A generalization of an extended stochastic integral
We propose a generalization of an extended stochastic integral to the case of integration with respect to a broad class of random processes. In particular, we obtain conditions for the coincidence of the considered integral with the classical Itô stochastic integral.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509406481874944 |
|---|---|
| author | Berezansky, Yu. M. Tesko, V. A. Березанський, Ю. М. Теско, В. А. |
| author_facet | Berezansky, Yu. M. Tesko, V. A. Березанський, Ю. М. Теско, В. А. |
| author_sort | Berezansky, Yu. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:51:39Z |
| description | We propose a generalization of an extended stochastic integral to the case of integration with respect to a broad class of random processes. In particular, we obtain conditions for the coincidence of the considered integral with the classical Itô stochastic integral. |
| first_indexed | 2026-03-24T02:40:36Z |
| format | Article |
| fulltext |
UDC 517.9
S. Albeverio (Inst. Angewandte Math., Univ. Bonn, Germany),
Yu. M. Berezansky, V. A. Tesko (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
A GENERALIZATION OF AN EXTENDED
STOCHASTIC INTEGRAL∗
UZAHAL\NENNQ ROZÍYRENOHO
STOXASTYÇNOHO INTEHRALA
We propose a generalization of an extended stochastic integral in the case of integration with respect to a wide
class of random processes. In particular, we obtain conditions for the coincidence of the considered integral
with the classical Itô stochastic integral.
Zaproponovano uzahal\nennq rozßyrenoho stoxastyçnoho intehrala na vypadok intehruvannq vidnosno
ßyrokoho klasu vypadkovyx procesiv. Zokrema, oderΩano umovy, za qkyx vkazanyj intehral zbiha[t\sq
z klasyçnym stoxastyçnym intehralom Ito.
1. Introduction. It is well known that the extended (Hitsuda – Skorokhod) stochastic
integral (that is a natural generalization of the classical Itô integral) plays an important
role in the Gaussian and Poissonian analysis. The notion of such integrals was intro-
duced approximately at the same time in the works of several mathematicians: M. Hit-
suda [1], Yu. L. Daletsky and S. N. Paramonova [2, 3], A .V. Skorokhod [4], Yu. M. Ka-
banov and A .V. Skorokhod [5], Yu. M. Kabanov [6] and later for the Gamma-process by
N. A. Kachanovsky [7, 8]. The definitions of the extended stochastic integral proposed
in the mentioned works are equivalent but their forms are different (see, e.g., [9, 10] for
details).
In this work the notion of an extended stochastic integral is introduced in terms of a
rigging of a Fock space. Under a functional realization of Fock space using a Wiener –
Itô – Segal-type isomorphism (see, e.g., [11, 12]) we obtain a general definition of an
extended stochastic integral in terms of an L2-space and its rigging. Note that in the
Gaussian and Poissonian cases this definition coincides with the corresponding definitions
given in [1, 4 – 6].
Such an approach to the construction of the extended stochastic integral is, on the
one hand, simple, and, on other hand, very general and applicable to many stochastic
processes. It is based on the theory of generalized functions of infinitely many variables
(see the corresponding surveys [11, 12] and, in particular, the papers [13 – 15]).
One of the main ingredients is the realization of the conditional expectations as or-
thogonal projectors in an L2-space (see, e.g., [16]). In this way for a square integrable
martingale M(t) one can write M(t) = E(t)M , t ∈ [0,∞), where E(t) is some resolu-
tion of identity in the L2-space [17 – 20] and M is a fixed vector from L2. Since in the
theory of stochastic processes it is an accepted assumption that M(t) is right-continuous,
we will assume that E(t) is right-continuous (instead usual for functional analysis of
left-continuous).
In the last part of this paper we find conditions under which the extended stochastic
integral is an extension of the Itô integral. Here we also recall the theory of multiple
spectral integrals for the symmetric complex-valued functions (this theory is based on
some general results of spectral theory, see [20]) and describe an interconnection of such
integrals with multiple Itô integrals.
The authors hope that a similar construction can be developed for the more com-
plicated cases of stochastic integration connected with Gamma, Pascal, and Meixner pro-
∗ This work was partly supported by DFG 436 UKR 113/78/0-1.
c© S. ALBEVERIO, YU. M. BEREZANSKY, V. A. TESKO, 2007
588 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
A GENERALIZATION OF AN EXTENDED STOCHASTIC INTEGRAL 589
cesses. But in these cases it is necessary to use more complicated “extended Fock spaces”;
some results connected with such spaces are given in [21 – 30].
Let us describe the general idea of our construction. Let (Ω,A, P ) be a probability
space with a flow of σ-subalgebras {At}t∈R+ . Let
{
M(t)
}
t∈R+
be a normal martingale
with respect to the flow {At}t∈R+ with the chaotic representation property. This property
means that the mapping
F � f = (fn)∞n=0 �→ If =
∞∑
n=0
In(fn) ∈ L2(Ω,A, P ) =: L2
is well-defined and unitary. Here F is a Fock space constructed over L2(R+, dt) (dt is
the Lebesgue measure) and In(fn) is an n-multiple stochastic integral with respect to M .
By definition the Itô integral
∫
R+
F (t)dM(t) of a simple At-adapted function (con-
structed using the characteristic functions κ∆j
of sets ∆j)
F (t) =
n∑
j=1
Fjκ∆j
(t), ∆j = (sj , tj ], Fj ∈ L2,
is defined by the equality∫
R+
F (t)dM(t) :=
n∑
j=1
Fj(M(tj) −M(sj)) ∈ L2.
It is easy to verify that the I−1-image of this integral has the form
I−1
∫
R+
F (t)dM(t)
=
n∑
j=1
I−1(Fj)♦κ∆j ∈ F ,
where ♦ is the Wick multiplication in the space F (see, for example, [11]).
According to this equality it is reasonable to define the “Itô integral” of a simple
function
f(t) =
n∑
j=1
fjκ∆j (t), fj ∈ F ,
on the Fock space F by the formula
SI(f) :=
n∑
j=1
fj♦κ∆j
∈ F .
Using the heuristic representation
κ∆ =
∫
∆
δtdt,
we obtain (at least heuristically)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
590 S. ALBEVERIO, YU. M. BEREZANSKY, V. A. TESKO
SI(f) =
n∑
j=1
fj♦κ(∆j) =
n∑
j=1
fj♦
∫
∆j
δtdt =
n∑
j=1
∫
∆j
fj♦δtdt =
=
∞∫
0
( n∑
j=1
fjκ∆j (t)
)
♦δtdt =
∞∫
0
f(t)♦δtdt =
∞∫
0
a+(δt)f(t)dt.
Here a+(δt) is the creation operator in a Fock space, δt is the δ-function concentrated at
t ∈ R+.
This heuristic construction gives a reason to take the formula
∞∫
0
a+(δt)f(t)dt (1.1)
as the definition of an extended stochastic integral in a Fock space. In Section 3 we prove
that this integral exists as a Bochner integral of the vector-valued function R+ � t �→
�→ a+(δt)f(t) with values in some negative Fock space F− ⊃ F . In Section 5 we show
that the image of this integral under several functional realizations of Fock space F is an
extension of the Itô integral.
This paper presents the above described results. Other results connected with sub-
ject of this article are given, e.g., in [9, 10, 31 – 36] (see also references therein). The
preliminary version of this paper was published in the preprint [37].
2. Preliminaries. In this section we recall some well known objects (a Fock space
and its riggings) and functional realizations of these objects: Jacobi fields acting on a
Fock space and the general theory of generalized functions of infinitely many variables
(see, e.g., [38, 20, 39, 11, 12] for more details).
2.1. A general symmetric Fock space and its rigging. A more detailed account of
the results of this subsection is contained in [40, 11]. In what follows we will use the
notation
Np := {p, p+ 1, . . .}, p ∈ Z,
where Z is the set of all entire numbers.
We consider a fixed family (Hp)p∈N0 of real separable Hilbert spaces Hp; H0 will
also be denoted by H . This family is such that for all p ∈ N0 the space Hp+1 is densely
embedded in Hp, and this embedding is quasinuclear, i.e., of Hilbert – Schmidt type (the
Hilbert – Schmidt norm will be denoted by ‖·‖HS). Without loss of generality we assume
that ‖ · ‖Hp
≤ ‖ · ‖Hp+1 . We can construct the nuclear rigging of the space H0
Φ
′
:= ind lim
p∈N0
H−p ⊃ H−p ⊃ H0 ⊃ Hp ⊃ pr lim
p∈N0
Hp =: Φ, (2.1)
where H−p, p ∈ N1, is the dual space to Hp with respect to the zero space H0. We denote
by 〈· , ·〉 the dual pairing between the elements of H−p and Hp (this pairing is generated
by the scalar product in H0). It is possible to construct for any n ∈ N0 the nuclear chain
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
A GENERALIZATION OF AN EXTENDED STOCHASTIC INTEGRAL 591
Fn(Φ
′
) ⊃ Fn(H−p) ⊃ Fn(H0) ⊃ Fn(Hp) ⊃ Fn(Φ),
‖ ‖ ‖
H⊗̂n
−p,C H⊗̂n
0,C H⊗̂n
p,C
Fn(Φ) := pr lim
p∈N0
Fn(Hp), Fn(Φ
′
) := ind lim
p∈N0
Fn(H−p).
(2.2)
Here and below the symbol ⊗̂ denotes the symmetric tensor product (⊗ is the ordinary
tensor product), the subindex C denotes a complexification. We denote by (· , ·)Fn(H0)
the complex pairing between elements of Fn(H−p) and Fn(Hp) (for real pairings we
preserve the notation 〈· , ·〉). Note that for n = 0 all spaces in (2.2) coincide with C.
For each p ∈ Z we introduce a weighted symmetric Fock space F(Hp, τ) with a fixed
weight τ = (τn)∞n=0, τn > 0, by setting
F(Hp, τ) :=
∞⊕
n=0
Fn(Hp)τn =
=
{
f = (fn)∞n=0
∣∣ fn ∈ Fn(Hp), ‖f‖2
F(Hp,τ) =
∞∑
n=0
‖fn‖2
Fn(Hp)τn<∞
}
. (2.3)
We will often use the following weight: fix K > 1 and put
τ(q) = ((n!)2Kqn)∞n=0, q ∈ N0 (0! = 1). (2.4)
Using rigging (2.1) and the weight (2.4) we construct the nuclear rigging
F(Φ
′
) ⊃ F(−p,−q) ⊃ F (H0) ⊃ F(p, q) ⊃ F(Φ),
F(Φ) := pr lim
p,q∈N0
F(p, q), F(Φ
′
) := ind lim
p,q∈N0
F(−p,−q).
(2.5)
Here
F(−p,−q) := F(H−p, (K−qn)∞n=0), F(p, q) := F(Hp, ((n!)2Kqn)∞n=0),
F (H0) := F(H0, (n!)∞n=0).
(2.6)
The first two spaces from (2.6) are dual with respect to the space F (H0). We point out
that in (2.5) and (2.6) the zero space F (H0) is Fock space (2.3) with the weight τn = n!.
It is obvious that the set Ffin (Φ) of all finite sequences (ϕn)∞n=0, ϕn ∈ Fn(Φ), is dense
in each space of (2.5).
The complex pairing between elements of F(−p,−q) and F(p, q) (generated by the
scalar product in F (H0)) will be denoted by 〈〈· , ·〉〉 (or (· , ·)F (H0)). This pairing is given
by the formula
〈〈ξ, f〉〉 =
∞∑
n=0
〈ξn, f̄n〉n!,
ξ = (ξn)∞n=0 ∈ F(−p,−q), f = (fn)∞n=0 ∈ F(p, q),
(2.7)
where the overbar denotes complex conjugation.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
592 S. ALBEVERIO, YU. M. BEREZANSKY, V. A. TESKO
2.2. Jacobi fields. We recall some results concerning the theory of Jacobi fields in
a Fock space (see [38, 20, 39, 41] for more details). This theory gives a possibility to
pass from an abstract Fock space to the functional Hilbert space L2(Q,B(Q), ρ) on some
space Q with respect to a probability measure ρ on the Borel σ-algebra B(Q). We recall
that the theory of Jacobi fields was created under the influence of the works of M. Krein
(see, e.g., [42, 43]) about Jacobi matrices.
Consider Fock space (2.1) with p = 0 and weight τn = 1, n ∈ N0, i.e., the space
F(H) =
∞⊕
n=0
Fn(H), (2.8)
where we set H = H0. As usually Ffin (H) denotes the set of finite vectors from F(H);
the vector Ω = (1, 0, . . .) ∈ Ffin (H) is called the vacuum. Let H1 = H+ be a fixed
space from chain (2.1); the embedding H+ ↪→ H be quasinuclear (as we have demanded
above). Consider in the space F(H) a family J = (J(ϕ))ϕ∈H+ of operator-valued Jacobi
matrices
J(ϕ) =
b0(ϕ) a0(ϕ) 0 0 0 . . .
a0(ϕ) b1(ϕ) a∗1(ϕ) 0 0 . . .
0 a1(ϕ) b2(ϕ) a∗2(ϕ) 0 . . .
· · · · · . . .
, ϕ ∈ H+, (2.9)
with entries
an(ϕ) : Fn(H) → Fn+1(H), bn(ϕ) = (bn(ϕ))∗ : Fn(H) → Fn(H),
a∗n(ϕ) = (an(ϕ))∗ : Fn+1(H) → Fn(H), n ∈ N0.
(2.10)
Assume that the following conditions on (2.9) are fulfilled.
a) For any ϕ ∈ H+ operators (2.10) are bounded and real (i.e., act from real sub-
spaces of Fn(H), Fn+1(H) into real ones).
b) The dependence of the elements of J(ϕ) on ϕ ∈ H+ is linear and continuous in
the following sense: the operators
H+ � ϕ �→ an(ϕ)fn ∈ Fn+1(H+),
H+ � ϕ �→ bn(ϕ)fn ∈ Fn(H+), fn ∈ Fn(H+),
H+ � ϕ �→ a∗n(ϕ)fn+1 ∈ Fn(H+), fn+1 ∈ Fn+1(H+), n ∈ N0,
(2.11)
are linear and bounded (this can be seen as a condition of “smoothness” of entries from
(2.9): the vectors from H+ are more “smooth” then vectors from H).
Every matrix (2.9) gives rise to a Hermitian operator A(ϕ) on space F(H) (2.8): for
f = (fn)∞n=0 ∈ Dom (A(ϕ)) := Ffin (H+) we put
(A(ϕ)f)n := (J(ϕ)f)n = an−1(ϕ)fn−1 + bn(ϕ)fn + a∗n(ϕ)fn+1, (2.12)
n ∈ N0, a−1(ϕ) = 0.
c) The operators A(ϕ), ϕ ∈ H+, are essentially selfadjoint and their closures Ã(ϕ)
are strongly commuting.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
A GENERALIZATION OF AN EXTENDED STOCHASTIC INTEGRAL 593
d) (regularity) For each n ∈ N1, a real linear operator Vn,n : Fn(H+) → Fn(H+)
defined by the formula
Vn,n(ϕ1⊗̂ . . . ⊗̂ϕn) := (J(ϕ1) . . . J(ϕn)Ω)n = an−1(ϕ1) . . . a0(ϕn)1, (2.13)
ϕ1, . . . , ϕn ∈ H+,
is continuous and invertible; we also put V0,0 := 1.
Above described family J = (J(ϕ))ϕ∈H+ of matrices is by definition a (commuting)
Jacobi field. Our first aim is to construct the generalized eigenvector expansion for the
family A =
(
Ã(ϕ)
)
ϕ∈H+
of the corresponding selfadjoint operators acting on the Fock
space F(H) (about the general theory of such expansions see, e.g., [40, 44]).
For the investigation of the spectral theory of the family A we start from giving a
quasinuclear rigging of real Hilbert spaces
H− ⊃ H ⊃ H+. (2.14)
After this we construct the following rigging of the space F(H), using weighted spaces
of form (2.3):
F(H−, τ
−1) ⊃ F(H) ⊃ F(H+, τ) ⊃ Ffin (H+),
H− = H−1, τ = (τn)∞n=0, τn ≥ 1, τ−1 = (τ−1
n )∞n=0.
(2.15)
We suppose that the embedding F(H+, τ) ↪→ F(H) is quasinuclear, i.e., that the weight
is such that
∞∑
n=0
‖O‖2n
HSτ
−1
n < ∞, (2.16)
where O is the embedding operator H+ ↪→ H [40, 44, 11].
The main result about generalized eigenvector expansion is as follows.
LetA = (Ã(ϕ))ϕ∈H+ be a Jacobi field. ForA there exists a Borel probability measure
ρ on the space H− (the spectral measure) such that the Fourier transform
F(H) ⊃ F(H+, τ) � f = (fn)∞n=0 �→ (Ff)(·) = (f, P (·))F(H) =
=
∞∑
n=0
(fn, Pn(·))Fn(H) ∈ L2(H−,B(H−), ρ) =: (L2
H−) (2.17)
after being extended by continuity to the whole space F(H) is a unitary operator acting
from the space F(H) to the space (L2
H−
).
In (2.17) for any x ∈ H−, P (x) = (Pn(x))∞n=0, Pn(x) ∈ Fn(H−), is a real-valued
sequence that is a joint solution of the system of the following operator-difference equa-
tions:
(a∗n−1(ϕ))+Pn−1(x) + (bn(ϕ))+Pn(x) + (an(ϕ))+Pn+1(x) = (x, ϕ)HPn(x),
(2.18)
n ∈ N0, x ∈ H−, ϕ ∈ H+; P−1(x) = 0, P0(x) = 1.
Here we denote by C+ the operator adjoint to C with respect to the zero spaces H , i.e., if
C : Fk(H+) → Fj(H+) is continuous, then C+ : Fj(H−) → Fk(H−) and is connected
with C by the equality
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
594 S. ALBEVERIO, YU. M. BEREZANSKY, V. A. TESKO
(Cfk, gj)Fj(H) = (fk, C
+gj)Fk(H), fk ∈ Fk(H+), gj ∈ Fj(H−), j, k ∈ N0.
(2.19)
Equality (2.18) and relation (2.19) show that the sequence P (x) = (Pn(x))∞n=0,
Pn(x) ∈ Fn(H−) is the solution (in a generalized sense) of the equation
J(ϕ)P (x) = (x, ϕ)HP (x), ϕ ∈ H+, x ∈ H−, (2.20)
i.e., is a generalized eigenvector of operator Ã(ϕ) with “eigenvalue” (x, ϕ)H . Pn(x) is
in some sense “a polynomial of degree n with respect to infinite-dimensional variable
x ∈ H−”. For more details on the construction and properties of P (x) see [38, 20, 39].
2.3. Two classical examples of Jacobi fields. 1. Free field (see, e.g., [38, 40, 20,
45]). In this case dimH = ∞, H+ is arbitrary with quasinuclear embedding H+ ↪→ H .
Matrix J(ϕ) (2.9) has the form: for any ϕ ∈ H+
J(ϕ) = J+(ϕ) + J−(ϕ), (2.21)
where for fn ∈ Fn(H), n ∈ N0,
J+(ϕ)fn =
√
n+ 1ϕ⊗̂fn : Fn(H) → Fn+1(H), J−(ϕ) = (J+(ϕ))∗, (2.22)
i.e., J+(ϕ), J−(ϕ) are classical creation and annihilation operators. The conditions a) –
d) are fulfilled and the operators Vn,n have the form
Vn,n =
√
n! Id, n ∈ N1. (2.23)
The spectral measure ρ is equal to the Gaussian measure gS on the space H− with the
zero mean and the correlation operator S = O+OI : H− → H−, where O : H+ ↪→ H ,
O+ : H ↪→ H−, I : H− → H+ are canonical operators connected with chain (2.14).
Fourier transform (2.17) is the classical Wiener – Itô – Segal transformation.
2. Poisson field [46, 38, 20, 45, 41, 47]. In this case H = L2
Re(R,B(R), ν) =:
=: L2
Re(R, ν), where R is a topological abstract space with a σ-finite Borel measure ν
on B(R). Let H+ be a certain fixed real Hilbert space embedded into H densely and
quasinuclearly (for the construction of such spaces see [44, 12]). Jacobi matrix J(ϕ)
(2.9) has now the form
J(ϕ) = J+(ϕ) +B(ϕ) + J−(ϕ), ϕ ∈ H+, (2.24)
i.e., it is equal to some perturbation of matrix (2.21) by a diagonal matrix B(ϕ). This
matrix B(ϕ) is equal to the second (differential) quantization of the operator b(ϕ) of
multiplication by a bounded function ϕ in the space H+, i.e., for any fn ∈ Fn(H)
B(ϕ)fn = bn(ϕ)fn = (b(ϕ) ⊗ Id ⊗ . . .⊗ Id)fn + (Id ⊗ b(ϕ) ⊗ Id ⊗ . . .⊗ Id)fn + . . .
. . .+ (Id ⊗ . . .⊗ Id ⊗ b(ϕ))fn ∈ Fn(H), n ∈ N1, B(ϕ)f0 = 0. (2.25)
The conditions a) – d) also are fulfilled. As in example 1 the operator Vn,n has
form (2.23). The spectral measure ρ is now a centered Poisson measure with intensity
ν (ν may be atomic). The measure ρ is defined by its Fourier transform∫
H−
ei(x,ϕ)Hdρ(x) = exp
∫
R
(eiϕ(q) − 1 − iϕ(q))dν(q)
, ϕ ∈ H+. (2.26)
Fourier transform (2.17) is the Wiener – Itô – Segal type transform of Poisson mea-
sures.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
A GENERALIZATION OF AN EXTENDED STOCHASTIC INTEGRAL 595
2.4. Transfer from the Fock space to a space of functions in the general case.
In Subsections 2.2, 2.3 the unitary operator F : F(H) → L2(Ω,A, P ) is a Fourier trans-
form generated in F(H) by some Jacobi field. In this subsection we will propose some
more general construction of the unitary operator F : F(H) → L2(Ω,A, P ), using the
orthogonal approach to the theory of generalized functions of infinitely many variables
(see, e.g., [11, 12] and references therein).
Let Q be a (separable) metric space, ρ be a fixed Borel finite measure on B(Q), and
L2(Q,B(Q), ρ) =: (L2
Q)
be the corresponding space of square integrable functions. By C(Q) we denote the linear
space of all complex-valued locally bounded (i.e., bounded on every ball inQ) continuous
functions on Q. We will understand C(Q) as a linear topological space with convergence
uniform on every ball from Q.
Let B0 be a neighborhood of zero in the space H0,C = F1(H0) and let
Q×B0 � {x, λ} �→ h(x, λ) ∈ C
be a given function. We assume that for each x ∈ Q h(x, ·) is analytic in a neighborhood
of zero in H0,C, and, for each λ ∈ B0, h(·, λ) ∈ C(Q). Moreover, h(·, λ) is locally
bounded uniformly with respect to λ from any closed ball inside of B0 and h(x, 0) = 1
for all x from Q.
It follows from [11], Sections 2.3, that, for each point x ∈ Q, there exists a neighbor-
hood of zero B1(x) ⊂ B0 in the space H1,C, such that
h(x, λ) =
∞∑
n=0
1
n!
〈
λ⊗n, hn(x)
〉
, hn(x) ∈ Fn(H−1), h0(x) = 1, (2.27)
for all λ fromB(x). Moreover, the last series converges uniformly on any closed ball from
B(x). Suppose that for all x ∈ Q there exists a general neighborhood of zero B1 ⊂ B0
with this property.
It is possible to construct a mapping of type (2.17) using instead of Pn(x) the func-
tions hn(x) from (2.27). For this aim it is necessary to impose some conditions on h. So,
we will assume that for all n ∈ N0 the estimate∥∥ ‖hn(·)‖Fn(H−1)
∥∥
(L2
Q)
≤ LCnn! (2.28)
with some constants L > 0, C > 0 is fulfilled.
It follows from (2.28) that for any fn ∈ Fn(H1) the functions
Q � x �→
〈
fn, hn(x)
〉
∈ C (2.29)
belong to the space (L2
Q). We suppose that the set of all functions (2.29), where fn ∈
∈ Fn(H1), n ∈ N0, is dense in (L2
Q) and that they are orthogonal in the following sense:∫
Q
〈
fn, hn(x)
〉
〈gm, hm(x)〉dρ(x) = δn,mn!〈fn, ḡn〉, n, m ∈ N0. (2.30)
It is possible to prove that condition (2.30) of orthogonality is fulfilled if estimate
(2.28) holds with H−1 replaced by H−p with some p ∈ N1 and the following equality:
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
596 S. ALBEVERIO, YU. M. BEREZANSKY, V. A. TESKO∫
Q
h(x, f)h(x, g)dρ(x) = exp〈f, g〉 (2.31)
is fulfilled. Here f, g ∈ ΦC ⊂ H1,C are such that ‖f‖Hp,C
< r, ‖g‖Hp,C
< r, where
r > 0 is sufficiently small (the proof of the latter results is contained in [12], Section 3,
see also [11], Section 7).
Fix function h(x, λ) (2.27) with above-mentioned properties and introduce a mapping
of the form (2.17) but taking instead of Pn(x) the functions hn(x). So, we put
F (H0) ⊃ Ffin (Φ) � f = (fn)∞n=0 �→ (Ihf)(·) =
∞∑
n=0
〈fn, hn(·)〉 ∈ (L2
Q). (2.32)
Orthogonality (2.30) and the density of Ffin (Φ) in F (H0) mean that after extending
by continuity to the whole space F (H0) map (2.32) turns into the unitary operator Ih
that maps the whole space F (H0) onto whole (L2
Q). In this way we get a functional
realization of a Fock space.
The map Ih transfers rigging (2.5) onto the following rigging of the space (L2
Q):
ind lim
p,q∈N0
H(−p,−q) = (H)
′ ⊃ H(−p,−q) ⊃ (L2
Q) ⊃ H(p, q) ⊃ H = pr lim
p,q∈N0
H(p, q).
(2.33)
Here H(p, q) := IhF(p, q) is a Hilbert space with topology inducted by the topology of
F(p, q), H(−p,−q) is the negative space with respect to the zero space (L2
Q) and the
positive space H(p, q).
Remark 2.1. Note that function (2.29) belongs to the space C(Q) (see, e.g., [11],
Lemma 3.2) and for
fn = ϕ(1)⊗̂ . . . ⊗̂ϕ(n), ϕ(1), . . . , ϕ(n) ∈ H1,
we have〈
ϕ(1)⊗̂ . . . ⊗̂ϕ(n), hn(x)
〉
=
∂n
∂z1 . . . ∂zn
h(x, z1ϕ(1) + . . .+ znϕ
(n))
∣∣∣∣
z1=...=zn=0
,
(2.34)
for all x ∈ Q.
Moreover, one can show (see, e.g., [11] for more details) that for K > 1 sufficiently
large (we recall that K is the constant in (2.3), this constant is used in the definition of
F(p, q)) the mapping
F(p, q) � (fn)∞n=0 �→ f(·) :=
∞∑
n=0
〈fn, hn(·)〉 ∈ C(Q)
is well-defined, continuous and injective. Therefore the space H(p, q) is embedded in the
space C(Q), and one can understand H(p, q) as the Hilbert space of continuous functions
H(p, q) =
{
f ∈ C(Q)
∣∣ ∃(fn)∞n=0 ∈ F(p, q) : f(x) =
∞∑
n=0
〈fn, hn(x)〉, x ∈ Q
}
with the Hilbert norm
‖f‖H(p,q) =
∥∥∥∥∥
∞∑
n=0
〈fn, hn(·)〉
∥∥∥∥∥
H(p,q)
=
∥∥(fn)∞n=0
∥∥
F(p,q)
.
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A GENERALIZATION OF AN EXTENDED STOCHASTIC INTEGRAL 597
Remark 2.2. It is clear that the mapping
F(−p,−q) ⊃ F (H0) � f = (fn)∞n=0 �→ Ihf =
∞∑
n=0
〈fn, hn〉 ∈ H(−p,−q)
is isometric and after closure by continuity is a unitary isomorphism between F(−p,−q)
and H(−p,−q) (we preserve the notation Ih for the closure). As a result the space of
generalized functions H(−p,−q) can be presented in the form
H(−p,−q) = Ih(F(−p,−q)) =
=
{
ξ =
∞∑
n=0
〈ξn, hn〉
∣∣∣∣ (ξn)∞n=0 ∈ F(−p,−q), ‖ξ‖H(−p,−q) = ‖(ξn)∞n=0‖F(−p,−q)
}
.
(2.35)
Here
〈ξn, hn〉 := lim
k→∞
〈f (k)
n , hn〉 ∈ H(−p,−q), n ∈ N0, (2.36)
where the sequence (f (k)
n )∞k=0 ⊂ Fn(H0) converges to ξn ∈ Fn(H−p) in the topology of
Fn(H−p) (note that we understand the limit in (2.36) as a limit in H(−p,−q)). One can
show (see [12]) that in H(−p,−q)
〈ξm, hm〉 = ∂+(ξm)1, ξm ∈ Fm(H−p), m ∈ N0,
where
∂+(ξm) := Iha+(ξm)I−1
h : H(−p,−q) → H(−p,−q) (2.37)
is a linear continuous operator that is the image of the creation operator
a+(ξm) : F(−p,−q) → F(−p,−q), p, q ∈ N1.
We recall that by definition the operator a+(ξm) acts on any vector η = (ηn)∞n=0 ∈
∈ F(−p,−q) by the formula
a+(ξm)η = a+(ξm)(η0, η1, . . .) := (0, . . . , 0︸ ︷︷ ︸
m
, ξm⊗̂η0, ξm⊗̂η1, . . .), (2.38)
∥∥a+(ξm)η
∥∥
F(−p,−q)
≤ K− qm
2 ‖ξm‖Fm(H−p)‖η‖F(−p,−q) (2.39)
(this estimate follows from (2.3) with τ given by (2.4)).
The dual pairing 〈〈· , ·〉〉 between elements of H(−p,−q) and H(p, q), p, q ∈ N1, from
rigging (2.33) that is generated by the scalar product in (L2
Q) has the form
〈〈ξ, f〉〉 =
〈〈 ∞∑
n=0
〈ξn, hn〉,
∞∑
n=0
〈fn, hn〉
〉〉
=
∞∑
n=0
〈ξn, f̄n〉n!,
ξ =
∞∑
n=0
〈ξn, hn〉 ∈ H(−p,−q), f =
∞∑
n=0
〈fn, hn〉 ∈ H(p, q).
(2.40)
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598 S. ALBEVERIO, YU. M. BEREZANSKY, V. A. TESKO
2.5. Connection between Subsection 2.4 and 2.2. Let we have Jacobi field (2.9)
and construct its spectral representation using chain (2.15). Fourier transform F (2.17)
transfers the space F(H0) onto the space L2(H−,B(H−), ρ) = (L2
H−
). As it follows
from the property of F , the series
∞∑
n=0
1√
n!
〈
λ⊗n, Pn(x)
〉
, λ ∈ H+,C,
converges in the topology of (L2
H−
), and its sum is a holomorphic function with respect
to λ. Moreover, according to [38]
‖ ‖Pn(·)‖Fn(H−)‖(L2
H−
) ≤ Cn
√
n!, n ∈ N0,
for some constant C > 0. Therefore, if we put
h(x, λ) :=
∞∑
n=0
1
n!
〈
λ⊗n, hn(x)
〉
, hn(x) :=
√
n!Pn(x), (2.41)
we can understand Fourier transform (2.17) as a particular case of transform (2.32). In
this case, it is not necessarily to verify that function (2.41) satisfies all assumptions formu-
lated in Subsection 2.4 because for the sequence
(
hn(x) =
√
n!Pn(x)
)∞
n=0
orthogonality
relation (2.30) holds, and this gives a possibility to repeat the corresponding parts of the
construction in Subsection 2.4.
Note that it is possible to calculate the generating function h(x, λ) for the classical
examples of Jacobi fields (2.9) (see, e.g., [14, 11, 12] and Section 6).
3. On extended stochastic integral in a Fock space and in its functional realiza-
tion. In this section we give an exact definition of extended stochastic integral (1.1). The
probability sense of such integral will be discussed in Section 5.
3.1. On extended stochastic integral in a Fock space. Here and below we restrict
ourself to special form of rigging (2.1). Namely, fix a constant T ∈ (0,∞). Let
H0 := L2
Re
(
[0, T ),B([0, T )),m
)
=: L2
Re
(
[0, T ),m
)
,
where m is the Lebesgue measure on [0, T ), i.e., dm(t) = dt . It is clear that the space
Fn(H0), n ∈ N1, is isomorphic to the space L̂2([0, T ),m⊗n) of all complex-valued
symmetric functions from L2([0, T )n,m⊗n). Now
‖fn‖2
Fn(H0)
=
∫
[0,T )n
∣∣fn(t1, . . . , tn)
∣∣2dt1 . . . dtn =
= n!
T∫
0
tn∫
0
. . .
t2∫
0
∣∣fn(t1, . . . , tn)
∣∣2dt1
. . . dtn−1dtn.
Introduce rigging (2.1) of the form:
Φ
′
:= ind lim
p∈N0
H−p ⊃ H−p ⊃ H0 ⊃ Hp ⊃ pr lim
p∈N0
Hp =: Φ, (3.1)
where
Hp := W 2
p ([0, T ),m), p ∈ N0,
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A GENERALIZATION OF AN EXTENDED STOCHASTIC INTEGRAL 599
are the real Sobolev spaces, Φ
′
and H−p are the spaces dual to Φ and Hp with respect to
the zero space H0 correspondingly (see, e.g., [40, 44] for more details). Using (3.1) we
construct two parameter rigging corresponding to (2.5),
F(Φ
′
) ⊃ F(−p,−q) ⊃ F (H0) ⊃ F(p, q) ⊃ F(Φ). (3.2)
Let K be some Hilbert space. ByL2([0, T );K) we denote the Hilbert space of (vector-
valued) functions
[0, T ) � t �→ f(t) ∈ K, ‖f‖2
L2([0,T );K) =
∫
[0,T )
∥∥f(t)
∥∥2
Kdt < ∞,
with the corresponding scalar product.
The general definition of an extended stochastic integral is the following:
The extended stochastic integral (in a Fock space) of a function
ξ ∈ L2([0, T );F(−p,−q)), p, q ∈ N1,
is defined by the formula
Sext(ξ) =
∫
[0,T )
a+(δt)ξ(t)dt ∈ F(−p,−q). (3.3)
Here we understand the right-hand side as a Bochner integral of the vector-valued func-
tion
[0, T ) � t �→ a+(δt)ξ(t) ∈ F(−p,−q), (3.4)
were δt is the delta-function concentrated at t.
The correctness of this definition from the following statement follows.
Proposition 3.1. If ξ ∈ L2
(
[0, T );F(−p,−q)
)
, p, q ∈ N1, then the function (3.4)
is integrable in the Bochner sense on [0, T ).
Proof. Let ξ ∈ L2([0, T );F(−p,−q)). Using (2.39) and the estimate
‖δt‖F1(H−p) ≤ c, t ∈ [0, T ),
with some c > 0 (see, e.g., [44]) we obtain∫
[0,T )
‖a+(δt)ξ(t)‖F(−p,−q)dt ≤ K− q
2
∫
[0,T )
‖δt‖F1(H−p)‖ξ(t)‖F(−p,−q)dt ≤
≤ K− q
2
∫
[0,T )
‖δt‖2
F1(H−p)dt
1
2
∫
[0,T )
‖ξ(t)‖2
F(−p,−q)dt
1
2
≤
≤ cK− q
2T
1
2
∫
[0,T )
‖ξ(t)‖2
F(−p,−q)dt
1
2
< ∞,
whence the necessary statement follows.
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600 S. ALBEVERIO, YU. M. BEREZANSKY, V. A. TESKO
3.2. Some properties of the introduced integral. We shall prove the important prop-
erties of the extended stochastic integral Sext. Let fn(· ; ·1, . . . , ·n) ∈ F1(H0)⊗Fn(H0),
n ∈ N1. We denote by f̂n+1(t1, . . . , tn+1) the symmetrization of fn with respect to n+1
variables, i.e.,
f̂n+1(t1, . . . , tn+1) :=
1
n+ 1
n+1∑
k=1
fn(tk; t1, . . . , tk�, . . . , tn+1) (3.5)
for m⊗(n+1)-almost all (t1, . . . , tn+1) ∈ [0, T )n+1. We put f̂1(t) := f0(t) for all t ∈
∈ [0, T ).
Theorem 3.1. Let f(·) =
(
fn(·)
)∞
n=0
∈ L2
(
[0, T );F (H0)
)
and∑∞
n=0
‖f̂n+1‖2
Fn+1(H0)
(n+ 1)! < ∞. Then
Sext(f) = S(f) := (0, f̂1, . . . , f̂n, . . .) (3.6)
in the space F(−p,−q), p, q ∈ N1.
Proof. It is sufficient to show that
〈〈Sext(f), ψ〉〉 = 〈〈S(f), ψ〉〉
for each ψ =
(
0, . . . , 0︸ ︷︷ ︸
k
, ϕ⊗k, 0, 0, . . .
)
, ϕ ∈ Φ, k ∈ N0. Applying to a function
[0, T ) � t �→ f(t) = (fn(t))∞n=0 ∈ F (H0) ⊂ F(−p,−q)
the operator a+(δt) and using (2.38) we obtain
a+(δt)f(t) =
(
0, δt⊗̂f0(t), δt⊗̂f1(t), . . .
)
∈ F(−p,−q). (3.7)
Using (3.7) we have
〈〈Sext(f), ψ〉〉 =
〈 ∫
[0,T )
a+(δt)f(t)dt, ψ
〉
=
∫
[0,T )
〈〈a+(δt)f(t), ψ〉〉 dt =
= k!
∫
[0,T )
〈
δt⊗̂fk−1(t), ϕ⊗k
〉
dt = k!
∫
[0,T )
ϕ(t)
〈
fk−1(t), ϕ⊗(k−1)
〉
dt =
= k!
∫
[0,T )
ϕ(t)
∫
[0,T )k−1
fk−1(t; t1, . . . , tk−1)ϕ⊗(k−1)(t1, . . . , tk−1)dt1 . . . dtk−1
dt =
= k!
∫
[0,T )k
fk−1(t; t1, . . . , tk−1)ϕ⊗k(t, t1, . . . , tk−1)dt1 . . . dtk−1dt =
= k!
(
fk−1, ϕ
⊗k
)
H⊗k
0,C
= k!
(
f̂k, ϕ
⊗k
)
H⊗k
0,C
= k!
〈
f̂k, ϕ
⊗k
〉
=
〈
S(f), ψ
〉
.
The theorem is proved.
Let D ⊂ L2([0, T );F (H0)) ⊂ L2
(
[0, T );F(−p,−q)
)
, p, q ∈ N1, be the class of all
functions
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A GENERALIZATION OF AN EXTENDED STOCHASTIC INTEGRAL 601
[0, T ) � t �→ f(t) = (fn(t))∞n=0 ∈ F (H0) (3.8)
from L2
(
[0, T );F (H0)
)
such that for m-almost all t ∈ [0, T ) and m⊗n-almost all
(t1, . . . , tn) ∈ [0, T )n
fn(t) = fn(t; t1, . . . , tn) = κ(0,t]n(t1, . . . , tn)fn(t; t1, . . . , tn), n ∈ N1, (3.9)
where κα(·) is the characteristic function of a Borel set α ∈ B([0, T )n), κ(0,0]n := 0.
Theorem 3.2. If f ∈ D ⊂ L2
(
[0, T );F (H0)
)
then
‖S(f)‖F(H0) = ‖f‖L2([0,T );F (H0)). (3.10)
Proof. For f(·) =
(
fn(·)
)∞
n=0
∈ D we have
‖f‖2
L2([0,T );F (H0))
=
∫
[0,T )
∥∥f(t)
∥∥2
F (H0)
dt =
∫
[0,T )
∞∑
n=0
∥∥fn(t)
∥∥2
Fn(H0)
n! dt =
=
∞∑
n=0
n!
∫
[0,T )
∥∥fn(t)
∥∥2
Fn(H0)
dt =
=
∞∑
n=0
n!
∫
[0,T )
∫
[0,T )n
∣∣fn(t; t1, . . . , tn)
∣∣2dt1) . . . dtn
dt =
=
∞∑
n=0
n!
∫
[0,T )
∫
[0,t)n
∣∣fn(t; t1, . . . , tn)
∣∣2dt1 . . . dtn
dt =
=
∞∑
n=0
(n!)2
T∫
0
t∫
0
tn∫
0
. . .
t2∫
0
∣∣fn(t; t1, . . . , tn)
∣∣2dt1 . . . dtn−1dtn
dt =
=
∞∑
n=0
((n+ 1)!)2
T∫
0
tn+1∫
0
. . .
t2∫
0
∣∣f̂n+1(t1, . . . , tn+1)
∣∣2dt1 . . . dtndtn+1 =
=
∞∑
n=0
(n+ 1)!
∫
[0,T )n+1
∣∣f̂n+1(t1, . . . , tn+1)
∣∣2dt1 . . . dtn+1 =
=
∞∑
n=0
‖f̂n+1‖2
Fn+1(H0)
(n+ 1)! =
∥∥S(f)
∥∥2
F (H0)
.
3.3. On extended stochastic integral in a functional realization of the Fock space.
We will pass now to the construction of the “Ih-image” (the definition of Ih is given by
(2.32)) of extended stochastic integral (3.3). We consider instead of rigging (2.5), (2.6) of
the Fock space F (H0) the Ih-image
H(−p,−q) ⊃ (L2
Q) ⊃ H(p, q)
of this rigging (see (2.33)).
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602 S. ALBEVERIO, YU. M. BEREZANSKY, V. A. TESKO
Note that the above-mentioned space L2
(
[0, T );K
)
(K is a separable Hilbert space)
can be understood as a tensor product L2
(
[0, T ),m
)
⊗K, therefore
1 ⊗ Ih : L2([0, T );F(−p,−q)) → L2([0, T );H(−p,−q)), p, q ∈ N1,
is the unitary operator. This remark and (3.3) give the following definition.
The extended stochastic integral of a function
ξ ∈ L2([0, T );H(−p,−q)), p, q ∈ N1, (3.11)
is defined by the formula
Sext,h(ξ) =
∫
[0,T )
∂+(δt)ξ(t)dt ∈ H(−p,−q). (3.12)
Here we understand the right-hand side as a Bochner integral of the vector-valued func-
tion
[0, T ) � t �→ ∂+(δt)ξ(t) ∈ H(−p,−q), (3.13)
where ∂+(δt) is the Ih-image of the creation operator a+(δt), i.e., ∂+(δt) := Iha
+(δt)I−1
h .
The existence of a Bochner integral in (3.12) follows from Proposition 3.1 because if
ξ belongs L2([0, T );H(−p,−q)) then (1 ⊗ Ih)−1ξ belongs L2
(
[0, T );F(−p,−q)
)
and∫
[0,T )
∥∥∂+(δt)ξ(t)
∥∥2
H(−p,−q)
dt =
∫
[0,T )
∥∥Iha
+(δt)I−1
h ξ(t)
∥∥2
H(−p,−q)
dt =
=
∫
[0,T )
∥∥a+(δt)I−1
h ξ(t)
∥∥2
F(−p,−q)
dt < ∞.
We also point out that from (3.11), (3.12) and (3.3) we have
Sext,h(ξ) = IhSext((1 ⊗ Ih)−1ξ), ξ ∈ L2([0, T );H(−p,−q), p, q ∈ N1. (3.14)
Assume now that f(·) =
∑∞
n=0
〈fn(·), hn〉 ∈ L2([0, T ); (L2
Q)) and
∞∑
n=0
‖f̂n+1‖2
Fn+1(H0)
(n+ 1)! < ∞.
Using (3.14), (3.6) and (2.32) we obtain
Sext,h(f) = IhSext
(
(1 ⊗ Ih)−1f
)
= IhS
(
(1 ⊗ Ih)−1f
)
=
= Ih(0, f̂1, f̂2, . . .) =
∞∑
n=1
〈
f̂n, hn
〉
∈ (L2
Q). (3.15)
Moreover, if
f ∈ Dh := (1 ⊗ Ih)D ⊂ L2
(
[0, T ); (L2
Q)
)
=
= L2([0, T ),m) ⊗ (L2
Q) ⊂ L2([0, T );H(−p,−q)) (3.16)
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(D is defined by (3.8), (3.9)) then it follows from (3.10) and (3.15) that∥∥Sext,h(f)
∥∥
(L2
Q)
= ‖f‖L2([0,T );(L2
Q)).
Formulas (3.11) – (3.16) and corresponding assertions constitute, in particular, the ver-
sion of Theorem 3.1 and Theorem 3.2 in the language of functional realizations of Fock
space.
Remark 3.1. It is easy to understand that the constructions of this Section are pre-
served for the case T = ∞ if we take as Hp the weighted Sobolev space W 2
p
(
[0,∞), (1+
+ t2)pdm(t)
)
(such a construction is described in [37]). Now Φ is the Schwartz space of
infinite differentiable rapidly decreasing real-valued functions on [0,∞).
4. Martingales and their construction. Multiple spectral integrals. 4.1. Resolu-
tion of identity and martingales. We recall at first some generalization of the notion of
martingale and the integration with respect to such martingales of scalar-valued functions
[17 – 20].
Let Ω be some space of points ω, endowed by a σ-algebra A and a probability mea-
sure P defined on A, i.e., (Ω,A, P ) is a probability space. Let (At)t∈[0,T ) be a flow
of σ-subalgebras At of A with the properties: As ⊂ At if s ≤ t, s, t ∈ [0, T ), and⋂
t<u<T Au = At, T ≤ ∞. All the algebras A, At are supposed to be complete with
respect to the measure P . So, we have a filtration (At)t∈[0,T ), which is right continuous
for every t ∈ [0, T ).
Introduce the complex Hilbert space L2(Ω,A, P ) =: L2 and its subspaces
L2(Ω,At, P ) =: L2
t , t ∈ [0, T ). Denote by E(t) the orthogonal projector in the space L2
onto L2
t :
E(t)L2 = L2
t , t ∈ [0, T ); E(0) := 0. (4.1)
For the subspace L2
t we evidently have
L2
s ⊂ L2
t , E(s) ≤ E(t), s ≤ t, s, t ∈ [0, T ). (4.2)
The inclusion in (4.2) shows that for all s, t ∈ [0, T )
E(s)E(t) = E
(
min{s, t}
)
. (4.3)
As a result we constructed the operator-valued function E(t) with the properties of
a resolution of identity in L2. This function we will be called a quasiresolution of the
identity because it can be E
(
[0, T )
)
< 1. It is possible to understand E(t) as a projector-
valued measure B([0, T )) � α �→ E(α) on the σ-algebra B([0, T )) of Borel subsets of
[0, T ): for this we set E((s, t]) := E(t) − E(s) and extend this definition to all Borel
subsets of [0, T ). For details on such a procedure see [48], Chapter 6, [49], Chapter 6,
[44], Chapter 13, [40], Chapter 3.
Let MT be some vector from L2, then the vector-valued function
[0, T ) � t �→ M(t) := E(t)MT ∈ L2
t ⊂ L2 (4.4)
is a uniformly square-integrable martingale on the probability space (Ω,A, P ) with re-
spect to the filtration (At)t∈[0,T ), i.e., M(t) is a martingale with respect to (At)t∈[0,T )
and
∥∥M(t)
∥∥
L2 ≤ c, t ∈ [0, T ), for some constant c > 0 (see, e.g., [50 – 52, 31, 53] for
the corresponding definition).
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604 S. ALBEVERIO, YU. M. BEREZANSKY, V. A. TESKO
Conversely, every uniformly square-integrable martingale M(t) with respect to
(At)t∈[0,T ), has form (4.4). In fact, we construct by the filtration (At)t∈[0,T ) the cor-
responding quasiresolution of identity E(t), t ∈ [0, T ), in L2. Then for each f ∈ L2
the vector E(t)f is equal to the conditional expectation E{f |At}, but for a uniformly
square-martingale there exists a vector MT ∈ L2 such that M(t) = E{MT |At} (see,
e.g., [51], Chapter 1, § 1). The latter equality is equivalent to (4.4).
A slight generalization of (4.4) is the following. Let H be a complex Hilbert space
and E(t), t ∈ [0, T ), T ≤ ∞, be some quasiresolution of the identity in H, i.e., a
operator-valued function (or the corresponding operator-valued measure E(α)) with all
properties of right continuous resolutions of identity in H, but for which E([0, T )) ≤ 1.
Let MT ∈ H be fixed. Then the vector-valued function
[0, T ) � t �→ M(t) := E(t)MT ∈ H, (4.5)
is by definition, an abstract martingale.
For a Borel function [0, T ) � t �→ f(t) ∈ C we introduce an abstract stochastic
integral with respect to martingale (4.5) by the formula
∫
[0,T )
f(t)dM(t) :=
∫
[0,T )
f(t)dE(t)
MT , (4.6)
where in the right-hand side we have an ordinary spectral integral. The well-known prop-
erties of spectral integrals (see, e.g., [48, 49, 44]) give the corresponding properties of
integral (4.6).
Note one simple property of the definitions introduced above. Let U be some unitary
operator acting from H onto another Hilbert space K. Then Z(t) = UM(t), t ∈ [0, T ),
is also an abstract martingale in the space K because
Z(t) = UM(t) = G(t)ZT ; G(t) = UE(t)U−1, ZT = UMT ∈ K, (4.7)
and G(t) is a quasiresolution of identity in the space K.
Applying the operator U to equality (4.6) we get an abstract stochastic integral with
respect to the abstract martingale Z(t):
U
∫
[0,T )
f(t)dM(t)
=
∫
[0,T )
f(t)dZ(t) =
∫
[0,T )
f(t)dG(t)
ZT . (4.8)
4.1. On multiple spectral integrals. In this subsection we recall a generalization of
above constructions (4.5) – (4.8) for the introduction of multiple spectral integrals with
respect to an abstract n-dimensional martingale for complex-valued symmetric functions
of n ∈ N1 variables t1, . . . , tn ∈ [0, T ) (see [20] for details).
At first we construct some class of n-dimensional resolutions of identity using a tensor
product. Namely, let E(t), t ∈ [0, T ), be some quasiresolution of identity in a complex
Hilbert space H. Introduce for each n ∈ N1 in the complex Hilbert space H⊗n the
quasiresolution of identity E⊗n by setting for Borel rectangles ∆1 × . . .× ∆n
E⊗n(∆1 × . . .× ∆n) := E(∆1) ⊗ . . .⊗ E(∆n). (4.9)
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A GENERALIZATION OF AN EXTENDED STOCHASTIC INTEGRAL 605
This projector-valued function of rectangles (projector in the space H⊗n) can be extended
to some quasiresolution of identity E⊗n, which we also denote by E⊗n (see, e.g., [40]).
LetMT be some vector from the space H, andM(t) = E(t)MT be the corresponding
martingale. Then we define an abstract n-dimensional martingale B([0, T )n) � α �→
�→ M(α) ∈ H⊗n by the formula
Mn(α) := E⊗n(α)M⊗n
T , α ∈ B([0, T )n). (4.10)
We will pass now to the construction of some n-dimensional quasiresolution of iden-
tity E⊗̂n acting in the symmetric tensor product H⊗̂n ⊂ H⊗n. Denote by B̂([0, T )n) ⊂
⊂ B([0, T )n) the σ-algebra spanned by all rectangles ∆1 × . . .×∆n, where ∆1, . . . ,∆n
are disjoint Borel subsets of [0, T ). We put for these rectangles ∆1 × . . .× ∆n
E⊗̂n(∆1×. . .×∆n) = E(∆1)⊗̂ . . . ⊗̂E(∆n) :=
1
n!
∑
σ∈Sn
E⊗n(∆σ(1)×. . .×∆σ(n)) =
=
1
n!
∑
σ∈Sn
E(∆σ(1)) ⊗ . . .⊗ E(∆σ(n)), (4.11)
where Sn is the group of all permutation σ
(
1, . . . , n) = (σ(1), . . . , σ(n)
)
of {1, . . . , n}
(n! values of the index σ). It is possible to prove [20] that this projector-valued function of
rectangles can be extended to some quasiresolution of identity E⊗̂n(α), α ∈ B̂([0, T )n),
in the space H⊗̂n. According to (4.11) it is possible to say that the quasiresolution of
identity E⊗̂n, acting in the space H⊗̂n, is a symmetrization of the quasiresolution of
identity E⊗n.
Introduce the “diagonal” set d ⊂ [0, T )n:
d =
⋃
{j1,j2}⊂
{
1,...,n}
{t ∈ [0, T )n
∣∣ tj1 = tj2
}
. (4.12)
It follows from (4.9) and (4.11) that for a Borel complex-valued symmetric function f(t),
t ∈ [0, T )n, vanishing in some neighborhood of the set d, we have∫
[0,T )n
f(t)dE⊗̂n(t) =
( ∫
[0,T )n
f(t)dE⊗n(t)
)
� H⊗̂n. (4.13)
Above described construction gives the possibility to introduce an abstract symmetric
n-dimensional martingale M̂n similar to (4.10). This martingale will be understood as a
vector-valued measure defined on B̂
(
[0, T )n
)
with values in H⊗̂n:
B̂([0, T )n) � α �→ M̂n(α) := E⊗̂n(α)M⊗n
T ∈ H⊗̂n, (4.14)
where MT ∈ H.
Apply equality (4.13) to M⊗n
T . Using the definition of integral by martingales of type
(4.6) and (4.14), (4.10) we find the following relation for an above appearing symmetric
function f vanishing in some neighborhood of diagonal d (4.12):∫
[0,T )n
f(t)dM̂n(t) =
∫
[0,T )n
f(t)dMn(t). (4.15)
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606 S. ALBEVERIO, YU. M. BEREZANSKY, V. A. TESKO
The integrals ∫
[0,T )n
f(t)dM̂n(t) :=
( ∫
[0,T )n
f(t)dE⊗̂n(t)
)
M⊗n
T (4.16)
will be called multiple spectral integrals with respect to the symmetric n-dimensional
martingale M̂n.
It is possible to apply in case (4.14), (4.16) a construction of the form (4.7), (4.8).
Namely, let U be some unitary operator acting from H⊗̂n onto another Hilbert space K.
Then
B̂([0, T )n) � α �→ Ẑn(α) := UM̂n(α) ∈ K, (4.17)
is an abstract symmetric martingale and for a measurable with respect to B̂
(
[0, T )n
)
func-
tions f vanishing in some neighborhood of diagonal d we have
U
( ∫
[0,T )n
f(t)dM̂n(t)
)
=
∫
[0,T )n
f(t)dẐn(t). (4.18)
4.2. The multiple spectral integral in an n-particle Fock space. The result of Sub-
section 4.1 is concerned with a general quasiresolution of the identity E acting in a com-
plex Hilbert space H. In this subsection we will consider a more special situation when
H is equal to
H = L2([0, T ),m) = H0,C, 0 < T ≤ ∞
(m is the Lebesgue measure), and the quasiresolution of identity in this space has the
form:
B([0, T )) � α �→ E(α)f := καf ∈ L2([0, T ),m), f ∈ L2([0, T ),m), (4.19)
where κα denotes the characteristic function of the set α. In other words, our E is the res-
olution of identity of the operator of multiplication by t in the space H0,C. Construct ac-
cording to (4.9) and (4.11) the corresponding resolutions of identity E⊗n and E⊗̂n. They
act in the spaces H⊗n = L2
(
[0, T )n,m⊗n
)
and H⊗̂n = Fn(H0) = L̂2
(
[0, T )n,m⊗n
)
respectively.
We will prove an essential formula which represents a function from the space Fn(H0)
as an action of the spectral integral with respect to E⊗̂n on a certain function from
Fn(H0). Namely, let some positive essentially bounded function MT ∈ L2
(
[0, T ),m
)
be fixed. Construct the martingale M̂n by formulas (4.14) and (4.11) from the one-
dimensional resolution of identity (4.19) and MT .
Lemma 4.1. For an arbitrary symmetric function fn ∈ Fn(H0) = L̂2
(
[0, T )n,
m⊗n
)
the following representation is valid
fn(τ) =
1
M⊗n
T (τ)
( ∫
[0,T )n
fn(t)dM̂n(t)
)
(τ) (4.20)
for m⊗n-almost all τ ∈ [0, T )n. Here the integral is the multiple spectral integral with
respect to the symmetric n-dimensional martingale M̂n.
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Proof. From (4.16) and (4.13) we conclude that∫
[0,T )n
fn(t)dM̂n(t) =
( ∫
[0,T )n
f(t)dE⊗̂n(t)
)
M⊗n
T =
( ∫
[0,T )n
fn(t)dE⊗n(t)
)
M⊗n
T
(4.21)
for any fn ∈ Fn(H0) = L̂2
(
[0, T )n,m⊗n
)
, additionally equal to zero in some neigh-
borhood of d. But the Lebesgue measure m is non atomic, therefore the latter func-
tions are dense in the whole space Fn(H0). Then equality (4.21) is valid for arbitrary
fn ∈ Fn(H0).
The operator-valued function E(t) is the resolution of identity of the operator of
multiplication by t in the space L2
(
[0, T ),m
)
, therefore the spectral integral∫
[0,T )n
fn(t)dE⊗n(t) is the operator of multiplication by the function fn ∈ Fn(H0) =
= L̂2
(
[0, T )n,m⊗n
)
and the right-hand side in (4.21) is equal to fn(τ)M⊗n
T (τ). This
gives (4.20).
The lemma is proved.
Remark 4.1. Assume that T ∈ (0,∞), then m([0, T )) < ∞ and we can put MT =
= 1. In this case formula (4.20) have a simpler view: for m⊗n-almost all τ ∈ [0, T )n
fn(τ) =
∫
[0,T )n
fn(t)dM̂n(t)
(τ). (4.22)
5. The connection of the extended stochastic integral with the classical Itô in-
tegral. Multiple Itô integral and its spectral representation. In Section 3 we have
defined extended stochastic integral Sext = S (3.6) for vector-valued function ξ(t) with
values in the Fock space F (H0). One can easily “rewrite” this integral in form Sext,h
(3.15), when the values of such function ξ(t) belong to (L2
Q). We remind that H0 =
= L2
Re([0, T ),m),
(
Q,B(Q), ρ
)
is the probability space and (L2
Q) = L2(Q,B(Q), ρ)
is the Ih-image of the Fock space F (H0), where Ih : F (H0) → (L2
Q) is unitary opera-
tor (2.32).
In this section we find a condition on h(x, λ) (2.27) under which the extended stochas-
tic integral Sext,h is equal to an ordinary Itô integral constructed by a certain normal mar-
tingale. Moreover, we obtain conditions of coincidence of the multiple spectral integral
with a multiple Itô integral.
5.1. Preliminaries. We will apply the results of Subsection 4.1. Namely, let T ∈
∈ (0,∞), H be equal to F (H0). The quasiresolution of identity E(t) in this space has
the form
[0, T ) ∈ t �→ E(t)f =
(
f0,κ(0,t]f1, . . . ,κ(0,t]nfn, . . .
)
∈ F (H0),
f = (fn)∞n=0 ∈ F (H0),
where κα, as usual, denotes the characteristic function of the set α; κ(0,0]n := 0, n ∈ N1.
Let MT = (0, 1, 0, 0, . . .) be a fixed vector from H = F (H0). Then
M(t) := E(t)MT = (0,κ(0,t], 0, 0, . . .), t ∈ [0, T ), (5.1)
is an abstract martingale in the Fock space F (H0).
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608 S. ALBEVERIO, YU. M. BEREZANSKY, V. A. TESKO
If we apply to (5.1) unitary operator (2.32) U = Ih, which transfers the Fock space
F (H0) onto (L2
Q), we get as a result (according to (4.7)) the abstract martingale
Z(t) = Z(t, x) = IhM(t) =
〈
κ(0,t], h1(x)
〉
, t ∈ [0, T ),
in the space (L2
Q).
So, we have constructed the required martingale Z(t). Using orthogonality relation
(2.30) for hn and (2.32) it is easy to check that for 0 ≤ s < t < T∥∥Z(t) − Z(s)
∥∥2
(L2
Q)
=
∥∥ 〈
κ(s,t], h1
〉 ∥∥2
(L2
Q)
=
∥∥κ(s,t]
∥∥2
L2([0,T ),m)
= t− s. (5.2)
In addition, the condition h0(x) = 1, x ∈ Q, is fulfilled (see (2.27)). Therefore, in
accordance with (2.30) for all t ∈ [0, T )∫
Q
Z(t, x)dρ(x) =
∫
Q
〈
κ(0,t], h1(x)
〉
dρ(x) =
=
∫
Q
〈
κ(0,t], h1(x)
〉 〈
κ(0,t], h0(x)
〉
dρ(x) = 0. (5.3)
Let (At)t∈[0,T ) be the flow of σ-algebras At generated by the process
{
Z(t) |t ∈
∈ [0, T )
}
, i.e., for every t ∈ [0, T ) At is the σ-algebra on Q generated by the sets{
x ∈ Q
∣∣Z(s, x) ∈ α
}
, α ∈ B(C), 0 ≤ s ≤ t. This flow is right continuous because
E(t) has such a property. We assume that A0 is complete with respect to the measure ρ
and B(Q) coincides with the smallest σ-algebra generated by
⋃
t∈[0,T ) At.
In the sequel, we will assume that the process
{
Z(t)
∣∣ t ∈ [0, T )]
}
is a normal mar-
tingale with respect to the flow of σ-algebras At, i.e., that
{
Z(t)
∣∣ t ∈ [0, T )]
}
and{
Z2(t) − t
∣∣ t ∈ [0, T )
}
are martingales with respect to (At)t∈[0,T ).
Note that if Z has independent increments then Z is a normal martingale. This follows
from the properties (5.2), (5.3) and the property of Z having independent increments.
5.2. The classical Itô integral with respect to the normal martingale Z. Multiple
Itô integrals. Let T ∈ (0,∞) be fixed. We denote by DI the set of B([0, T )) × B(Q)-
measurable functions
[0, T ) ×Q � {t, x} �→ f(t, x) ∈ C, (5.4)
which are At-adapted and belong to the space L2
(
[0, T ); (L2
Q)
)
. We recall that func-
tion (5.4) is At-adapted if for each t ∈ [0, T ) the function
Q � x �→ f(t, x) ∈ C
is At-measurable.
We note that in terms of the resolution of identity function (5.4) is At-adapted if
f(t) = E(t)f(t) for each t ∈ [0, T ), where E(t) is the resolution of identity generated by
the σ-algebra At (recalling that E(t) is the projector in the space (L2
Q) onto its subspace
consisting of all functions from (L2
Q), which are measurable with respect to At).
The Itô integral of the integrand f(t) = f(t, x)
SI(f) =
T∫
0
f(t)dZ(t) (5.5)
with respect to the normal martingale Z(t) is defined as the unique linear isometric map-
ping
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A GENERALIZATION OF AN EXTENDED STOCHASTIC INTEGRAL 609
L2
(
[0, T ); (L2
Q)
)
⊃ DI � f �→ SI(f) ∈ (L2
Q) (5.6)
such that
SI(gκ(s,t]) = g
(
Z(t) − Z(s)
)
= g
〈
κ(s,t], h1
〉
, 0 ≤ s < t < T, (5.7)
for any As-measurable function g ∈ (L2
Q).
We stress that the isometry of the mapping (5.6) means that the following equality
holds: ∥∥∥∥∥∥
T∫
0
f(t)dZ(t)
∥∥∥∥∥∥
2
(L2
Q)
=
T∫
0
∥∥f(t)
∥∥2
(L2
Q)
dt, f ∈ DI. (5.8)
For a proof of existence and the properties of such an Itô integral SI(f) we refer to the
books [50 – 53, 19].
Let us recall some results concerning the definition and properties of the multiple Itô
integrals for the integrands that are complex-valued symmetric functions. Namely, let
fn ∈ Fn(H0) = L̂2
(
[0, T )n,m⊗n
)
, n ∈ N1. For such a function fn we can form the
iterated Itô integral
T∫
0
tn∫
0
. . .
t2∫
0
fn(t1, . . . , tn)dZ(t1)
. . . dZ(tn−1)dZ(tn) (5.9)
because at each Itô integration with respect to dZ(ti) the integrand is At-adapted and
square integrable with respect to dρ(x) × dm(ti), i ∈ {1, . . . , n}. Moreover, applying n
times equality (5.8) we obtain∥∥∥∥∥∥
T∫
0
tn∫
0
. . .
t2∫
0
fn(t1, . . . , tn)dZ(t1) . . . dZ(tn−1)
dZ(tn)
∥∥∥∥∥∥
2
(L2
Q)
=
=
T∫
0
∥∥∥∥∥∥
tn∫
0
. . .
t2∫
0
fn(t1, . . . , tn)dZ(t1) . . . dZ(tn−1)
∥∥∥∥∥∥
2
(L2
Q)
dtn = . . .
. . . =
T∫
0
tn∫
0
. . .
t2∫
0
∣∣fn(t1, . . . , tn)
∣∣2dt1 . . . dtn−1dtn =
1
n!
‖fn‖2
Fn(H0)
.
Hence, the mapping
Fn(H0) � fn �→ n!
T∫
0
tn∫
0
. . .
t2∫
0
fn(t1, . . . , tn)dZ(t1) . . .
. . . dZ(tn−1)dZ(tn) ∈ (L2
Q), n ∈ N1,
is linear and continuous.
For a function fn ∈ Fn(H0), n ∈ N1, we define an Itô multiple stochastic integral
Sn(fn) by
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610 S. ALBEVERIO, YU. M. BEREZANSKY, V. A. TESKO
Sn(fn) := n!
T∫
0
tn∫
0
. . .
t2∫
0
fn(t1, . . . , tn)dZ(t1)
. . . dZ(tn−1)dZ(tn). (5.10)
The properties of iterated Itô integrals (5.9) give the corresponding properties of the mul-
tiple stochastic integrals Sn(fn). For more details on the construction and properties of
the integrals Sn(fn) see [54, 19], in the Gaussian and Poissonian cases see [31, 32, 55,
56]. We note that in the special case
fn = κ∆1⊗̂ . . . ⊗̂κ∆n
∈ Fn(H0), n ∈ N1,
where ∆j = (aj , bj ] ⊂ [0, T ), j ∈ {1, . . . , n}, are disjoint, it is possible to calculate the
integrals Sn(fn) and get:
Sn(κ∆1⊗̂ . . . ⊗̂κ∆n) = 〈κ∆1 , h1〉 . . . 〈κ∆n , h1〉 . (5.11)
5.3. When the image of a multiple spectral integral is an Itô multiple stochastic in-
tegral. We will now continue the investigations of Subsection 4.3 concerning the prop-
erties of multiple spectral integrals.
At first, we recall the results of Subsection 4.3. Let H = H0,C = L2
(
[0, T ),m
)
,
T < ∞. Using resolution of identity E (4.19) of the operator of multiplication by t in
the space L2
(
[0, T ),m
)
and the function MT = 1 we construct by (4.14), (4.11) the
martingale M̂n with values in H⊗̂n = Fn(H0) = L̂2
(
[0, T )n,m⊗n
)
. According to
Remark 4.1 for an arbitrary function fn ∈ Fn(H0) the following representation holds:
fn =
∫
[0,T )n
fn(t)dM̂n(t). (5.12)
Apply operator Ih (2.32) to equality (5.12) (we consider fn as a vector (0, . . . , 0, fn,
0, 0, . . .) from F (H0)). Using formulas (2.32) and (4.18) (with U = Ih) we get
〈fn, hn(x)〉 =
∫
[0,T )n
fn(t)dẐn(t, x), (5.13)
in the space (L2
Q), where
Ẑn(α) = IhM̂n(α), α ∈ B̂([0, T )n),
is a symmetric martingale.
Our aim is to find conditions on h(x, λ) under which image (5.13) of multiple spectral
integral (5.12) coincides with Itô multiple stochastic integral (5.10).
Theorem 5.1. For an arbitrary function fn ∈ Fn(H0), n ∈ N1, the equality
Sn(fn) =
∫
[0,T )n
fn(t)dẐn(t) (5.14)
holds if and only if
Ih(κ∆1⊗̂ . . . ⊗̂κ∆n) = Ih(κ∆1) . . . Ih(κ∆n) (5.15)
for all disjoint intervals ∆j = (aj , bj ], j ∈ {1, . . . , n}, from [0, T ).
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A GENERALIZATION OF AN EXTENDED STOCHASTIC INTEGRAL 611
Proof. Let us assume that equality (5.15) take place, i.e.,〈
κ∆1⊗̂ . . . ⊗̂κ∆n , hn
〉
= 〈κ∆1 , h1〉 . . . 〈κ∆n , h1〉 (5.16)
for all disjoint interval ∆j = (aj , bj ], j ∈ {1, . . . , n}, from [0, T ). Using (5.16), (5.13)
and (5.11) we conclude:
Sn(κ∆1⊗̂ . . . ⊗̂κ∆n
) =
∫
[0,T )n
(
κ∆1⊗̂ . . . ⊗̂κ∆n
)
(t)dẐn(t). (5.17)
Since the vectors κ∆1⊗̂ . . . ⊗̂κ∆n
form a total set in Fn(H0) we obtain (5.14) from (5.17).
Conversely, let (5.14) takes place. Then from (5.13), (2.32) and (5.11) we obtain (5.15).
The theorem is proved.
We have the following statement.
Theorem 5.2. If for all x ∈ Q and for any ϕ1, . . . , ϕn ∈ Φ such that suppϕi ∩
∩ suppϕj = ∅ if j �= i, i, j ∈ {1, . . . , n},
∂nh(x, z1ϕ1 + . . .+ znϕn)
∂z1 . . . ∂zn
∣∣∣∣
z1=...=zn=0
=
=
∂
∂z1
h(x, z1ϕ1)
∣∣∣∣
z1=0
. . .
∂
∂zn
h(x, znϕn)
∣∣∣∣
zn=0
(5.18)
then for all disjoint ∆j = (aj , bj ] ⊂ [0, T ), j ∈ {1, . . . , n},
Ih(κ∆1⊗̂ . . . ⊗̂κ∆n
) = Ih(κ∆1) . . . Ih(κ∆n
), n ∈ N1.
Proof. Let us assume that (5.18) is fulfilled. Then using (2.34) and (5.18) we obtain〈
ϕ1⊗̂ . . . ⊗̂ϕn, hn(x)
〉
= 〈ϕ1, h1(x)〉 . . . 〈ϕn, h1(x)〉 , n ∈ N1, (5.19)
for all x ∈ Q and any ϕ1, . . . , ϕn ∈ Φ (under the conditions of the theorem).
It is well known that for all disjoint ∆1, . . . ,∆n ⊂ [0, T ) there exist ϕj,ε ∈ Φ, ε > 0,
such that suppϕj,ε ⊂ ∆j and ϕj,ε → κ∆j
in H0,C = L2([0, T ),m) as ε → 0. So, using
(5.19) we have
Ih(κ∆1⊗̂ . . . ⊗̂κ∆n
) = lim
ε→0
Ih(ϕ1,ε⊗̂ . . . ⊗̂ϕn,ε) =
= lim
ε→0
Ih(ϕ1,ε) . . . Ih(ϕn,ε) = Ih(κ∆1) . . . Ih(κ∆n
).
5.4. The coincidence of the extended stochastic integral with the Itô integral for
adapted processes. In the classical Gaussian and Poissonian analysis the extended sto-
chastic integral is a generalization of the Itô integral: they are equal for adapted processes.
Therefore, there is a natural question about conditions on the unitary map Ih : F (H0) →
→ (L2
Q) such that Sext,h = SI. As an answer we have the following statement.
Theorem 5.3. If the unitary map Ih : F (H0) → (L2
Q) is such that
Ih(κ∆1⊗̂ . . . ⊗̂κ∆n) = Ih(κ∆1) . . . Ih(κ∆n), i.e.,〈
κ∆1⊗̂ . . . ⊗̂κ∆n , hn
〉
= 〈κ∆1 , h1〉 . . . 〈κ∆n , h1〉 , n ∈ N1,
(5.20)
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612 S. ALBEVERIO, YU. M. BEREZANSKY, V. A. TESKO
for all disjoint intervals ∆j = (aj , bj ], j ∈ {1, . . . , n}, from [0, T ) then DI = Dh (the set
Dh is defined by (3.16)). In this case for the extended integral Sext,h, defined by (3.15),
we have
Sext,h(f) = SI(f), f ∈ Dh = DI. (5.21)
Conversely, if Dh ⊂ DI and (5.21) takes place then (5.20) is fulfilled and Dh = DI.
Proof. Let (5.20) takes place. In order to prove the equality Dh = DI, it is sufficient
to check that
E
{
〈fn, hn〉 |At
}
=
〈
κ(0,t]nfn, hn
〉
, t ∈ [0, T ), (5.22)
for arbitrary fn ∈ Fn(H0), n ∈ N1. Here E{f |At} denotes the conditional expectation
of a random variable f with respect to the σ-algebras At (note that E{ · |At} is the pro-
jector in the space (L2
Q) onto its subspace consisting of all functions from (L2
Q) which
are measurable with respect to At).
Since the functions
fn = κ∆1⊗̂ . . . ⊗̂κ∆n , ∆i ∩ ∆j = ∅, i �= j,
form a total set in Fn(H0), it is sufficient to check (5.22) for such functions.
Using (5.20), the equality
E
{ 〈
κ(0,s], h1
〉
| At
}
=
〈
κ(0,t], h1
〉
, 0 < t ≤ s < T
(recall that Z(t) =
〈
κ(0,t], h1
〉
, t ∈ [0, T ) is a martingale) and the properties of the
conditional expectation, we get
E{fn|At} = E
{〈
κ∆1⊗̂ . . . ⊗̂κ∆n
, h1
〉
| At
}
=
= E
{
〈κ∆1 , h1〉 . . . 〈κ∆n
, h1〉 |At
}
=
= E
n∏
j=1
( 〈
κ∆j∩(0,t], h1
〉
+
〈
κ∆j∩(t,T ), h1
〉 )∣∣At
=
=
〈
κ∆1∩(0,t], h1
〉
. . .
〈
κ∆n∩(0,t], h1
〉
=
〈
κ∆1∩(0,t]⊗̂ . . . ⊗̂κ∆n∩(0,t], hn
〉
=
=
〈
κ(0,t]n(κ∆1⊗̂ . . . ⊗̂κ∆n), hn
〉
=
〈
κ(0,t]nfn, hn
〉
.
So, the necessary equality Dh = DI is proved.
Let us prove (5.21). The mappings
L2
(
[0, T ); (L2
Q)
)
⊃ DI � f �→ SI(f) ∈ (L2
Q),
L2
(
[0, T ); (L2
Q)
)
⊃ Dh � f �→ Sext,h(f) ∈ (L2
Q)
are linear and continuous. Therefore, it is sufficient to show that (5.21) takes place for the
functions
fn(·) =
〈
κ∆1⊗̂ . . . ⊗̂κ∆n , hn
〉
κ∆(·) ∈ Dh = DI, n ∈ N1, (5.23)
where ∆j = (aj , bj ] ⊂ [0, T ), j ∈ {1, . . . , n}, are disjoint and ∆ = (a, b] ⊂ [0, T ),
a > bj , j ∈ {1, . . . , n} (we note that functions (5.23) form a total set in Dh = DI).
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A GENERALIZATION OF AN EXTENDED STOCHASTIC INTEGRAL 613
For such ∆j the function〈
κ∆1⊗̂ . . . ⊗̂κ∆n , hn
〉
= 〈κ∆1 , h1〉 . . . 〈κ∆n , h1〉
is Aa-measurable because each functions
〈
κ∆j
, hj
〉
, j ∈ {1, . . . , n}, is Aa-measurable.
Therefore, according to (5.7) and (5.20), for function (5.23) we get
SI(f) =
T∫
0
〈fn(t), hn〉 dZ(t) =
T∫
0
〈
κ∆1⊗̂ . . . ⊗̂κ∆n
, hn
〉
κ∆(t)dZ(t) =
=
〈
κ∆1⊗̂ . . . ⊗̂κ∆n , hn
〉
〈κ∆, h1〉 =
〈
κ∆1⊗̂ . . . ⊗̂κ∆n⊗̂κ∆, hn+1
〉
= Sext,h(f).
The first part of the theorem is proved.
For the proof of its second part consider the function
[0, T ) � t �→
〈
κ∆1⊗̂ . . . ⊗̂κ∆n−1 , hn−1
〉
κ∆n(t) ∈ (L2
Q),
where ∆j = (aj , bj ], j ∈ {1, . . . , n}, are disjoint and an > max{a1, . . . , an−1}. Evi-
dently, these functions belong to the set Dh ⊂ DI because condition (3.9) is fulfilled: if
t ∈ [0, T ) \ ∆n, then the function
fn−1(t) := (κ∆1⊗̂ . . . ⊗̂κ∆n−1)κ∆n
(t) ∈ Fn−1(H0)
is equal to zero; if t ∈ ∆n, then the multiplication by fn−1(t) on κ(0,t]n−1 does not
change this function (∆1, . . . ,∆n−1 ⊂ (0, t]). Therefore, according to (3.15) we have
Sext,h(
〈
κ∆1⊗̂ . . . ⊗̂κ∆n−1 , hn−1
〉
κ∆n
(·)) =
〈
κ∆1⊗̂ . . . ⊗̂κ∆n
, hn
〉
. (5.24)
On the other hand the function〈
κ∆1⊗̂ . . . ⊗̂κ∆n−1 , hn−1(·)
〉
is Aan -measurable, therefore using (5.7) we obtain
SI(
〈
κ∆1⊗̂ . . . ⊗̂κ∆n−1 , hn−1
〉
κ∆n(t)) =
〈
κ∆1⊗̂ . . . ⊗̂κ∆n−1 , hn−1
〉
〈κ∆n , h1〉 .
(5.25)
From (5.21), (5.25) and (5.24) we conclude that〈
κ∆1⊗̂ . . . ⊗̂κ∆n−1 , hn−1
〉
〈κ∆n , h1〉 =
〈
κ∆1⊗̂ . . . ⊗̂κ∆n , hn
〉
, n ∈ N2, (5.26)
in the space (L2
Q). Taking in (5.26) n = 2, 3, . . . we get step by step equality (5.20).
The theorem is proved.
6. Classical examples. 1. Gaussian white noise analysis. Let Q = H−1 =
= W 2
−1
(
R+, (1 + t2)1dt
)
, R+ = [0,∞), ρ = γ be the Gaussian measure, which is
completely characterized by its Fourier transform∫
H−1
exp(i 〈x, λ〉)dγ(x) = exp
(
−1
2
〈λ, λ〉
)
, λ ∈ H1 = W 2
1
(
R+, (1 + t2)1dt
)
.
The function
h(x, λ) := exp
(
〈x, λ〉 − 1
2
〈λ, λ〉
)
=
∞∑
n=0
1
n!
〈
λ⊗n, hn(x)
〉
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614 S. ALBEVERIO, YU. M. BEREZANSKY, V. A. TESKO
is the generating function for the Hermite polynomials hn(x). In this case, the unitary
mapping (2.32) is the classical Winer – Itô – Segal isomorphism.
One can verify that the function h(x, λ) satisfies (5.18) and the process{
B(t) = B(t, ·) :=
〈
κ(0,t], h1(·)
〉
=
〈
κ(0,t], ·
〉 ∣∣ t ∈ R+
}
,
is a normal martingale with respect to the flow (At)t∈R+ of σ-algebras At generated
by the set
{
x ∈ Q | B(s, x) ∈ α
}
, α ∈ B(R), 0 ≤ s ≤ t. Hence it follows from
Theorems 5.3 and 5.2 that the Itô integral coincides with the corresponding extended
stochastic integral
SI(f) = Sext,h(f) :=
∞∑
n=1
〈
f̂n, hn
〉
for f(·) =
∑∞
n=0
〈fn(·), hn〉 ∈ DI.
2. Poissonian white noise analysis. Let Q = H−1 = W 2
−1
(
R+, (1 + t2)1dt
)
, ρ = π
be the centered Poisson measure with intensity dt, which is completely characterized by
its Fourier transform∫
H−1
exp(i 〈x, λ〉)dπ(x) = exp
〈
1, eiλ − 1 − iλ
〉
, λ ∈ H1.
In this case the function h(x, λ) has the form
h(x, λ) := exp
(
〈x+ 1, log(1 + λ)〉 − 〈1, λ〉
)
=
∞∑
n=0
1
n!
〈
λ⊗n, hn(x)
〉
,
where hn(x) are the Charlier polynomials. It is known that the function h(x, λ) satisfies
all assumption formulated in Subsection 2.4. Therefore we have the statement that the Itô
integral with respect to the process{
C(t) = C(t, ·) :=
〈
κ(0,t], h1(·)
〉
=
〈
κ(0,t], ·
〉 ∣∣ t ∈ R+
}
,
coincide with the corresponding extended stochastic integral.
3. Let us fix a function
R+ � t �→ θ(t) ∈ C
such that
|θ(t)| ≤ c, t ∈ R+,
for some constant c > 0 and
‖θϕ‖Hp ≤ cp‖ϕ‖Hp , ϕ ∈ Hp := W 2
p
(
R+, (1 + t2)pdt
)
,
for some constant cp > 0, p ∈ N1.
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A GENERALIZATION OF AN EXTENDED STOCHASTIC INTEGRAL 615
We consider the probability measure µθ on B(H−1) given by the Fourier transform
(see, e.g., [15])
∫
S−2
exp(i 〈x, λ〉)dµθ(x) = exp
∫
R+
( ∞∑
n=2
(iλ(t))nθn−2(t)
n!
)
dt
=
= exp
〈
1, (eiλθ − 1 − iλθ)θ−2
〉
,
where
(eiλθ − 1 − iλθ)θ−2 :=
∞∑
n=2
(iλ)nθn−2
n!
∈ H1.
For
θ(t) = 0, t ∈ R+,
µ0 is the standard Gaussian measure, for
θ(t) = 1, t ∈ R+,
µ1 is the centered Poissonian measure.
Let us put Q = H−1, ρ = µθ. It follows from results of [15] that the function
hθ(x, λ) := exp
(〈
x, θ−1 log(1 + θλ)
〉
+
〈
1, θ−2 log(1 + θλ) − θ−1λ
〉)
=
=
∞∑
n=0
1
n!
〈
λ⊗n, hθ
n(x)
〉
,
θ−1 log(1 + θλ) :=
∞∑
n=1
(−1)n−1θn−1λn
n
∈ H−1 = H−,
satisfies all assumptions of Subsection 2.4 required for a function h. Therefore the map-
ping
F (H0) � f = (fn)∞n=0 �→ (Ihθf)(·) :=
∞∑
n=0
〈
fn, h
θ
n(·)
〉
∈ (L2
H−)
is well-defined and unitary. Under this mapping rigging (2.5) of the Fock space F (H0)
transforms into a rigging of the corresponding (L2
H−
). Note that for θ = 0 the mapping
Ihθ is the classical Wiener – Itô – Segal isomorphism. It follows from results of [15] that
the function hθ satisfies (5.18) and the process{
hθ(t) = hθ(t, ·) :=
〈
κ(0,t], h
θ
1(·)
〉 ∣∣ t ∈ R+
}
,
is a normal martingale with respect to the flow (At)t∈R+ of σ-algebras At generated by
hθ(t). Hence it follows from Theorems 5.3 and 5.2 that the Itô integral SI(f), f(·) =
=
∑∞
n=0
〈
fn(·), hθ
n
〉
∈ DI, coincides with the corresponding extended stochastic inte-
gral
Shθ (f) :=
∞∑
n=1
〈
f̂n, h
θ
n
〉
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
616 S. ALBEVERIO, YU. M. BEREZANSKY, V. A. TESKO
Acknowledgments. The authors are very grateful to Dr. M. O. Kachanovsky for his
remarks and useful discussions of the results of the paper.
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Received 10.01.2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
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| spelling | umjimathkievua-article-33342020-03-18T19:51:39Z A generalization of an extended stochastic integral Узагальнення розширеного стохастичного інтеграла Berezansky, Yu. M. Tesko, V. A. Березанський, Ю. М. Теско, В. А. We propose a generalization of an extended stochastic integral to the case of integration with respect to a broad class of random processes. In particular, we obtain conditions for the coincidence of the considered integral with the classical Itô stochastic integral. Запропоновано узагальнення розширеного стохастичного інтеграла на випадок інтегрування відносно широкого класу випадкових процесів. Зокрема, одержано умови, за яких вказаний інтеграл збігається з класичним стохастичним інтегралом Іто. Institute of Mathematics, NAS of Ukraine 2007-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3334 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 5 (2007); 588–617 Український математичний журнал; Том 59 № 5 (2007); 588–617 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3334/3412 https://umj.imath.kiev.ua/index.php/umj/article/view/3334/3413 Copyright (c) 2007 Berezansky Yu. M.; Tesko V. A. |
| spellingShingle | Berezansky, Yu. M. Tesko, V. A. Березанський, Ю. М. Теско, В. А. A generalization of an extended stochastic integral |
| title | A generalization of an extended stochastic integral |
| title_alt | Узагальнення розширеного стохастичного інтеграла |
| title_full | A generalization of an extended stochastic integral |
| title_fullStr | A generalization of an extended stochastic integral |
| title_full_unstemmed | A generalization of an extended stochastic integral |
| title_short | A generalization of an extended stochastic integral |
| title_sort | generalization of an extended stochastic integral |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3334 |
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