.
In order to express the Itô integral I(f) in terms of the extended Fock space Fext we
accept the following convention.
Convention 4.1. When we consider elements of the space L2(R+;Fn,ext) ∼=
∼= Fn,ext ⊗ L2(R+) we always select a representative that vanishes on the set
dn+1 :=
{
(t1, . . . , tn; t) ⊂ Rn+1
+
∣∣∣∃tj = t
}
.
Such a choice of representative will not affect our discussion because in compliance with
(3.3) we have Fn,ext ⊗ L2(R+) ⊆ L̂2(Rn+, ρn) ⊗ L2(R+) and (ρn ⊗m)(dn+1) = 0,
where m denotes the Lebesgue measure on R+.
Now we have the following statement (cf. (2.11)).
Theorem 4.3. For each f(·) = (fn(·))∞n=0 ∈ L2
a(R+;Fext),
I(f) = (0, f̂0, f̂1, . . .) ∈ Fext, (4.6)
where f̂n ∈ Fn+1,ext is the symmetrizations of fn(t1, . . . , tn; t) with respect to n + 1
variables.
Proof. The correctness of the definition of f̂n was proved in [25], Lemma 3.2.
Equality (4.6) is based on (3.1) and easily follow from Theorem 4.1 and [25] (see the
proof of Theorem 3.1 therein).
5. Extended stochastic integrals on the extended Fock space and its riggings. In
this section we define and study generalizations of Itô integral (4.6). These generalizations
are constructed by analogy with the case of Fock spaces (see Definitions 2.2 and 2.3).
5.1. An extended stochastic integral on Fext. Taking into account Theorem 4.3
the simplest way to define an extended stochastic integral on Fext is the following.
Definition 5.1. For a function f(·) =
(
fn(·))∞n=0 ∈ L2(R+;Fext
)
such that
(0, f̂0, f̂1, . . . ) ∈ Fext or, equivalently,
∞∑
n=0
|f̂n|2Fn+1,ext
(n+ 1)! <∞
we define its extended stochastic integral by the formula
Iext(f) := (0, f̂0, f̂1, . . . ) ∈ Fext. (5.1)
Thus the extended stochastic integral Iext is defined as an unbounded operator
Iext : L2(R+;Fext)→ Fext, f(·) = (fn(·))∞n=0 7→ Iext(f) := (0, f̂0, f̂1, . . . )
with the dense in L2(R+;Fext) domain
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
STOCHASTIC INTEGRAL OF HITSUDA – SKOROKHOD TYPE ON THE EXTENDED FOCK SPACE 757
Dom (Iext) :=
{
f(·) = (fn(·))∞n=0 ∈ L2(R+;Fext)
∣∣∣ (0, f̂0, f̂1, . . . ) ∈ Fext
}
.
It follows immediately from Theorem 4.3 and Definition 5.1 that if f is integrable
by Itô (i.e., f ∈ L2
a(R+;Fext)) then f is integrable in the extended sense (i.e., f ∈
∈ Dom (Iext)) and Iext(f) = I(f). Hence Iext is an extension of the Itô integral I.
Let us establish an analog of Theorem 2.2. To this end at first we introduce an
annihilation operator a−(δ·). According to [25] if gn ∈ Fn,ext then gn can be considered
as an element of Fn−1,ext ⊗ L2(R+) and, moreover,
|gn|Fn−1,ext⊗L2(R+) ≤ |gn|Fn,ext .
Due to this fact the following definition is correct.
Definition 5.2. An annihilation operator a−(δ·) is defined as an unbounded
operator
a−(δ·) : Fext → L2(R+;Fext),
g = (gn)∞n=0 7→ a−(δ·)g := ((n+ 1)gn+1(·))∞n=0,
(a−(δt)g)n(t1, . . . , tn−1) = ngn(t1, . . . , tn−1, t)
(5.2)
with the dense in Fext domain
Dom(a−(δ·)) :=
{
g = (gn)∞n=0 ∈ Fext
∣∣∣∣∣
∞∑
n=0
|gn(·)|2Fn−1,ext⊗L2(R+)n!n <∞
}
.
From the corresponding statement in [25] we obtain.
Theorem 5.1. The extended stochastic integral Iext : L2(R+;Fext) → Fext and
the annihilation operator a−(δ·) : Fext → L2(R+;Fext) are adjoint one to another. In
particular, these operators are closed.
Remark 5.1. It is possible to consider I and Iext on intervals [0, t], t ∈ R+, using
functions f(·)1[0,t](·) instead of f(·) in the corresponding definitions. But in this case
it is necessary to keep in mind that the domain of Iext depends on t and (even in the
case η = 0) it is possible that f is integrable in the extended sense on R+ but is not
integrable on [0, t]. Note that the extended stochastic integral on riggings of Fext (see
below) has no this lack.
5.2. An extended stochastic integral on the “regular” rigging of Fext. The space
Fext has the following “lacks”: the extended stochastic integral Iext : L2(R+,Fext) →
→ Fext is an unbounded operator (and, moreover, the domain of Iext(◦1[0,t]) depends
on t); there is no a multiplication on Fext that is naturally connected with Iext. This
constricts an area of possible applications of Iext. In this subsection we consider a natural
in a sense extension of Fext that has no the mentioned lacks.
Let q ∈ N,
Fext(q) :=
∞⊕
n=0
Fn,ext (n!)22qn
be a Hilbert space of sequences f = (fn)∞n=0, fn ∈ Fn,ext, such that
‖f‖2Fext(q)
=
∞∑
n=0
|fn|2Fn,ext
(n!)22qn <∞.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
758 N. A. KACHANOVSKY, V. A. TESKO
We consider the (“regular” in a terminology of [25]) rigging of Fext
F−ext = ind lim
q∈N
Fext(−q) ⊃ Fext(−q) ⊃ Fext ⊃ Fext(q) ⊃ pr lim
q∈N
Fext(q) = F+
ext,
(5.3)
where the space
Fext(−q) =
∞⊕
n=0
Fn,ext 2−qn, ‖f‖2Fext(−q) =
∞∑
n=0
|fn|2Fn,ext
2−qn <∞
is dual of Fext(q) with respect to Fext.
The extended stochastic integral Iext : L2(R+;Fext(−q))→ Fext(−q) and the anni-
hilation operator a−(δ·) : Fext(q)→ L2(R+;Fext(q)) can be defined by formulas (5.1)
and (5.2) respectively, one can show by analogy with [25] that now these operators are
adjoint one to another and continuous. Moreover, Iext and a−(δ·) can be continued to
adjoint one to another linear continuous operators acting from F−ext ⊗ L2(R+) to F−ext
and from F+
ext to F+
ext ⊗ L2(R+) correspondingly. Elements of the Wick calculus on
F−ext can be defined and applied by analogy with the Gaussian analysis.
But rigging (5.3) is not suit in order to define Iext as a Bochner integral by analogy
with (2.17) (because δt 6∈ L2
C(R+)), this can lead to inconvenience in some applications.
In the forthcoming subsection we consider the analog of rigging (2.3) that is similar to
(5.3) but has no such a lack.
5.3. The “nonregular” rigging of Fext and elements of the Wick calculus. Exclu-
ding from T some indexes (and preserving for this modified set of indexes the notation
T ) we can formulate the following statement that is a suitable reformulation of Proposi-
tion 2.3 in [25].
Proposition 5.1. For each τ ∈ T there exists q0 = q0(τ) ∈ N such that for all
q ∈ Nq0 := {q0, q0 + 1, . . .} the dense and continuous embedding F(τ, q) ↪→ Fext takes
place.
In what follows, we accept on default τ ∈ T and q ∈ Nq0 . Due to Proposition 5.1
one can construct a rigging of the extended Fock space Fext
Fext,− ⊃ Fext(−τ,−q) ⊃ Fext ⊃ F(τ, q) ⊃ F+, (5.4)
where Fext(−τ,−q), Fext,− = ind limτ∈T,q∈Nq0 Fext(−τ,−q) are the dual spaces of
F(τ, q), F+ with respect to Fext correspondingly. It is not difficult to show that
Fext(−τ,−q) =
∞⊕
n=0
D
(n)
−τ,C2−qn, D
b⊗0
−τ,C := C,
where D(n)
−τ,C, n ∈ N are the negative spaces of the chain
ind lim
τ∈T
D
(n)
−τ,C =: D′(n)
C ⊃ D(n)
−τ,C ⊃ Fn,ext ⊃ D
b⊗n
τ,C ⊃ D
b⊗n
C := pr lim
τ∈T
D
b⊗n
τ,C.
Hence Fext(−τ,−q) consists of sequences ξ = (ξn)∞n=0, ξn ∈ D
(n)
−τ,C such that
‖ξ‖2Fext(−τ,−q) =
∞∑
n=0
|ξn|2D(n)
−τ,C
2−qn <∞.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
STOCHASTIC INTEGRAL OF HITSUDA – SKOROKHOD TYPE ON THE EXTENDED FOCK SPACE 759
Remark 5.2. It is easy to see that if η = 0 then F(τ, q) ⊂ Fext(q), therefore
Fext(−q) ⊂ Fext(−τ,−q). But in the general case there are no such embeddings, this
is connected with the structure of norms in Fn,ext.
Let us denote by 〈〈· , ·〉〉Fext the dual pairing between elements of Fext(−τ,−q)
and F(τ, q) (just as Fext,− and F+), this pairing is generated by the scalar product in
Fext. The spaces Fext(−τ,−q) and Fext,− have a complicated structure as against usual
symmetric Fock spaces. However since the positive spaces in riggings (5.4) and (2.3)
coincide, there exists a uniquely defined isomorphism
U : Fext,− → F−
such that for all ξ ∈ Fext,− and all ϕ ∈ F+
〈〈ξ, ϕ〉〉Fext = 〈〈Uξ, ϕ〉〉F .
It is clear that U =
⊕∞
n=0 Un, where each Un : D′(n)
C → D′ b⊗nC is defined by
〈ξn, ϕn〉Fn,ext = 〈Unξn, ϕn〉, ξn ∈ D′
(n)
C , ϕn ∈ D
b⊗n
C .
One can show [25] that the restrictions of Un on D(n)
−τ,C are isometrical isomorphisms
between D(n)
−τ,C and D
b⊗n
−τ,C, therefore the restrictions of U on Fext(−τ,−q) are isometri-
cal isomorphisms between Fext(−τ,−q) and F(−τ,−q). In what follows, it is conveni-
ent for us to understand Fext,− and Fext(−τ,−q) as the U−1-images of F− and
F(−τ,−q) respectively.
Above mentioned realization of the space Fext,− is convenient for developing of a
Wick calculus on it. We do not discuss this in details, but we give a definition of a Wick
product on Fext,−. For given ξ = (ξn)∞n=0, ζ = (ζn)∞n=0 ∈ Fext,− a Wick product
ξ♦extζ ∈ Fext,− is defined by
ξ♦extη := U−1
(
Uξ♦Uζ
)
=
( n∑
m=0
ξm � ζn−m
)∞
n=0
,
where for each ξn ∈ D′(n)
C and each ζm ∈ D′(m)
C
ξn � ζm := U−1
n+m
(
Unξn ⊗̂ Umζm
)
.
The correctness of this definition (and, moreover, the fact that ♦ext is a continuous
operation in the topology of Fext,−) follows from results of [25]. We note also that if
η = 0 (the Gaussian and Poissonian cases) then the product � moves to the symmetric
tensor product ⊗̂ and ♦ext moves to ♦.
In order to describe an important property of the product � we adopt the following
convention.
Convention 5.1. Elements of the space Fn,ext ⊗ Fm,ext are equivalence classes,
and considering such elements we always choose representatives that vanish on the set{
(t1, . . . , tn; tn+1, . . . , tn+m) ∈ Rn+m
+
∣∣∃i ∈ {1, . . . , n},
j ∈ {n+ 1, . . . , n+m} : ti = tj
}
.
The following statement follows from [25], Lemma 4.1.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
760 N. A. KACHANOVSKY, V. A. TESKO
Proposition 5.2. Let fn ∈ Fn,ext, gm ∈ Fm,ext. Then
fn � ηm = f̂ngm ∈ Fn+m,ext,
where f̂ngm is the symmetrization of fn ⊗ gm with respect to n+m variables.
Finally, let us consider the creation operator aext,+(δt) that is defined by
aext,+(δt) : Fext(−τ,−q)→ Fext(−τ,−q), aext,+(δt) := U−1a+(δt)U, (5.5)
where a+(δt) is given by (2.7). It is easy to show that the operator aext,+(δt) is dual
of the annihilation operator a−(δt) : F(τ, q) → F(τ, q) with respect to the zero space
Fext:
〈〈aext,+(δt)ξ, ϕ〉〉Fext = 〈〈ξ, a−(δt)ϕ〉〉Fext , ξ ∈ Fext(−τ,−q), ϕ ∈ F(τ, q).
(5.6)
Moreover, a trivial calculation gives
aext,+(δt)ξ = ξ♦ext(0, δt, 0, 0, . . .), ξ ∈ Fext(−τ,−q).
5.4. An extended stochastic integral on the “nonregular” rigging of Fext. It
follows from Theorem 4.3 and Proposition 5.2 that the Itô integral I(f) of a simple
function
f(·) =
n−1∑
k=0
f(k)1(tk,tk+1](·) ∈ L2
a(R+;Fext)
has the form
I(f) =
n−1∑
k=0
f(k)♦ext(0,1(tk,tk+1], 0, 0, . . .) ∈ Fext.
Using the same arguments as in Subsection 2.2 it is natural to give the following definition
of the extended stochastic integral on the extended Fock spaces.
In what follows, let us fix τ ∈ T such that∫
R+
|δt|2D−τ dt = c(τ) <∞ (5.7)
and q ∈ Nq0(τ) = {q0(τ), q0(τ) + 1, . . .}, where q0(τ) ∈ N is given in Proposition 5.1.
The existence of τ with the required property is proved in, e.g., [42], Chapter XIV.
Definition 5.3. The extended stochastic integral of a function
ξ ∈ L2(R+;Fext(−τ,−q))
is defined by the formula
Îext(ξ) :=
∫
R+
aext,+(δt)ξ(t) dt ∈ Fext(−τ,−q) (5.8)
as a Bochner integral of the vector-valued function
R+ 3 t 7→ aext,+(δt)ξ(t) = ξ(t)♦ext(0, δt, 0, 0, . . .) ∈ Fext(−τ,−q).
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
STOCHASTIC INTEGRAL OF HITSUDA – SKOROKHOD TYPE ON THE EXTENDED FOCK SPACE 761
The correctness of this definition from the following statement follows.
Proposition 5.3. For all ξ ∈ L2(R+;Fext(−τ,−q)) integral (5.8) is well-defined
as a Bochner one and is continuous as an operator acting from L2(R+;Fext(−τ,−q))
to Fext(−τ,−q).
Proof. Let ξ ∈ L2(R+;Fext(−τ,−q)). Using the estimate
‖aext,+(δt)ξ(t)‖Fext(−τ,−q) ≤ 2−
q
2 |δt|D−τ ‖ξ(t)‖Fext(−τ,−q)
(this inequality follows from (5.5) and results of [22]) we obtain
∥∥∥∥∥∥∥
∫
R+
aext,+(δt)ξ(t) dt
∥∥∥∥∥∥∥
Fext(−τ,−q)
≤
∫
R+
‖aext,+(δt)ξ(t)‖Fext(−τ,−q) dt ≤
≤ 2−q/2
∫
R+
|δt|D−τ ‖ξ(t)‖Fext(−τ,−q) dt ≤
≤ 2−q/2
∫
R+
|δt|2D−τ dt
1/2∫
R+
‖ξ(t)‖2Fext(−τ,−q) dt
1/2
=
= 2−q/2c(τ)1/2‖ξ‖L2(R+,Fext(−τ,−q)) <∞,
whence the necessary statement follows.
Remark 5.3. It follows from [25] that in the case where (5.7) does not hold,
integral (5.8) is well-defined as a Pettis one. Namely, for all ξ ∈ L2(R+,Fext(−τ,−q)),
τ ∈ T, q ∈ Nq0(τ), a function
ζ : R+ → Fext(−τ,−q), t 7→ ζ(t) := aext,+(δt)ξ(t)
is Pettis integrable, i.e.,
the function 〈〈ζ(·), ϕ〉〉Fext is measurable for any ϕ ∈ Fext(τ, q);
〈〈ζ(·), ϕ〉〉Fext ∈ L1(R+, dt) for all ϕ ∈ Fext(τ, q).
The corresponding Pettis integral of ξ is defined as a unique element of the space
Fext(−τ,−q), denoted by
∫
R+
ζ(t) dt, such that〈〈 ∫
R+
ζ(t) dt, ϕ
〉〉
Fext
=
∫
R+
〈〈ζ(t), ϕ〉〉Fext dt.
Let us point out a relation between the extended stochastic integral Îext and the
annihilation operator a−(δ·).
Theorem 5.2. The integral Îext : L2(R+;Fext(−τ,−q)) → Fext(−τ,−q) is
adjoint of the annihilation operator a−(δ·) : F(τ, q) → L2(R+,F(τ, q)) in the sense
that 〈〈
Îext(ξ), ϕ
〉〉
Fext
= 〈〈ξ, a−(δ·)ϕ〉〉L2(R+;Fext) (5.9)
for all ξ ∈ L2(R+;Fext(−τ,−q)) and all ϕ ∈ F(τ, q).
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
762 N. A. KACHANOVSKY, V. A. TESKO
Proof. Using (5.8) and (5.6) we obtain
〈〈Iext(ξ), ϕ〉〉Fext =
〈〈 ∫
R+
aext,+(δt)ξ(t) dt, ϕ
〉〉
Fext
=
∫
R+
〈〈aext,+(δt)ξ(t), ϕ〉〉Fextdt =
=
∫
R+
〈〈ξ(t), a−(δt)ϕ〉〉Fextdt = 〈〈ξ, a−(δ·)ϕ〉〉L2(R+;Fext)
for all ξ ∈ L2(R+;Fext(−τ,−q)) and all ϕ ∈ F(τ, q).
From this statement and Theorem 5.1 we obtain the following corollary.
Corollary 5.1. Let f ∈ L2(R+;Fext) ⊂ L2(R+,Fext(−τ,−q)) be integrable in
the sense of Definition 5.1. Then the extended stochastic integral Îext(f) that is defined
by (5.8) coincides with the extended stochastic integral Iext(f) from Definition 5.1, i.e.,
Îext(f) = Iext(f), f ∈ Dom (Iext)
(this explains why we use the same name for these integrals).
In order to rewrite integral (5.8) by analogy with (5.1) we define
ξ̂n := U−1
n+1
(
Pr((Un ⊗ 1)ξn)
)
∈ D(n+1)
−τ,C
for all ξn ∈ D(n)
−τ,C ⊗ L2(R+), where Pr denotes the symmetrization operator. The next
statement follows from [25], Theorem 4.4.
Theorem 5.3. Let ξ = (ξn)∞n=0 ∈ L2(R+,Fext(−τ,−q)) (now ξn ∈ D
(n)
−τ,C ⊗
⊗ L2(R+)). Then
Îext(ξ) = (0, ξ̂0, ξ̂1, . . .) ∈ Fext(−τ,−q).
Remark 5.4. All results of this subsection can be rewritten with obvious modifi-
cations for the space Fext,− instead of Fext(−τ,−q).
Remark 5.5. Using the generalized Wiener – Itô – Segal isomorphism I : Fext →
→ (L2), defined by (3.4), one can reformulate all the above definitions and statements
in terms of test and generalized functions on D′ whose dual paring is generated by the
scalar product in the space (L2). In such a way one obtains a natural generalization of
the Itô integral with respect to Meixner processes, see [25] for more details.
Acknowledgment. We are very grateful to Professor Yu. M. Berezansky for helpful
discussions.
1. Hitsuda M. Formula for Brownian partial derivatives // Proc. Second Japan – USSR Symp. Probab.
Theory. – 1972. – 2. – P. 111 – 114.
2. Daletsky Yu. L., Paramonova S. N. Stochastic integrals with respect to a normally distributed additive
set function // Dokl. Akad. Nauk SSSR. – 1973. – 208, № 3. – P. 512 – 515 (in Russian).
3. Daletsky Yu. L., Paramonova S. N. A certain formula of the theory of Gaussian measures, and the
estimation of stochastic integrals // Theory Probab. Appl. – 1974. – 19, № 4. – P. 844 – 849 (in Russian).
4. Kabanov Yu. M., Skorokhod A. V. Extended stochastic integrals // Proc. School-Semin. Theory Stochast.
Process. – Vilnus: Inst. Phys. Math. – 1975. – I. – P. 123 – 167 (in Russian).
5. Skorokhod A. V. On a generalization of the stochastic integral // Teor. Verojatnost. i Prim. – 1975. – 20,
№ 2. – P. 223 – 238 (in Russian).
6. Kabanov Yu. M. Extended stochastic integrals // Teor. Verojatnost. i Prim. – 1975. – 20, № 4. – P. 725 – 737
(in Russian).
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
STOCHASTIC INTEGRAL OF HITSUDA – SKOROKHOD TYPE ON THE EXTENDED FOCK SPACE 763
7. Itô K. Multiple Wiener integral // J. Math. Soc. Jap. – 1951. – 3. – P. 157 – 169.
8. Itô K. Spectral type of the shift transformation of differential processes with stationary increments //
Trans. Amer. Math. Soc. – 1956. – 81. – P. 253 – 263.
9. Ma J., Protter P., Martin J. S. Anticipating integrals for a class of martingales // Bernoulli. – 1998. – 4,
№ 1. – P. 81 – 114.
10. Kachanovsky N. A. A generalized stochastic derivative connected with coloured noise measures // Meth.
Funct. Anal. and Top. – 2004. – 10, № 4. – P. 11 – 29.
11. Albeverio S., Berezansky Yu. M., Tesko V. A generalization of an extended stochastic integral // Ukr.
Math. J. – 2007. – 59, № 5. – P. 645 – 677.
12. Emery M. On the Azema martingales // Lect. Notes Math. – 1989. – 1372. – P. 66 – 87.
13. Dellacherie C., Maisonneuve B., Meyer P.-A. Probabilités et Potentiel. Chapitres XVII a XXIV. Processus
de Markov (fin). Compléments de calcul stochastique. – Paris: Hermann, 1992.
14. Emery M. On the chaotic representation property for martingales // Probab. Theory and Math. Statist.
(St.Petersburg, 1993). – Amsterdam: Gordon and Breach, 1996.
15. Meyer P.-A. Quantum probability for probabilists // Lect. Notes Math. – 1993. – 1538. – 287 p.
16. Attal S., Belton A. C. R. The chaotic-representation property for a class of normal martingales // Probab.
Theory and Relat. Fields. – 2007. – 139, № 3-4. – P. 543 – 562.
17. Hida T., Kuo H.-H., Potthoff J., Streit L. White noise. An infinite-dimensional calculus. – Dordrecht:
Kluwer Acad. Publ. Group, 1993. – 516 p.
18. Kuo H.-H. White noise distribution theory. – Boca Raton: CRC Press, 1996. – 378 p.
19. Nualart D. The Malliavin calculus and related topics. – New York: Springer, 1995. – 266 p.
20. Koshmanenko V. D., Samoilenko Yu. S. The isomorphism between Fok space and a space of functions
of infinitely many variables // Ukr. Math. J. – 1975. – 27, № 5. – P. 669 – 674.
21. Berezansky Yu. M., Kondratiev Yu. G. Spectral methods in infinite-dimensional analysis. – Dordrecht:
Kluwer Acad. Publ., 1995. – Vols 1, 2. – 576 + 432 p. (Russian edition: Kiev: Naukova Dumka, 1988).
22. Berezansky Yu. M., Tesko V. A. Spaces of fundamental and generalized functions associated with generali-
zed translation // Ukr. Math. J. – 2003. – 55, № 12. – P. 1907 – 1979.
23. Kachanovsky N. A. On the extended stochastic integral connected with the gamma-measure on an
infinite-dimensional space // Meth. Funct. Anal. and Top. – 2002. – 8, № 2. – P. 10 – 32.
24. Kachanovsky N. A. On extended stochastic integral and the Wick calculus on the connected with the
Gamma-measure spaces of regular generalized functions // Ukr. Math. J. – 2005. – 57, № 8. – P. 1030 –
1057.
25. Kachanovsky N. A. On an extended stochastic integral and the Wick calculus on the connected with the
generalized Meixner measure Kondratiev-type spaces // Meth. Funct. Anal. and Top. – 2007. – 13, № 4.
– P. 338 – 379.
26. Kondratiev Yu. G., Da Silva J. L., Streit L., Us G. F. Analysis on Poisson and gamma spaces // Infin.
Dimens. Anal. Quantum Probab. Relat. Top. – 1998. – 1, № 1. – P. 91 – 117.
27. Berezansky Yu. M., Mierzejewski D. A. The structure of the extended symmetric Fock space // Meth.
Funct. Anal. and Top. – 2000. – 6, № 4. – P. 1 – 13.
28. Berezansky Yu. M. Spectral approach to white noise analysis // Proc. Symp. Dynam. Complex and
Irregular Systems (Bielefeld, 1991); Bielefeld Encount. Math. Phys. – River Edge, NJ: World Sci. Publ.,
1993. – 8. – P. 131 – 140.
29. Berezansky Yu. M., Tesko V. A. One approach to a generalization of white noise analysis // Operator
Theory: Adv. and Appl. – 2009. – 190. – P. 123 – 139.
30. Rodionova I. V. Analysis connected with generating functions of exponential type in one and infinite
dimensions // Meth. Funct. Anal. and Top. – 2005. – 11, № 3. – P. 275 – 297.
31. Kondratiev Yu. G., Lytvynov E. W. Operators of gamma white noise calculus // Infin. Dimens. Anal.
Quantum Probab. Relat. Top. – 2000. – 3, № 3. – P. 303 – 335.
32. Berezansky Yu. M., Mierzejewski D. A. The chaotic decomposition for the gamma field // Funct. Anal.
and Appl. – 2001. – 35, № 4. – P. 305 – 308.
33. Berezansky Yu. M. Pascal measure on generalized functions and the corresponding generalized Meixner
polynomials // Meth. Funct. Anal. and Top. – 2002. – 8, № 1. – P. 1 – 13.
34. Berezansky Yu. M., Mierzejewski D. A. The construction of the chaotic representation for the gamma
field // Infin. Dimens. Anal. Quantum Probab. Relat. Top. – 2003. – 6, № 1. – P. 33 – 56.
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
764 N. A. KACHANOVSKY, V. A. TESKO
35. Berezansky Yu. M., Lytvynov E., Mierzejewski D. A. The Jacobi field of a Levy process // Ukr. Math. J.
– 2003. – 55, № 5. – P. 853 – 858.
36. Lytvynov E. W. Orthogonal decompositions for Levy processes with an applications to the Gamma,
Pascal and Meixner processes // Infin. Dimens. Anal. Quantum Probab. Relat. Top. – 2003. – 6, № 1. –
P. 73 – 102.
37. Lytvynov E. W. Polynomials of Meixner’s type in infinite dimensions — Jacobi fields and orthogonality
measures // J. Funct. Anal. – 2003. – 200, № 1. – P. 118 – 149.
38. Lytvynov E. W. Functional spaces and operators connected with some Levy noises // Proc. 5-th Int.
ISAAC Congr. (July 2005, Catania, Italy). – (arxiv.org/abs/math.PR/0608380).
39. Nualart D., Schoutens W. Chaotic and predictable representations for Lévy processes // Stochast. Process.
and Appl. – 2000. – 90, № 1. – P. 109 – 122.
40. Benth F. E., L/okka A. Anticipative calculus for Lévy processes and stochastic differential equations //
Stochast. Int. J. Probab. and Stochast. Process. – 2004. – 76, № 3. – P. 191 – 211.
41. Nunno G. D., /Oksendal B., Proske F. White noise analysis for Lévy processes // J. Funct. Anal. – 2004.
– 206, № 1. – P. 109 – 148.
42. Berezansky Yu. M., Sheftel Z. G., Us G. F. Functional analysis. – Basel etc.: Birkhäuser Verlag, 1996. –
Vols. 1, 2. – 423 + 293 p. (Russian edition: Kiev: Vyshcha Shkola, 1990).
43. Kondratiev Yu. G., Leukert P., Streit L. Wick calculus in Gaussian analysis // Acta Appl. Math. – 1996.
– 44, № 3. – P. 269 – 294.
44. Lindstrom T., /Oksendal B., Uboe J. Wick multiplication and Itô – Skorokhod stochastic differential
equations // Ideas and Meth. Math. Anal., Stochast., and Appl. – Cambridge: Cambridge Univ. Press,
1992. – P. 183 – 206.
45. Gikhman I. I., Skorokhod A. V. Theory of random processes. – Moscow: Nauka, 1975. – Vol. 3. – 496 p.
(in Russian).
46. Protter Ph. E. Stochastic integration and differential equations. Second ed. – Berlin: Springer, 2004. –
415 p.
47. Gaveau B., Trauber P. L’intégrale stochastique comme opérateur de divergence dans l’espace fonctionnel
// J. Funct. Anal. – 1982. – 46, № 2. – P. 230 – 238.
48. Kuo H.-H. Stochastic integration via white noise analysis // Nonlinear Anal. – 1997. – 30, № 1. –
P. 317 – 328.
49. Us G. F. Towards a coloured noise analysis // Meth. Funct. Anal. and Top. – 1997. – 3, № 2. – P. 83 – 99.
50. Gelfand I. M., Vilenkin N. Ya. Generalized functions // Applications of Harmonic Analysis. – New York;
London: Acad. Press, 1964 (1977). – 384 p.
51. Meixner J. Orthogonale Polynomsysteme mit einem besonderen Gestalt der erzeugenden Funktion // J.
London Math. Soc. – 1934. – 9, № 1. – P. 6 – 13.
52. Skorokhod A. V. Integration in Hilbert space. – New York; Heidelberg: Springer, 1974. – 177 p.
Received 04.03.09
ISSN 1027-3190. Укр. мат. журн., 2009, т. 61, № 6
|
| id | umjimathkievua-article-3056 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:35:26Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a6/c0ce41257067b9f19afc21bc4ca245a6.pdf |
| spelling | umjimathkievua-article-30562020-03-18T19:44:24Z Stochastic integral of Hitsuda–Skorokhod type on the extended Fock space Стохастичний інтеграл типу Хітцуди - Скорохода на розширеному просторі Фока Kachanovskii, N. A. Tesko, V. A. Качановський, М. О. Теско, В. А. We review some recent results related to stochastic integrals of the Hitsuda–Skorokhod type acting on the extended Fock space and its riggings. Наведено огляд деяких останніх результатів, пов'язаних із стохастичними інтегралами типу Хітцуди-Скорохода, що діють на розширеному просторі Фока та його оснащеннях. Institute of Mathematics, NAS of Ukraine 2009-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3056 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 6 (2009); 733-764 Український математичний журнал; Том 61 № 6 (2009); 733-764 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3056/2861 https://umj.imath.kiev.ua/index.php/umj/article/view/3056/2862 Copyright (c) 2009 Kachanovskii N. A.; Tesko V. A. |
| spellingShingle | Kachanovskii, N. A. Tesko, V. A. Качановський, М. О. Теско, В. А. Stochastic integral of Hitsuda–Skorokhod type on the extended Fock space |
| title | Stochastic integral of Hitsuda–Skorokhod type on the extended Fock space |
| title_alt | Стохастичний інтеграл типу Хітцуди - Скорохода на розширеному просторі Фока |
| title_full | Stochastic integral of Hitsuda–Skorokhod type on the extended Fock space |
| title_fullStr | Stochastic integral of Hitsuda–Skorokhod type on the extended Fock space |
| title_full_unstemmed | Stochastic integral of Hitsuda–Skorokhod type on the extended Fock space |
| title_short | Stochastic integral of Hitsuda–Skorokhod type on the extended Fock space |
| title_sort | stochastic integral of hitsuda–skorokhod type on the extended fock space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3056 |
| work_keys_str_mv | AT kachanovskiina stochasticintegralofhitsudaskorokhodtypeontheextendedfockspace AT teskova stochasticintegralofhitsudaskorokhodtypeontheextendedfockspace AT kačanovsʹkijmo stochasticintegralofhitsudaskorokhodtypeontheextendedfockspace AT teskova stochasticintegralofhitsudaskorokhodtypeontheextendedfockspace AT kachanovskiina stohastičnijíntegraltipuhítcudiskorohodanarozširenomuprostorífoka AT teskova stohastičnijíntegraltipuhítcudiskorohodanarozširenomuprostorífoka AT kačanovsʹkijmo stohastičnijíntegraltipuhítcudiskorohodanarozširenomuprostorífoka AT teskova stohastičnijíntegraltipuhítcudiskorohodanarozširenomuprostorífoka |