Generalized stochastic derivatives on spaces of nonregular generalized functions of Meixner white noise
We introduce and study generalized stochastic derivatives on Kondratiev-type spaces of nonregular generalized functions of Meixner white noise. Properties of these derivatives are quite analogous to properties of stochastic derivatives in the Gaussian analysis. As an example, we calculate the genera...
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| author | Kachanovskii, N. A. Качановський, М. О. |
| author_facet | Kachanovskii, N. A. Качановський, М. О. |
| author_sort | Kachanovskii, N. A. |
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| description | We introduce and study generalized stochastic derivatives on Kondratiev-type spaces of nonregular generalized functions of Meixner white noise. Properties of these derivatives are quite analogous to properties of stochastic derivatives in the Gaussian analysis. As an example, we calculate the generalized stochastic derivative of a solution of a stochastic equation with Wick-type nonlinearity. |
| first_indexed | 2026-03-24T02:37:58Z |
| format | Article |
| fulltext |
UDC 517.9
N. A. Kachanovsky (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
GENERALIZED STOCHASTIC DERIVATIVES ON SPACES
OF NONREGULAR GENERALIZED FUNCTIONS
OF MEIXNER WHITE NOISE
UZAHAL|NENI STOXASTYÇNI POXIDNI
NA POV’QZANYX IZ BILYM ÍUMOM MAJKSNERA
PROSTORAX NEREHULQRNYX UZAHAL|NENYX FUNKCIJ
We introduce and study generalized stochastic derivatives on the Kondratiev-type spaces of nonregular
generalized functions of Meixner white noise. Properties of these derivatives are quite analogous to the
properties of the stochastic derivatives in the Gaussian analysis. As an example we calculate the
generalized stochastic derivative of the solution of some stochastic equation with a Wick-type
nonlinearity.
Vvodqt\sq ta vyvçagt\sq uzahal\neni stoxastyçni poxidni na pov’qzanyx iz bilym ßumom Majk-
snera prostorax typu Kondrat\[va nerehulqrnyx uzahal\nenyx funkcij. Vlastyvosti cyx
poxidnyx analohiçni vlastyvostqm stoxastyçnyx poxidnyx u haussivs\komu analizi. Qk pryklad
obçysleno uzahal\nenu stoxastyçnu poxidnu rozv’qzku pevnoho stoxastyçnoho rivnqnnq z neli-
nijnistg typu Vika.
Introduction. Let S ′ be the Schwartz distributions space, µ be the Gaussian mea-
sure on S ′. As is well known, every square integrable function f L∈ ′2( ),S µ can be
presented in the form
f = H fn
n
n
, ( )
=
∞
∑
0
, (0.1)
where H fn
n
n
, ( ){ } =
∞
0
are the generalized Hermite polynomials, f
n n( ) ˆ
∈H � , H (in
the simplest case) is the complexification of L2( )R , �̂ denotes a symmetric tensor
product. A stochastic derivative D S L H S: , , ,( ) ( ( ))L L2 2′ → ′µ µ can be defined on
its domain
f L n n f n
n
n∈ ′ < ∞
=
∞
∑2 2
1
( ), : ! ( )S
H
µ �
by the formula
( )( )( )D f g 1 : = n H f gn
n
n
−
=
∞
∑ 1
1
1
, ,( ) ( ) ∀ ∈g( )1 H ,
where
f gn n( ) ( ) ˆ
, 1 1∈ −H � is defined by
f g hn n( ) ( ) ( ), ,1 1− : =
f g hn n( ) ( ) ( ), ˆ1 1� − ∀ ∈− −h n n( ) ˆ1 1H �
( here 〈⋅ ⋅〉, denotes the scalar product in H �̂n
) .
In the paper [1] Fred E. Benth extended the derivative D on the Kondratiev space
of nonregular generalized functions ( )′ −S 1 ( elements of ( )′ −S 1 can be presented in
the similar to (0.1) form, but the kernels { } =
∞f n
n
( )
0 are singular ) . This generalization
is useful for different applications. For example, as distinct from L2( ),′S µ in the
space ( )S −1 one can introduce the Wick product ◊ by setting for the Hermite
polynomials
© N. A. KACHANOVSKY, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6 737
738 N. A. KACHANOVSKY
H f H gn
n
m
m, ,( ) ( )◊ : =
H f gn m
n m
+ , ˆ( ) ( )� ,
and D is a differentiation with respect to ◊ : for all F, G ∈ ( )S −1 D ( )F G◊ =
= ( )D F G◊ + F G◊ ( )D . Using this result (and another properties of D ) one can
study properties of solutions of stochastic equations with Wick type nonlinearity. Ano-
ther possible applications are connected with the fact that the stochastic derivative is
the adjoint operator to the extended (Skorokhod) stochastic integral.
In the papers [2, 3] the author generalized the results of [1] to the spaces of genera-
lized functions of the so-called Gamma white noise analysis (i.e., instead of the Gaussi-
an measure the introduced in [4] Gamma measure γ on S ′ was used). Since the
Gamma measure has no the so-called Chaotic Representation Property (i.e., a function
f L∈ ′2( ),S γ can not be presented in complete analogy with (0.1), generally speaking)
and has some another peculiarities, the corresponding spaces have a more complicated
than in the Gaussian analysis structure: nevertheless a natural and rich in content ana-
log of the Gaussian theory is possible.
A next natural step consists in the construction of a theory of stochastic diffe-
rentiation on the generalized functions spaces of the so-called Meixner analysis. In
fact, the (introduced in [5]) generalized Meixner measure µ on the Schwartz distribu-
tions space ′D (the base measure of the Meixner analysis) is a direct generalization of
“classical” measures on ′D , such as the Gaussian, Poisson and Gamma measures.
This measure is very general, but still has some “classical” properties (for example, the
orthogonal polynomials in L D2( ),′ µ are Schefer (generalized Appell in another
terminology) ones), therefore a consrtuctive theory is still possible.
In this paper we construct and study generalized stochastic derivatives on the
Kondratiev-type (finite order) spaces of nonregular generalized functions of Meixner
white noise. These spaces (cf. [6]) are less wide (and therefore more convenient for
applications) than the “classical” Kondratiev spaces of generalized functions (e.g., [7 –
10]), but have all necessary for our considerations properties of Kondratiev spaces (in
particular, one can consider the Wick calculus and stochastic equations with a Wick
type nonlinearity). Generalized stochastic derivatives on the Kondratiev-type space of
regular generalized functions will be considered in the forthcoming paper.
The paper is organized in the following manner. In the first section we give a ne-
cessary information about the generalized Meixner measure, spaces of test and genera-
lized functions, the stochastic integration and the Wick calculus. The second section is
devoted to generalized stochastic derivatives on the spaces of nonregular generalized
functions.
1. Preliminaries. Let σ be a measure on ( ), ( )R R+ +B (here B denotes the
Borel σ -algebra) satisfying the following assumptions:
1) σ is absolutely continuous with respect to the Lebesgue measure and the density
is an infinite differentiable function on R+ ;
2) σ is a nondegenerate measure, i.e., for each nonempty open set O ⊂ +R
σ ( )O > 0.
Remark 1.1. Note that these assumptions are the “simplest sufficient ones” for
our considerations; actually it is possible to consider a much more general σ.
By D denote 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
of a nuclear space (by analogy with, e.g., [11]): D = pr limτ τ∈T H , where T is the
set of all pairs τ = ( , )τ τ1 2 , τ1 ∈N , τ2 is an infinite differentiable function on R+
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
GENERALIZED STOCHASTIC DERIVATIVES … 739
such that τ2 1( )t ≥ ∀ ∈ +t R ; H τ =
H ( , )τ τ1 2
is the Sobolev space on R+ of order
τ1 weighted by the function τ2, i.e., the scalar product in H τ is given by the
formula
( , )f g τ : =
( , )f g H τ
= f t g t f t g t t dtk k
k
( ) ( ) ( ) ( ) ( ) ( )( ) ( )+
=
∑∫
+ 1
2
1τ
τ σ
R
.
Hence in what follows, we understand D as the corresponding topological space.
Let
H τ,C : = H Hτ τ� i be the complexification of H τ (here and below by the
subindex C denote complexifications of spaces). By ⋅ τ denote the corresponding to
the scalar product ( , )⋅ ⋅ τ norm in H τ,C , i.e., f τ
2 = ( , )f f τ .
Let us consider the (nuclear) chain (the rigging of L2( ),R+ σ — the space of
square integrable with respect to σ real-valued functions on R+ )
′D =
ind lim
′∈
− ′
τ
τ
T
H ⊃ H −τ ⊃ L2( ),R+ σ = : H ⊃
⊃ H τ ⊃
pr lim ′∈ ′τ τT H = D, (1.1)
where H −τ , D ′ are the dual to H τ , D with respect to H spaces correspondingly.
By ⋅ −τ and ⋅ 0 denote the norms in H −τ and H . Let 〈⋅ ⋅〉, be the generated by
the scalar product in H dual pairing between elements of D ′ and D (and also H −τ
and H τ ). The notation ⋅ τ , ⋅ 0 , ⋅ −τ , ( , )⋅ ⋅ τ , and 〈⋅ ⋅〉, will be preserved for
tensor powers and complexifications of spaces.
Remark 1.2. Note that all scalar products and pairings in this paper are real,
i.e., they are bilinear functionals. In particular, 〈⋅ ⋅〉, is a real pairing in complexi-
fications of spaces.
Let us fix arbitrary functions α, β : R + → C that are smooth and satisfy
θ : = – α – β ∈ R, η : = αβ ∈ R+ ,
θ and η are bounded on R + (note that in a sense η is a “key parameter” and will be
mentioned very often below). Further, let ̃ ( , , )v α β ds be a probability measure on R
that is defined by its Fourier transform
e dsius ˜ ( , , )v α β
R
∫ =
= exp ( )
( ) ( )
!
( )− + + − + +…+
−
=
∞
− − −
=
∞
∑ ∑iu
m
iu
n
m
m
n
n n n
n
m
α β αβ αβ β β α α2
1
1
2 3 2
2
,
v( , , )α β ds : =
1
2s
ds˜ ( , , )v α β .
Definition 1.1. We say that the probability measure µ on the measurable space
( ),′D F (here F is the generated by cylinder sets σ-algebra on D ′ ) with the
Fourier transform
e dxi x
D
〈 〉
′
∫ , ( )ξ µ =
exp ( ) ( ( ), ( ), ) ( )( )
R R+
∫ ∫ − −( )
σ α β ξξdt t t ds e is tis tv 1
( here ξ ∈ D ) is called the generalized Meixner measure.
Theorem 1.1 [5]. The generalized Meixner measure µ is a generalized sto-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
740 N. A. KACHANOVSKY
chastic process with independent values in the sense of [12]. The Laplace transform
of µ is given in a neighborhood of zero U0 ⊂ DC by the following formula:
lµ λ( ) = e dxx
D
〈 〉
′
∫ , ( )λ µ = exp
( ( ) ( ))
R+
∫ ∑
−
=
∞
α βt t
m
m
m
1
1
×
× ( )
!
( ) ( ) ( ) ( ) ( )− + + … +( )
=
∞
− − −∑ λ β β α α σ
n
m
n n n
m
n
t t t t dt
1
2 3 2 , λ ∈U0 .
Remark 1.3. Accordingly to the classical classification [13] (see also [14, 5]) for
α = β = 0 (here and below all such equalities we understand σ - a.e. ) µ is the Gaus-
sian measure: for α ≠ 0 (here and below a ( ⋅ ) ≠ b ( ⋅ ) means that a – b ≠ 0 on
some measurable set M such that σ ( M ) > 0), β = 0 µ is the centered Poissonian
measure; for α = β ≠ 0 µ is the centered Gamma measure; for α ≠ β, α β ≠ 0,
α, β : R + → R µ is the centered Pascal measure; for α = β , Im ( α ) ≠ 0 µ is the
centered Meixner measure.
The following statement describes an important property of µ .
Lemma 1.1 [15]. There exists τ̃ ∈T such that the generalized Meixner measure
is concentrated on
H −τ̃ , i.e.,
µ τ( )˜H − = 1.
Remark 1.4. In what follows, we assume that µ is concentrated on H −τ for
all τ ∈T . In fact, it is sufficient to exclude from T the indexes τ such that µ is not
concentrated on H −τ .
Now by ( )L2 = L D2( , )′ µ denote the space of square integrable with respect to µ
complex-valued functions on D ′ . Let us construct orthogonal polynomials on ( )L2 .
Definition 1.2. We define a so-called Wick exponential (a generating function of
the orthogonal polynomials) by setting
: exp( ; ) :x λ
=df exp
( ) ( )
( ) ( ) ( ) ( ) ( )− + + + … +( )
+
∫ ∑
=
∞
− − −
R
λ λ α α β β σt t
n
t t t t dt
n
n
n n n
2
3
2 3 2
2
+
+ x
n
n
n
n n n, λ λ α α β β+ + + … +( )
=
∞
− − −∑
2
1 2 1 , (1.2)
where λ ∈ ⊂U0 DC , x D∈ ′ , U0 is some neighborhood of 0 ∈DC .
Remark 1.5. It was proved in [5] that
: exp( ; ) :
( ( ))
, ( )
x e
l
x
λ
λ
λ
µ
=
〈 〉Ψ
Ψ
with
Ψ ( λ ) = λ λ α α β β+ + + … +( )
=
∞
− − −∑
n
n
n n n
n2
1 2 1 ,
therefore : exp( ; ) :x ⋅ is a generating function of the so-called Schefer polynomials
(or the generalized Appell polynomials in another terminology). This fact gives us the
possibility to use in our considerations well-known results of the so-called “biorthogo-
nal analysis” (see, e.g., [6 – 10, 16 – 18] and references therein).
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
GENERALIZED STOCHASTIC DERIVATIVES … 741
It is clear (see also [5]) that : exp( ; ) :x ⋅ is a holomorphic at zero function on DC
for each x D∈ ′ . Therefore using the Cauchy inequalities (see, e.g., [19]) and the ker-
nel theorem (see, e.g., [11]) one can obtain the representation
: exp( ; ) :
!
( ),x
n
P x
n
n
nλ λ=
=
∞
∑ 1
0
� ,
P x Dn
n( )
ˆ
∈ ′
C
� x D∈ ′ , λ ∈DC .
Here (and below) ̂� denotes a symmetric tensor product, λ�0 = 1 even for λ ≡ 0.
Remark 1.6. It follows from the given in [5] recurrence formula for P xn( ) that
actually P x Dn
n( )
ˆ
∈ ′� for x D∈ ′ . Moreover, if τ ∈T is such that the Dirac delta-
function δ τ0 ∈ −H (it means that δ τs ∈ −H ∀ ∈ +s R , see, e.g., [11]) then for
x ∈ −H τ we have P xn
n( )
ˆ
∈ −H τ
� .
In what follows, we assume that this statement holds true for all τ ∈T . In fact, by
analogy with Remark 1.4 it is sufficient to exclude from T the indexes τ such that
δ τ0 ∉ −H .
Definition 1.3. We say that the polynomials 〈 〉P fn
n, ( ) ,
f Dn n( ) ˆ
∈
C
� , n ∈ +Z ,
are called the generalized Meixner polynomials.
Remark 1.7. Depending on α and β in (1.2) the generalized Meixner polynomi-
als can be the generalized Hermite polynomials ( α = β = 0 ) ; the generalized Charli-
er polynomials ( α ≠ 0, β = 0 ) ; the generalized Laguerre polynomials ( α = β ≠
0 ) ; the Meixner polynomials ( α ≠ β, αβ ≠ 0, α, β : R + → R ) ; the Meixner –
Pollaczek polynomials ( α = β , Im ( α ) ≠ 0 ) (see also Remark 1.3).
In order to formulate a statement on an orthogonality of the generalized Meixner
polynomials we need the following definition.
Definition 1.4. We define the scalar product 〈⋅ ⋅〉, ext on
D n
C
�̂ , n ∈N , by the
formula
〈 〉f gn n( ) ( ), ext : =
k l s j k l l l
l s l s n
s
k
s
kj j k
k k
k
n
l l s s, , : , , , ,
!
! !∈ = … > > >
+ + =
∑ … …
N 1 1 11 2
1 1
1
…
…
×
×
R+
+ +
∫ + + + +
s s
k k
k
k
f t t t t t tn
l
s s
l
s s s s
l1
1 1
1
1 1
1
1 1
…
…��� … …
��� ��
… …
� ���� ����… …
( )( ), , , , , , , , , , ×
×
g t t t t t t t tn
l
s s
l
s s s s
l
l
s
l
k k
k
( )( ), , , , , , , , , , ( ) ( )1 1 1
1 1
1
1 1
1
1 1
1
1
1…��� … …
��� ��
… …
� ���� ����
…… …+ + + +
− −η η ×
×
η η η η( ) ( ) ( ) ( )t t t ts
l
s s
l
s s
l
s s
l
k
k
k
k
1
2
1 2
2
1 1 11
1 1
1
1 1
+
−
+
−
+ + +
−
+ +
−
−
… … …… … ×
× σ σ( ) ( )dt dts sk1 1
… …+ + .
Denote by ⋅ ext the corresponding norm, i.e., f n( )
ext
2
= 〈 〉f fn n( ) ( ), ext . For n =
= 0 〈 〉f g( ) ( ),0 0
ext : = f g( ) ( )0 0 ∈ C, f ( )0
ext
= f ( )0 .
Example. It is easy to see that for n = 1
〈 〉f g( ) ( ),1 1
ext = 〈 〉f g( ) ( ),1 1 = f t g t dt( ) ( )( ) ( ) ( )1 1 σ
R+
∫ .
Further, for n = 2
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
742 N. A. KACHANOVSKY
〈 〉f g( ) ( ),2 2
ext = 〈 〉 +
+
∫f g f t t g t t t dt( ) ( ) ( ) ( ), ( , ) ( , ) ( ) ( )2 2 2 2 η σ
R
.
If η = 0 (this means that µ is the Gaussian or Poissonian measure, see Remark 1.3)
then 〈 〉f gn n( ) ( ), ext = 〈 〉f gn n( ) ( ), for all n ∈ Z + ; in general, 〈 〉f gn n( ) ( ), ext =
= 〈 〉f gn n( ) ( ), + … .
Theorem 1.2 [5]. The generalized Meixner polynomials are orthogonal in ( )L2
in the sense that
〈 〉〈 〉
′
∫ P x f P x g dxn
n
m
m
D
( ), ( ), ( )( ) ( ) µ = δmn
n nn f g! ,( ) ( )〈 〉ext . (1.3)
By H ext
( )n , n ∈ N , denote the closure of
D n
C
�̂ with respect to the norm ⋅ ext ,
H ext
( )0 : = C . Of course, H ext
( )n , n ∈ Z + , are Hilbert spaces; for the scalar products in
these spaces it is natural to preserve the notation 〈⋅ ⋅〉, ext .
Remark 1.8. It is not difficult to prove by analogy with [20] that the space H ext
( )n
is, generally speaking, the orthogonal sum of
H
C
�̂n ≡
L n2( ),
ˆ
R
C+ σ � and some ano-
ther Hilbert spaces (as a “limit case” one can consider η = 0, in this case H ext
( )n =
= H
C
�̂n ). In this sense H ext
( )n is an extension of H
C
�̂n .
One can give another explanation of the fact that H ext
( )n is a more wide space than
H
C
�̂n . Namely, let
F n n( ) ˆ
∈H
C
� ( F n( ) is an equivalence class in
H
C
�̂n
). We select
a representative (a function) ˜ ( ) ( )F Fn n∈ with a “zero diagonal”, i.e., ˜ ( )F n is such that
˜ ( , , )( )F t tn
n1 … = 0 if t ti j= for i ≠ j, where i, j ∈ { 1, … , n } . This function gene-
rates the equivalence class ˆ ( )F n in H ext
( )n that can be identified with F n( ) (see [15]
for details).
Let us recall the construction of the Kondratiev-type spaces of test and generalized
functions (see, e.g., [6 – 10, 16 – 18, 21]).
In the classical Gaussian and Poissonian analysis the Kondratiev-type spaces are
“based” on the tensor powers of complexification of chain (1.1):
′D n
C
�̂ ⊃
H −τ,
ˆ
C
�n ⊃
H
C
�̂n ⊃
H τ,
ˆ
C
�n ⊃ D n
C
�̂ , τ ∈ T . (1.4)
But in the light of orthogonality relation (1.3) now it will be more natural to use H ext
( )n
as “central spaces” (by analogy with the Gamma analysis, see, e.g., [22]). In order to
construct such chains we need the following proposition.
Proposition 1.1 [15]. There exists τ̃ ∈T such that for each n ∈N
H ˜,
ˆ
τ C
�n is
densely and continuously embedded in H ext
( )n and, moreover, for all
f n n( )
˜,
ˆ
∈H τ C
�
the estimate
f n( )
ext
2
≤ n c fn n! ( )
τ̃
2
(1.5)
with some c > 0 is valid.
Remark 1.9. Let H τ be continuously embedded in H τ̃ ( τ τ, ˜ ∈T , τ̃ from Pro-
position 1.1 ) . Then it easily follows from Proposition 1.1 that for each n ∈ N
H τ,
ˆ
C
�n
is densely and continuously embedded in H ext
( )n , and estimate (1.5) with τ instead of
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GENERALIZED STOCHASTIC DERIVATIVES … 743
τ̃ ( c depends on τ ) holds true. Therefore by analogy with Remarks 1.4, 1.6 one can
exclude from T indexes τ such that there is no a continuous embedding H τ in H τ̃ ,
and assume in what follows, that the results of Proposition 1.1 hold true for all τ
∈ T .
Now we can consider the chains
′D n
C
( ) ⊃
H −τ,
( )
C
n ⊃ H ext
( )n ⊃ H τ,
ˆ
C
�n ⊃ D n
C
�̂ , (1.6)
where
H −τ,
( )
C
n , ′D n
C
( ) =
ind lim ,
( )
τ τ∈ −T
nH
C
are the dual to
H τ,
ˆ
C
�n , D n
C
�̂ with res-
pect to H ext
( )n spaces correspondingly. For the generated by the scalar product in
H ext
( )n (real) dual pairings between elements of ′D n
C
( )
and D
n
C
�̂
( in the same way as
H −τ,
( )
C
n and
H τ,
ˆ
C
�n
) we preserve the notation 〈 〉⋅ ⋅, ext .
Of course, for n = 1 chain (1.6) has the form
′DC ⊃
H −τ,C ⊃ H ext
( )1 = H C ⊃
H τ,C ⊃ DC ,
i.e., this chain coincides with the complexification of chain (1.1). But for n > 1 and
η ≠ 0 chain (1.6) is not a tensor power of chain of type (1.1). Nevertheless, there ex-
ists the natural interconnection between chains (1.4) and (1.6). In fact, since ′D n
C
( ) in
the same way as
′D n
C
�̂ , n ∈ +Z , are the sets of linear continuous functionals on
D n
C
�̂ , there exist linear bijective operators U D Dn
n n: ( ) ˆ′ → ′
C C
� such that ∀ ∈ ′F Dn n
ext
( ) ( )
C
∀ ∈f Dn n( ) ˆ
C
�
〈 〉U F fn
n n
ext
( ) ( ), = 〈 〉F fn n
ext ext
( ) ( ), . (1.7)
By analogy, since for all τ ∈ T
H −τ,
( )
C
n and
H −τ,
ˆ
C
�n are the sets of linear continuous
functionals on H τ,
ˆ
C
�n , there exist linear isometrical bijective operators
Un
n n
, ,
( )
,
ˆ
:τ τ τH H− −→
C C
� such that
∀ ∈ −F n n
ext
( )
,
( )H τ C
∀ ∈f n n( )
,
ˆ
H τ C
� : 〈 〉U F fn
n n
,
( ) ( ),τ ext = 〈 〉F fn n
ext ext
( ) ( ), .
Proposition 1.2 [15]. For each τ ∈ T and each n ∈ +Z the restriction of the
operator Un on H −τ,
( )
C
n coincides with Un,τ.
Taking into acount Proposition 1.2, in what follows, we omit a subindex τ for ope-
rators Un,τ, i.e., we’ll write always Un for such operators.
Remark 1.10. We note that for n = 0 and n = 1, in the same way as for n ∈ +Z
and η = 0 Un = id ; but for n > 1 and η ≠ 0 Un
nH ext
( ) ≠
H
C
�̂n . This fact was
proved in [22] for η ≡ 1 (in the Gamma analysis), the proof in the general case can be
constructed by analogy.
Let P be the set of all continuous polynomials on ′D . It follows from results of
[16, 10] that any element of P can be presented in the form
f = 〈 〉
=
∑ P fn
n
n
N f
, ( )
0
,
f Dn n( ) ˆ
∈
C
� , N f ∈ +Z . (1.8)
We define on P a family of scalar products by setting for f g, ∈P , τ ∈T , q ∈N
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744 N. A. KACHANOVSKY
( , ) ,f g qτ : = ( !) ,( )( ) ( )
min( , )
n f gqn n n
n
N Nf g
2
0
2 τ
=
∑ ,
where f n( ) , g n( ) are the kernels from decompositions (1.8) for f and g respective-
ly. By ⋅ τ,q denote the corresponding norm, i.e., for f ∈ P of form (1.8) we have
f qτ,
2 = ( ), ,f f qτ = ( !) ( )n fqn n
n
N f
2 2
0
2
τ
=
∑ .
Definition 1.5. By ( )H τ q denote a Hilbert space that is the closure of P with
respect to the norm ⋅ τ,q . Let also
( ) ( ): limH Hτ τ= ∈pr q qN , ( D ) : =
pr lim , ( )τ τ∈ ∈T q qN H .
The spaces ( )H τ q , ( )H τ , ( D ) are called the Kondratiev-type test functions spaces.
It is clear that f q∈( )H τ if and only if f can be presented in the form
f = 〈 〉
=
∞
∑ P fn
n
n
, ( )
0
(1.9)
with f n n( )
,
ˆ
∈H τ C
� , and the series converges in the sense that
f qτ,
2 : =
f
q( )H τ
2 = ( !) ( )n fqn n
n
2 2
0
2
τ
=
∞
∑ < ∞ . (1.10)
Further, f ∈( )H τ if and only if f has form (1.9) and norm (1.10) is finite for all
q ∈N ; and f D∈( ) if and only if norm (1.10) for f is finite for all τ ∈T and q ∈N
(in this case, of course, the kernels from decomposition (1.9)
f Dn n( ) ˆ
∈
C
�
).
Remark 1.11. Let f, g q∈( )H τ . Then
( , )( )f g
qH τ
= ( !) ,( ) ( )n f gqn n n
n
2
0
2 ( )
=
∞
∑ τ
,
where f n( ) ,
g n n( )
,
ˆ
∈H τ C
� are the kernels from decompositions (1.9) for f and g re-
spectively; therefore the system of the generalized Meixner polynomials plays a role of
an orthogonal basis in ( )H τ q .
In order to define the Kondratiev-type spaces of generalized functions we need the
following proposition.
Proposition 1.3 [15]. There exists q0 ∈N such that for all natural q q≥ 0 and
for all τ ∈T the dense of continuous embedding ( ) ( )H τ q LO
2 takes place (we
remind that T is modified in accordance with Remarks 1.4, 1.6, 1.9).
Remark 1.12. Let N Nq q q
0 0 0 1: { , , }= + … ⊆ . Then we can reformulate Propo-
sition 1.3 as follows: for all q q∈N
0
and for all τ ∈T the dense and continuous em-
bedding ( ) ( )H τ q LO
2 takes place.
Now we can consider the chain
( )′ ′D ⊃ ( )H −τ ⊃ ( )H − −τ q ⊃ ( )L2 ⊃ ( )H τ q ⊃ ( )H τ ⊃ ( )D ,
q q∈N
0
, τ ∈T ,
where
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GENERALIZED STOCHASTIC DERIVATIVES … 745
( )H − −τ q , ( )H −τ =
ind lim ( )q qq∈ − −N
0
H τ , ( )′ ′D =
ind lim , ( )q T qq∈ ∈ − −N
0
τ τH
are the dual to ( )H τ q , ( )H τ , ( )D with respect to ( )L2 spaces correspondingly.
Definition 1.6. The spaces ( )H − −τ q , q q∈N
0
, τ ∈T , ( )H −τ , ( )′ ′D are call-
ed the Kondratiev-type spaces of nonregular generalized functions.
Let us recall a construction of orthogonal bases in the spaces ( )H − −τ q . For F n
ext
( )
*∈
∈
H – ,
( )
τ C
n we define 〈 〉P Fn
n, ( )
ext ∈ ( )H − −τ q as the limit in ( )H − −τ q of a sequence of
polynomials 〈 〉P Fn k
n, ( ) such that
H τ,
ˆ ( ) ( )
C
�n
k
n nF F' → ext as k → ∞ in
H – ,
( )
τ C
n (the
correctness of this definition was proved in [15]).
Theorem 1.3 [15]. A generalized function F q∈ − −( )H τ , τ ∈T , q q∈N
0
, if and
only if there exists a sequence
F n n
next
( )
– ,
( )∈( ) =
∞
H τ C 0
(1.11)
such that F can be presented in the form
F = 〈 〉
=
∞
∑ P Fn
n
n
, ( )
ext
0
, (1.12)
where the series converges in ( )H − −τ q , i.e., the norm
F q− −τ,
2 : =
F
q( )H − −τ
2 = 2
2
0
−
−
=
∞
∑ qn n
n
Fext ext
( )
,τ
< ∞ (1.13)
(here and below by ⋅ −τ,ext denote the norms in H – ,
( )
τ C
n
) . Furthermore, the system
{ }, : ,( ) ( )
– ,
( )〈 〉 ∈ ∈ +P F F nn
n n n
ext ext H τ C
Z plays a role of an orthogonal basis in ( )H − −τ q
in the sense that for F, G q∈ − −( )H τ
( , )( )F G
qH − −τ
=
n
qn n nF G
=
∞
−
−∑
0
2 ( )( ) ( )
,,ext ext extτ ,
where F n
ext
( ) ,
G n n
ext
( )
– ,
( )∈H τ C
are the kernels from decompositions (1.12) for F and
G correspondingly, ( , ) ,⋅ ⋅ −τ ext is the scalar product in
H – ,
( )
τ C
n .
Remark 1.13. It is easy to see that F ∈ −( )H τ ( correspondingly F D∈ ′ ′( ) ) if
and only if there exists sequence (1.11) such that F can be presented in form (1.12)
with finite norm (1.13) for some q ∈N ( correspondingly for some q ∈N and some
τ ∈T ) .
Remark 1.14. Note that one can introduce the spaces ( )D q : = pr lim ( )τ τ∈T qH ,
q ∈N , and the corresponding dual ones; but from “the point of view of this paper”
these spaces are completely analogous to the spaces ( )H τ q and ( )H − −τ q .
The generated by the scalar product in ( )L2 (real) dual pairing between elements
of ( )H − −τ q and ( )H τ q ( in the same way as ( )H −τ and ( )H τ , ( )′ ′D and ( )D )
will be denoted by 〈〈⋅ ⋅〉〉, . It was shown in [15] that for a generalized function F of
form (1.12) and a test function f of form (1.9)
〈〈 〉〉F f, =
n
n nn F f
=
∞
∑ 〈 〉
0
! ,( ) ( )
ext ext . (1.14)
Now let us recall elements of the Wick calculus.
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746 N. A. KACHANOVSKY
Definition 1.7. For F D∈ ′ ′( ) we define an integral S-transform ( )( )SF λ ( λ
belongs to some depending on F neighborhood of zero in D C ) by setting (see
(1.2))
( )( )SF λ : = 〈〈 ⋅ 〉〉F, :exp( ; ) :λ .
The S-transform is well-defined because for each F D∈ ′ ′( ) there exist τ ∈T and
q q∈N
0
such that F q∈ − −( )H τ ; and for λ ∈DC such that 2 2q λ τ < 1 we have
:exp( ; ) :⋅ λ ∈ ( )H τ q .
Remark 1.15. It is easy to see that
( )( )SF λ =
n
n nF
=
∞
∑ 〈 〉
0
ext ext
( ), λ� , (1.15)
where
F n n
ext
( )
– ,
( )∈H τ C
, n ∈ +N , are the kernels from decomposition (1.12) for F. In
particular, ( )( )SF 0 = Fext
( )0 , S1 ≡ 1.
Theorem 1.4 [16, 10]. The S - transform is a topological isomorphism between
the space ( )′ ′D and the algebra Hol0 of germs of holomorphic at zero functions
on DC .
Definition 1.8. For F , G D∈ ′ ′( ) and a holomorphic at ( )( )SF 0 function h :
C → C we define the Wick product F ◊ G ∈ ( )′ ′D and the Wick version of h
h F◊( ) ∈ ( )′ ′D by setting
F ◊ G : = S SF SG− ⋅1( ), h F◊( ) : = S h SF−1 ( ) .
The correctness of this definition from Theorem 1.4 follows.
Remark 1.16. It is easy to see that the Wick multiplication ◊ is commutative,
associative and distributive (over the field C ). Further, if h from Definition 1.8 is
presented in the form
h ( u ) =
n
n
nh u SF
=
∞
∑ −
0
0( ( ) )( ) (1.16)
then h F◊( ) = h F SFn
n
n
( ( ) )( )− ◊
=
∞∑ 0
0
, where F n◊ =
F F
n
◊ … ◊
times
� �� �� .
Let us write out the “coordinate form” of F ◊ G and h F◊( ) (this form is necessary
for calculations). Let for F Dk k
ext
( ) ( )∈ ′
C
, G Dm m
ext
( ) ( )∈ ′
C
F Gk m
ext ext
( ) ( )◊ : = U U F U Gk m k
k
m
m
+
– ( ) ( )( )ˆ1
ext ext� ∈ ′ +D k m
C
( )
(see (1.7)). It is obvious that the “multiplication” � is commutative, associative and
distributive (over the field C ). It was shown in [15] that
F ◊ G =
n
n
k
n
k n kP F G
=
∞
=
−∑ ∑
0 0
, ( ) ( )
ext ext� , (1.17)
h F◊( ) =
h P h F F
n
n
k
n
k
m m
m m m m n
k
k k
0
1 1
1
1 1
+ …
=
∞
= … ∈ +…+ =
∑ ∑ ∑, ( ) ( )
, , :
ext ext� �
N
, (1.18)
where F k
ext
( ) , G Dk k
ext
( ) ( )∈ ′
C
are the kernels from decompositions (1.12) for F and G cor-
respondingly, hk ∈C , k ∈ +Z , are the coefficients from decomposition (1.16) for h.
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GENERALIZED STOCHASTIC DERIVATIVES … 747
Remark 1.17. It follows from (1.17) that, in particular,
〈 〉 〈 〉◊P F P Gn
n
m
m, ,( ) ( )
ext ext = 〈 〉+P F Gn m
n m, ( ) ( )
ext ext� .
This formula can be used in order to define the Wick product (and then the Wick versi-
on of a holomorphic function as a series) without the S-transform. Formulas (1.17)
and (1.18) also can be used as definitions.
Let now F, G ∈ −( )H τ , τ ∈T . Since the space ( )H −τ is embedded in ( )′ ′D ,
the Wick product F ◊ G and the Wick version of a holomorphic at ( )( )SF 0 function
h h
◊ are well-defined as elements of ( )′ ′D and “coordinate representations” (1.17),
(1.18) hold true. Moreover, by Remark 1.13 and Proposition 1.2 the kernels from
decompositions (1.17) and (1.18) are elements of
H – ,
( )
τ C
n .
Theorem 1.5 [6]. Let F, G ∈ −( )H τ , τ ∈T , and h : C → C be a holomor-
phic at ( )( )SF 0 function. Then F ◊ G ∈ ( )H −τ a n d h F◊( ) ∈ ( )H −τ . More-
over, the Wick multiplication is continuous in the topology of ( )H −τ .
Finally, we recall the definition of the extended stochastic integral in the Meixner
analysis (see [15] for a detailed presentation).
Let F ∈ −( )H Hτ � C . It follows from Theorem 1.3 that F can be presented in
the form
F( )⋅ =
n
n
nP F
=
∞
⋅∑ 〈 〉
0
, ( )
ext, ,
F n n
ext, ⋅ ∈( )
– ,
( )H Hτ C C� (1.19)
with
F
q( )H H− −τ � C
2 = 2
2
0
−
⋅
=
∞
∑ qn n
n
F next,
( )
– ,
( )H Hτ C C�
< ∞
for some q ∈ N . For t ∈ [ 0, + ∞ ] we set
ˆ ( )F t
n
ext, [ , )0 : = U U Fn n
n
t+
−
⋅ ⋅[ ]1
1
01Pr ( )( ( ) )( )
ext, [ , ) ∈
H – ,
( )
τ C
n+1 , (1.20)
where Pr is the symmertization operator, Un , Un+1 are defined in (1.7), here and
below 1A denotes the indicator of a set A .
Let { }: ,M Ps s s= 〈 〉 ≥1 0 01[ , ) be the Meixner random process (this process is a locally
square integrable normal martingale with independent instruments, see [15, 5] for more
details).
Definition 1.9. Let F ∈ −( )H Hτ � C , t ∈ [ 0, + ∞ ] . We define an extended
stochastic integral with respect to the Meixner process
F s dMs
t
( ) ˆ ( )
0∫ ∈ −H τ by set-
ting
F s dMs
t
( ) ˆ
0
∫ : = 〈 〉+
=
∞
∑ P Fn t
n
n
1 0
0
, ˆ ( )
ext, [ , ) , (1.21)
where
ˆ ( )
– ,
( )F t
n n
ext, [ , )0
1∈ +H τ C
, n ∈ +Z , are constructed in (1.20) starting from the ker-
nels
F n n
ext, ⋅ ∈( )
– ,
( )H Hτ C C� from decomposition (1.19) for F.
It was proved in [15] that this definition is correct and integral (1.21) is a generali-
zation of the Itô stochastic integral with respect to M. The case F D∈ ′ ′( ) � H C is
completely analogous to the case F ∈ −( )H Hτ � C (see [15] for details).
The interconnection between the Wick calculus and the extended stochastic integra-
tion is given by the formula (see [15])
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748 N. A. KACHANOVSKY
F s dMs
t
( ) ˆ
0
∫ = F s M dss
t
( ) ( )◊ ′∫ σ
0
, (1.22)
where t ∈ [ 0, + ∞ ] , F ∈ −( )H Hτ � C (or F D∈ ′ ′( ) � H C ), { : ,′ = 〈 〉 ∈M Ps s1 δ
∈ ( )}H − ≥τ s 0 (here δ is the Dirac delta-function) is the Meixner white noise (the men-
tioned in Theorem 1.1 generalized stochastic process).
2. Generalized stochastic derivatives. In this section we introduce and study
generalized stochastic derivatives on the finite order spaces of generalized functions
( )H −τ (here and below τ ∈T ). As regards to the space ( )′ ′D one can easily repeat
all our considerations by analogy. In the end of the section we consider a one example:
calculate the generalized stochastic derivative of the solution of some stochastic equati-
on with a Wick-type nonlinearity.
First let η = 0 (this corresponds to the Gaussian and Poissonian cases). The gene-
ralization of the Hida stochastic derivative (see, e.g., [23]) ∂. can be defined now on
( )H −τ as follows.
Definition 2.1. Let F ∈ −( )H τ , η = 0. We define a generalized stochastic de-
rivative ∂ τ τ. ( ) – ,F ∈ −H H� C by setting
∂.F : = n P Fn
n
n
〈 〉−
=
∞
⋅∑ 1
1
, ( )( )
ext ≡ ( ) , ( )( )n P Fn
n
n
+ ⋅〈 〉+
=
∞
∑ 1 1
0
ext , (2.1)
where F n n
ext
( )
– ,
ˆ
– ,( )⋅ ∈ −H Hτ τC C
� �1 , n ∈ N , are obtained from the kernels from de-
composition (1.12) for F by “separating of a one argument” (note that now
H – ,
( )
τ C
n = H – ,
ˆ
τ C
�n ⊂ H H– ,
ˆ
– ,τ τC C
� �n−1
).
For a general η ≠ 0 this definition can not be accepted because for n > 1
H – ,
( )
τ C
n ⊄
H H– ,
( )
– ,τ τC C
n−1 � , therefore the kernels
F n n
ext
( )
– ,
( )∈H τ C
from (1.12) can not
be considered as elements of
H H– ,
( )
– ,τ τC C
n−1 � . In order to “go around” this problem
we accept the following definition.
Definition 2.2. For F n n
ext
( )
– ,
( )∈H τ C
we define F n n
ext
( )
– ,
( )
– ,( )⋅ ∈ −H Hτ τC C
1 � , n ∈ N ,
by the formula
F n
ext
( )( )⋅ : = U U Fn n
n
−
− ⋅[ ]1
1 ( )( ) ( )ext , (2.2)
where the isomorphisms Un
n n: – ,
( )
– ,
ˆ
H Hτ τC C
→ � , n ∈ +Z , are defined by (1.7).
Remark 2.1. Note that
F n
next
( )( )
– ,
( )
– ,
⋅ −H Hτ τC C
1 �
=
( )( ) ( )
– ,
ˆ
– ,
U Fn
n
next ⋅ −H Hτ τC C
� �
1 = U Fn
n
ext
( )
–τ
= F n
ext ext
( )
– ,τ
.
(2.3)
Remark 2.2. Since for η = 0 Un = id for all n ∈ +Z , a defined by (2.2)
F n
ext
( )( )⋅ coincides in this case with F n
ext
( )( )⋅ from Definition 2.1.
Now we are ready to give a definition of a generalized stochastic derivative on
( )H −τ .
Definition 2.3. Let F ∈ −( )H τ . W e define a generalized stochastic derivative
∂ τ τ. ( ) – ,F ∈ −H H� C by formula (2.1), where
F n n
ext
( )
– ,
( )
– ,( )⋅ ∈ −H Hτ τC C
1 � , n ∈ N ,
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GENERALIZED STOCHASTIC DERIVATIVES … 749
are constructed in (2.2) starting from the kernels
F n n
ext
( )
– ,
( )∈H τ C
from decomposi-
tion (1.12) for F.
Remark 2.3. Not that for η ≠ 0 the restriction of ∂. on ( )L2 does not coincide
with the constructed in [15] “Hida derivative” on ( )L2 . At the same time the “Hida
derivative” on ( )L2 can not be continued on ( )H −τ . This is an objective fact that is
connected with base properties of the Meixner measure.
Let us prove the correctness of Definition 2.3. Let F ∈ −( )H τ . Then there exists
q ∈ N such that F q∈ − − +( )H τ 1. Using this fact, (2.3), (2.1), and the simple estimate
max ( )[ ]n
nn∈
−
+
+Z 1 22 = 9 / 4 , we obtain
∂
τ τ
. ( ) – ,
F
qH H− − � C
2 =
( ) ( )( )
– ,
( )
– ,
n F
n
qn n
n+ ⋅
=
∞
− +∑ 1 22
0
1 2
ext H Hτ τC C�
=
=
n
n q n nn F
=
∞
− − + +∑ +
0
2 1 1 2
1 2 2[ ]( ) ( ) ( )
– ,ext extτ
≤
≤ 9 2 23
0
1 1 1 2
⋅ −
=
∞
− + + +∑q
n
q n nF( )( ) ( )
– ,ext extτ
≤ 9 2 3
1
2⋅ −
− +
q
qF – ,τ < ∞ .
Therefore ∂. is well-defined and, moreover, is a linear continuous operator acting
from ( )H −τ to ( ) – ,H H−τ τ� C .
Let us calculate the adjoint to ∂. operator ∂ τ τ τ⋅
∗ →: ( ) ( ),H H H� C . Let
f ( )⋅ =
m
m
mP f
=
∞
⋅∑ 〈 〉
0
, ( ) ∈ ( ) ,H Hτ τ� C , f m m
⋅ ∈( )
,
ˆ
,H Hτ τC C
� � (2.4)
(see (1.9)), F ∈ −( )H τ . We have (see (1.12), (1.14), (1.7), (2.2))
( ), ( )
( )
∂⋅ ⋅F f
L2 �H C
=
( ) , ( ) , ,( ) ( )
( )
n P F P f
n
n
n
m
m
m
L
+ ⋅
=
∞
+
=
∞
⋅∑ ∑〈 〉 〈 〉1
0
1
0 2
ext
�H C
=
= ( )! ( ),( ) ( )
( )n F f
n
n n
n+ ⋅
=
∞
+
⋅∑ 〈 〉1
0
1
ext
extH H� C
=
=
( )! ( ),( )( ) ( )
ˆn U F f
n
n
n n
n
+ ⋅
=
∞
+
+
⋅∑ 〈 〉1
0
1
1
ext
H H
C C
� �
=
= ( )! , Pr( ) ( )n U F f
n
n
n n+
=
∞
+
+∑ 〈 〉1
0
1
1
ext = ( )! , Pr( ) ( )n F f
n
n n+
=
∞
+∑ 〈 〉1
0
1
ext ext =
=
n
n
n
m
m
mP F P f
=
∞
=
∞
+∑ ∑〈 〉 〈 〉
0 0
1, , , Pr( ) ( )
ext = 〈〈 〉〉⋅
∗ ⋅F f, ( )∂ ,
where Pr is the symmetrization operator (more exactly, for f m m
⋅ ∈( )
,
ˆ
H τ C
� �
�
Hτ,
( )PrC f m ∈ H τ,
ˆ
C
�m+1 is a projection of f m
⋅
( ) on H τ,
ˆ
C
�m+1
). Therefore, for
f ( )⋅ ∈
( ) ,H Hτ τ� C of form (2.4)
∂⋅
∗ ⋅f ( ) =
m
m
mP f
=
∞
+∑ 〈 〉
0
1, Pr ( ) . (2.5)
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750 N. A. KACHANOVSKY
Remark 2.4. It is obvious that for η = 0 and f ( )⋅ ∈ ( ) ,H Hτ τ� C ∂⋅
∗ ⋅f ( ) =
= f s dMs( ) ˆ
R+
∫ . But for η ≠ 0 f s dMs( ) ˆ
R+
∫ ∉ ( )H τ , generally speaking ( and, of
course, ∂⋅
∗ ⋅�( ) does not coincide with �( ) ˆs dMs
R+
∫ ) . Note also that even the integra-
bility of f ( )⋅ by Itô is not sufficient for coincidence of ∂⋅
∗ ⋅f ( ) and f s dMs( ) ˆ
R+
∫ in
the case η ≠ 0.
Now let us obtain the analog of the Clack – Ocone theorem (see, e.g., [24 – 28]).
We recall that the task consists in finding of the explicit expression for m, in the
equality F = EF m dMs s+
+
∫ ˆ
R
( here and below E denotes the expectation: EF =
= 〈〈 〉〉F,1 = F( )0 = 〈 〉P F0
0, ( ) ∈ C, where F( )0 is the kernel from decomposition
(1.12) for F ) .
In the general case the following solution is possible. Let us introduce an operator
′⋅∂ ∈
L H H H(( ) ( ) ), – ,− − − −τ τ τq q � C ( q q∈N
0
, L ( ),H H1 2 denotes the set of linear
continuous operators acting from H1 to H2) by setting for F q∈ − −( )H τ
′⋅∂ F : = 〈 〉−
=
∞
⋅∑ P Fn
n
n
1
1
, ( )( )
ext ,
where F n n
ext
( )
– ,
( )
– ,( )⋅ ∈ −H Hτ τC C
1 � , n ∈ N , are constructed in (2.2) starting from the
kernels
F n n
ext
( )
– ,
( )∈H τ C
from decomposition (1.12) for F. Since (see (2.3))
′
− −
∂
τ τ
. ( ) – ,
F
qH H� C
2 =
n
q n nF n
=
∞
− −∑ ⋅ −
1
1 2
2 1
( ) ( )( )
– ,
( )
– ,
ext H Hτ τC C�
=
= 2 2
1
2q
n
qn nF
=
∞
−∑ ext ext
( )
– ,τ
≤ 2 2q
qF – ,τ − ,
this definition is correct.
Theorem 2.1 (cf. [29]). Let F ∈ −( )H τ . Then
F = EF F dMs s+ ′
+
∫ ∂ ˆ
R
.
Proof. In fact, using (1.20) and (2.2) we obtain
′
+
∫ ∂s sF dMˆ
R
=
n
n n n
nP U U F
=
∞
−
−∑ 〈 〉⋅
1
1
1, Pr ( )( ( ))( )
ext = 〈 〉
=
∞
∑ P Fn
n
n
, ( )
ext
1
.
The theorem is proved.
Unfortunately, the using of the operator ∂., generally speaking, is impossible. But
for a special particular case the following formal construction is possible (cf. [29]).
Let F ∈ −( )H τ be such that ∀ n ∈ N : U Fn
n
ext
( )
∈
H
C
�̂n (here F n
ext
( )
∈
H – ,
( )
τ C
n ,
n ∈ +Z , are the kernels from decomposition (1.12) for F ). We define
E( ). .∂ F F : = n P U U Fn n n
n
n
n〈 〉− −
−
⋅
=
∞
⋅ −∑ 1 1
1
0
1
1 1, ( )(( ) )( )
[ , ]ext (2.6)
(for η = 0 and under some additional conditions the right-hand side of (2.6) is the
conditional expectation of ∂.F with respect to the generated by M full σ - algebra
F. : = σ ( : )M uu ≤ ⋅ , cf. [29], therefore this notation is natural). Since
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GENERALIZED STOCHASTIC DERIVATIVES … 751
U U Fn n
n
n n−
−
⋅⋅ − −1
1
0
1 1 1
(( ) )( )
[ , ]
( )
– ,
( )ext
H Hτ C C�
=
( )( )
[ , ]
( )
– ,
ˆU Fn
n
n next ⋅ ⋅ − −
1
0 1 1H Hτ C C
� �
≤
≤ c U Fn
n
n n
( ) ( )( )( )
[ , ] ˆτ ext ⋅ ⋅ − −
1
0 1 1H H
C C
� �
≤
c U Fn
n
n( ) ( )( )( )
ˆτ ext ⋅ −H H
C C
� �
1 =
= c U Fn
n( ) ( )( )τ ext 0
< ∞ ,
each summand of series (2.6) is well-defined as an element of ( )H H−τ � C ; but the
series is formal in the sense that, generally speaking, it can diverge in ( )H H−τ � C .
Integrating (2.6) term by term we obtain the formal equality
R+
∫ E( ) ˆ∂s sF dM
sF = n P U U Fn n n
n
n
n〈 〉−
⋅
=
∞
⋅ −∑ , Pr ( )( (( ) ))( )
[ , ]
1
0
1
1 1ext . (2.7)
Lemma 2.1. Let
F n n( ) ˆ
∈H
C
� , n ∈ N . Then Pr ( )( )( )
[ , ]
F n
n⋅ ⋅ −1
0 1 = 1
n
F n( ) in
H
C
�̂n (here F n n( ) ˆ
( )⋅ ∈ −H H
C C
� �1 is obtained from F n( ) by “separating of a
one argument”, Pr is the symmetrization operator).
Proof. Let ˙ ( ) ( )F Fn n∈ be a representative of F n( ) (
˙ ( )F n is a function depen-
ding on n variables). Without loss of generality we may take ˙ ( )F n to be a symmetric
function (in fact, let M be the set of processions ( t1, … , tn ) such that ˙ ( , , )( )F t tn
n1 …
is not symmetric, then σ
�n M( ) = 0 ). We have
Pr ˙ ( , , ) ( , , )( )( )
[ , ]
F t t t tn
n t n
n
n1 0 1 11 1… …− − = 1 11 0 1 11
n
F t t t tn
n t n
n
n
˙ ( , , ) ( , , )( )
[ , ]
… …[ − − +
+ ˙ ( , , , ) ( , , , )( )
[ , ]
F t t t t t tn
n n t n n
n
n1 1 0 1 21
1
1… …− −
−
− + …
… + ˙ ( , , , ) ( , , )( )
[ , ]
F t t t t tn
n t nn2 1 0 21
1
1… … ]− .
If all t j , j ∈ { 1, … , n } , are different, then only one term in the right-hand side is not
equal to zero, hence by virtue of symmetry of ˙ ( )F n
Pr ˙ ( , , ) ( , , )( )( )
[ , ]
F t t t tn
n t n
n
n1 0 1 11 1… …− − = 1
1n
F t tn
n
˙ ( , , )( ) … .
The processions with coinciding arguments can be ignored because the measure σ�n
of the set of such processions is equal to zero.
The lemma is proved.
Using the rezult of this lemma, we can rewrite (2.7) in the form
R+
∫ E( ) ˆ∂s sF dM
sF = 〈 〉
=
∞
∑ P Fn
n
n
, ( )
ext
1
,
i.e.,
F =
E EF F dMs ss
+
+
∫
R
( ) ˆ∂ F .
By analogy with [1, 2] we consider now another stochastic differential operator
(this new operator is closely connected with ∂., see Proposition 2.1 below). We begin
from some technical preparation.
For
F m m
ext
( )
– ,
( )∈H τ C
and
f n n( )
,
ˆ
∈H τ C
� , m > n, we define a “pairing” ( )( ) ( ),F fm n
ext ext *∈
∈ H – ,
( )
τ C
m n− by the equality
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752 N. A. KACHANOVSKY
〈 〉−( )( ) ( ) ( ), ,F f gm n m n
ext ext ext =
〈 〉−F f gm n m n
ext ext
( ) ( ) ( ), �̂
∀ ∈− −g m n m n( )
,
ˆ
H τ C
� . (2.8)
Since
〈 〉−F f gm n m n
ext ext
( ) ( ) ( ), �̂ ≤
F f gm n m n
ext ext
( )
– ,
( ) ( )ˆ
τ τ
� − ≤
≤ F f gm n m n
ext ext
( )
– ,
( ) ( )
τ τ τ
− ,
( )( ) ( ),F fm n
ext ext is well-defined as an element of
H – ,
( )
τ C
m n− and, moreover,
( )( ) ( )
– ,
,F fm n
ext ext extτ
≤ F fm n
ext ext
( )
– ,
( )
τ τ
.
For m = n we set ( )( ) ( ),F fn n
ext ext : = 〈 〉F fn n
ext ext
( ) ( ), .
Definition 2.4. For an arbitrary
f n n( )
,
ˆ
∈H τ C
� we define an operator
( )( )( )D n nf� ∈ L H H(( ) ( )),− −τ τ by setting for F = 〈 〉=
∞∑ P Fm
m
m
, ( )
ext0
∈ ( )H −τ .
( )( )( )D n nF f : =
m
m
m n nm n
m
P F f
=
∞
+∑ + 〈 〉
0
( )!
!
, ,( )( ) ( )
ext ext ∈ ( )H −τ . (2.9)
Since for each F ∈ −( )H τ there exists q ∈ N such that F q∈ − − +( )H τ 2 , we have
( )( )( )
– ,
D n n
q
F f
τ −
2
=
m
m
m n n
q
m n
m
P F f
=
∞
+
−
∑ + 〈 〉
0
2
( )!
!
, ,( )( ) ( )
– ,
ext ext
τ
≤
≤
m
qm m n nm n
m
F f
=
∞
− +∑ +
0
2 2 2
2 ( )!
!
( )
– ,
( )
ext extτ τ
≤
≤ ( !) ( ) ( ) ( )
– ,
n f Fn
m
qm m n m n2 2
0
2 2
2 2
τ τ
=
∞
− + +∑ ext ext
=
= 2 22 2
0
2 2qn n
m
q m n m nn f F( !) ( ) ( )( ) ( )
– ,τ τ
=
∞
− − + +∑ ext ext
≤
≤ 2 2 2
2
2qn n
qn f F( !) ( )
– , ( )τ τ − − < ∞
(we used the estimate ( )!
!
m n
m
+ = n Cm n
m! + ≤ n m n!2 +
). Therefore this definition is
correct and, moreover, for each F ∈ −( )H τ ( ) ( ( ))( ) ,,
ˆ
D L H Hn nF � ∈ −τ τC
� .
Remark 2.5. Let D D:= 1. It is not difficult to show by the direct calculation
that for
g gn1
1 1( ) ( )
,, ,… ∈H τ C and F ∈ −( )H τ
( ( ( (( )( )))( ) )) ( )( ) ( ) ( )D D D… …
n
nF g g g
times
� ��� ��� 1
1
2
1 1 = ( )( )( ) ( ) ( )ˆ ˆ ˆD n
nF g g g1
1
2
1 1� � �… .
Theorem 2.2. For each F ∈ −( )H τ the kernels F n n
ext
( )
– ,
( )∈H τ C
, n ∈ +Z , from
decomposition (1.12) can be presented in the form
F n
ext
( ) =
1
n
Fn
!
( )E D . (2.10)
Proof. Using (2.9), for each f n n( )
,
ˆ
∈H τ C
� we obtain
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GENERALIZED STOCHASTIC DERIVATIVES … 753
E(( )( ))( )D n nF f = 〈〈 〉〉( )( )( ) ,D n nF f 1 = n F fn n! ,( )( ) ( )
ext ext = n F fn n! ,( ) ( )〈 〉ext ext ,
this equality can be formally rewritten in form (2.10).
The theorem is proved.
Let us calculate the adjoint to ( )( )( )D n nf� ,
f n n( )
,
ˆ
∈H τ C
� operator. For
F ∈ −( )H τ and g ∈( )H τ we have (see (2.9), (1.9), (1.14), (2.8), (1.12))
〈〈 〉〉( )( )( ) ,D n nF f g =
m
m
m n n
k
k
km n
m
P F f P g
=
∞
+
=
∞
∑ ∑+ 〈 〉 〈 〉
0 0
( )!
!
, , , ,( )( ) ( ) ( )
ext ext =
=
m
m n n mm n F f g
=
∞
+∑ + 〈 〉
0
( )! , ,( )( ) ( ) ( )
ext ext ext =
m
m n n mm n F f g
=
∞
+∑ + 〈 〉
0
( )! , ˆ( ) ( ) ( )
ext ext� =
=
k
k
k
m
m n
n mP F P f g
=
∞
=
∞
+∑ ∑〈 〉 〈 〉
0 0
, , , ˆ( ) ( ) ( )
ext � =
F g fn n, ( )( )( )D ∗ ,
therefore
( )( )( )D n ng f ∗ =
m
m n
n mP f g
=
∞
+∑ 〈 〉
0
, ˆ( ) ( )� , (2.11)
where g m m( )
,
ˆ
∈H τ C
� , m ∈ +Z , are the kernels from decomposition (1.9) for g.
Now we focus on the operator D = D1. The interconnection between D and ∂.
is given by the following proposition.
Proposition 2.1. For all F ∈ −( )H τ ,
f ( )
,
1 ∈H τ C
〈 〉⋅ ⋅∂ F f, ( )( )1 = ( )( )( )D F f 1 . (2.12)
Proof. For F = 〈 〉=
∞∑ P Fm
m
m
, ( )
ext0
∈ ( )H −τ , g = 〈 〉=
∞∑ P gn
n
n
, ( )
0
∈ ( )H τ ,
f ( )1 ∈
H τ,C we have (see (2.1), (1.14), (2.2), (2.9))
〈〈〈 〉 〉〉⋅ ⋅∂ F f g, ( ) ,( )1 = 〈〈〈 〉〉 〉⋅ ⋅∂ F g f, , ( )( )1 =
=
m
m
m
n
n
nm P F P g f
=
∞
+
=
∞
∑ ∑+ ⋅ ⋅〈 〉 〈 〉
0
1
0
11( ) , ( ) , , , ( )( ) ( ) ( )
ext =
=
n
n nn F g f
=
∞
+∑ + ⋅ ⋅〈 〉
0
1 11( )! ( ), , ( )( ) ( ) ( )
ext ext =
=
n
n
n nn U F g f
=
∞
+
+∑ + ⋅ ⋅〈 〉
0
1
1 11( )! ( ), , ( )( )( ) ( ) ( )
ext =
=
n
n
n nn U F g f
=
∞
+
+∑ + 〈 〉
0
1
1 11( )! , ˆ( ) ( ) ( )
ext � =
n
n nn F g f
=
∞
+∑ + 〈 〉
0
1 11( )! , ˆ( ) ( ) ( )
ext ext� =
=
n
n nn F f g
=
∞
+∑ + 〈 〉
0
1 11( )! , ,( )( ) ( ) ( )
ext ext ext =
=
m
m
m
n
n
nm P F f P g
=
∞
+
=
∞
∑ ∑+ 〈 〉 〈 〉
0
1 1
0
1( ) , , , ,( )( ) ( ) ( )
ext ext = 〈〈 〉〉( )( )( ) ,D F f g1 .
Since g ∈( )H τ is arbitrary, it follows from this calculation that (2.12) is true.
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754 N. A. KACHANOVSKY
Remark 2.6. Note that formally ∂.� = ( )( ).D � δ , where δ is the Dirac delta-
function.
Taking into consideration the result of Proposition 2.1, we preserve for D the
name “a generalized stochastic derivative”, cf. [1, 2, 3].
Remark 2.7. Comparing (2.5) and (2.11) (with n = 1) one can conclude that for
f ⋅ g( )( )1 ⋅ = f g� ( )1
∈ ( ) ,H Hτ τ� C (here f ∈( )H τ ,
g( )
,
1 ∈H τ C ) ∂⋅
∗ ⋅ ⋅f g( )( )1 =
= ( )( )( )D f g 1 ∗
.
Now let us study an interconnection between the stochastic differentiation and the
Wick calculus (cf. [1, 2]).
Theorem 2.3. The operator D � is a pre-image of the directional derivative of
S� under the S -transform, i.e., for all F ∈ −( )H τ ,
g ∈H τ,C
( )( )D F g = S D SFg
−1 ( ), (2.13)
where Dg denotes the directional derivative in the direction g. Formula (2.13)
can be accepted as a definition of D.
Proof. Let F = 〈 〉=
∞∑ P Fm
m
m
, ( )
ext0
∈ ( )H −τ ,
g ∈H τ,C . We have (see (1.15)
and Theorem 1.4)
( )( )SF λ =
m
m mF
=
∞
∑ 〈 〉
0
ext ext
( ), λ� ∈ Hol0
,
D SFg( )( )λ =
m
m mm F g
=
∞
−∑ 〈 〉
1
1
ext ext
( ), ˆλ� � =
=
m
m mm F g
=
∞
+∑ + 〈 〉
0
11( ) , ,( )( )
ext ext extλ� ∈ Hol0
.
Applying to D SFg( ) the inverse S -transform we obtain (see (2.9))
( )′ ′D � S D SFg
−1 ( ) =
m
m
mm P F g
=
∞
+∑ + 〈 〉
0
11( ) , ,( )( )
ext ext = ( )( )D F g .
But since ( ) ( )( ) ( )D HF g D∈ ⊂ ′ ′−τ , S D SFg
−1 ( ) is well-defined as an element of
( )H −τ . Hence (2.13) is proved.
Corollary. The operator D is a differentiation with respect to the Wick pro-
duct, i.e., for all F, G ∈ −( )H τ we have
D ( )F G◊ = ( ) ( )D DF G F G◊ + ◊ . (2.14)
Moreover, for each n ∈ +Z , F ∈ −( )H τ , and a holomorphic at ( )( )SF 0 function
h : C → C
D F n◊ = nF Fn◊ − ◊1 ( )D ,
(2.15)
D h F◊( ) = ′ ◊◊h F F( ) ( )D ,
where ′h is a usual derivative of h.
Proof. Using (2.13), for each g ∈H τ,C we obtain
( )( ) ( )D F G g◊ = S D S F Gg
− ◊1 ( )( ) = S D SF SGg
− ⋅1 ( ) =
= S D SF SG SF D SGg g
− ⋅ + ⋅1(( ) ( )) = S S F g SG SF S G g− ⋅ + ⋅1( ( ) ( ) )( ) ( )D D =
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GENERALIZED STOCHASTIC DERIVATIVES … 755
= ( ) ( )( ) ( )D DF g G F G g◊ + ◊ ,
i.e., (2.14) is proved. The first formula in (2.15) can be obtained from (2.14) by
induction (the case n = 0 is trivial), the second one is a consequence of the first one.
Finally, let us calculate a commutator between the extended stochastic integral and
the generalized stochastic derivative (known as a fundamental theorem of the Malliavin
calculus, cf. [30]).
Theorem 2.4. Let F ∈ −( )H Hτ � C . Then ∀ ∈ + ∞t [ , ]0
D F s dMs
t
( ) ˆ ( )
0
∫
� =
( ( ))( ) ˆ ( ) ( ) ( )D F s dM F s s dss
t t
� �
0 0
∫ ∫+ σ . (2.16)
Proof. By definition,
F s dMs
t
( ) ˆ
0
∫ = 〈 〉+
=
∞
∑ P Fn t
n
n
1 0
0
, ˆ ( )
ext, [ , ) ,
where ˆ ( )
– ,
( )F t
n n
ext, [ , )0
1∈ +H τ C
, n ∈ +Z , are constructed in (1.20) starting from the kernels
F n n
ext, ⋅ ∈( )
– ,
( )H Hτ C C� from decomposition (1.19) for F. Further,
D F s dMs
t
( ) ˆ ( )
0
∫
� =
( ) , ˆ ,( )( )n P Fn t
n
n
+ 〈 〉
=
∞
∑ 1 0
0
ext, [ , ) ext� .
On the other hand,
( ( ))( )D F ⋅ � = n P Fn
n
n
〈 〉− ⋅
=
∞
∑ 1
1
, ,( )( )
ext, ext� ,
( ( ))( ) ˆD F s dMs
t
�
0
∫ =
%
n P Fn
n
t
n
〈 〉⋅
=
∞
∑ , ,( )( )
ext, ext, [ , )� 0
1
,
F s s ds
t
( ) ( ) ( )� σ
0
∫ =
n
n
t
s
nP F s ds
=
∞
∑ ∫
0 0
, ( ) ( )( )
ext, � σ .
Therefore, it is sufficient to prove that for all n ∈ +Z
( ) ˆ ,( )( )n F t
n+ 1 0ext, [ , ) ext� =
%
n F F s dsn
t s
n
t
( )( ) ( ), ( ) ( )ext, ext, [ , ) ext,⋅ + ∫� �0
0
σ .
For n = 0 this equality is obviously true. Let n ∈N . It is sufficient to verify that for
all f n n( )
,
ˆ
∈H τ C
� and for all g ∈H τ,C
( ) ˆ , ,( )( ) ( )n F g ft
n n+ 〈 〉1 0ext, [ , ) ext ext =
%
n F g fn
t
n〈 〉⋅( )( ) ( ), ,ext, ext, [ , ) ext0 +
+
0
t
s
n nF g s ds f∫ ext,
ext
( ) ( )( ) ( ),σ . (2.17)
Using (1.20) we can rewrite the left-hand side of (2.17) as follows:
( ) ˆ , ,( )( ) ( )n F g ft
n n+ 〈 〉1 0ext, [ , ) ext ext = ( ) ˆ , ˆ( ) ( )n F g ft
n n+ 〈 〉1 0ext, [ , ) ext� =
= ( ) ( ) , ˆ( )( ) ( )n U F g fn
n
t
n+ ⋅〈 〉⋅1 1 0Pr ext, [ , ) � = ( ) ( ), ˆ( ) ( )n U F g fn
n
t
n+ ⋅〈 〉⋅1 1 0ext, [ , ) � =
=
( ) , ˆ ( ) ( )( ) ( )( )n U F g f s ds
t
n s
n n+ ∫ 〈 〉1
0
ext, � σ =
0
1
t
n s
n n
nU F f g s∫ 〈 ⋅ … ⋅ext,
( ) ( ), ( , , ) ( ) +
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
756 N. A. KACHANOVSKY
+ f s g f s g dsn n
n n
( ) ( )( , , ) ( ) ( , , ) ( ) ( )⋅ … ⋅ + … + … ⋅ ⋅− 〉2 1 1 σ =
=
0
t
n s
n nU F f g s ds∫ 〈 〉ext,
( ) ( ), ( ) ( )σ +
+
0
2 1 1
t
n s
n n n
n nU F f s g f s g ds∫ 〈 〉⋅ … ⋅ + … + … ⋅ ⋅−ext,
( ) ( ) ( ), ( , , ) ( ) ( , , ) ( ) ( )σ .
In the right-hand side of (2.17) we have
0
t
s
n nF g s ds f∫ ext,
ext
( ) ( )( ) ( ),σ =
0
t
s
n nF f g s ds∫ 〈 〉ext, ext
( ) ( ), ( ) ( )σ =
=
0
t
n s
n nU F f g s ds∫ 〈 〉ext,
( ) ( ), ( ) ( )σ ,
and by virtue of the symmetry of f n( ) and (1.20)
%
n F g fn
t
n〈 〉⋅( )( ) ( ), ,ext, ext, [ , ) ext0 = n U F g fn
n
t
n〈 〉− ⋅ ⋅Pr ext, ext [ , )( ( ) )( ) ( ), ( ) ,1 01 =
= n U F g fn
n
t
n〈 〉− ⋅ ⋅1 01( )( ) ( ), ( ),ext, ext [ , ) = n U F g f s ds
t
n s
n n
0
1∫ 〈 〉− ( )( ) ( ), , ( ) ( )ext, ext σ =
= n F g f s ds
t
s
n n
0
∫ 〈 〉( )( ) ( ), , ( ) ( )ext, ext ext σ = n F g f s ds
t
s
n n
0
∫ 〈 〉ext, ext
( ) ( ), ˆ ( ) ( )� σ =
= n U F g f s ds
t
n s
n n
0
∫ 〈 〉ext,
( ) ( ), ˆ ( ) ( )� σ =
0
2 1
t
n s
n n
nU F f s g∫ 〈 ⋅ … ⋅ ⋅ext,
( ) ( ), ( , , , ) ( ) +
+ f s g f s g dsn n
n n
( ) ( )( , , , ) ( ) ( , , , ) ( ) ( )⋅ … ⋅ ⋅ + … + ⋅ … ⋅ ⋅− 〉3 1 2 1 1 σ ,
thus (2.17) is proved.
By analogy with [1, 2] as an application of our results we will calculate the genera-
lized stochastic derivative of the solution of the stochastic equation
( )H −τ ' Ft = F h F dMs s
t
0
0
+ ◊∫ ( ) ˆ , (2.18)
where h : C → C is some entire function, F0 ∈C . Under certain conditions on h a
unique solution of (2.18) Ft ∈ −( )H τ exists. Applying D to (2.18) and taking into
account (2.16) and (2.15), for each g ∈H τ,C we obtain
( )( )D F gt =
D h F dM gs s
t
׺
( ) ˆ ( )
0
=
= ′ ◊ +◊ ◊∫ ∫h F F g dM h F g s dss s s
t
s
t
( ) ( )( ) ˆ ( ) ( ) ( )D
0 0
σ . (2.19)
Let φ λ λs
g
sS F g( ) : (( )( ))( )= D . Applying the S-transform to (2.19) and taking into ac-
count (1.22) we obtain
φ λt
g( ) = ′ +∫ ∫h SF s ds h SF g s dss s
g
t
s
t
(( )( )) ( ) ( ) ( ) (( )( )) ( ) ( )λ φ λ λ σ λ σ
0 0
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
GENERALIZED STOCHASTIC DERIVATIVES … 757
The solution of this equation is
φ λt
g( ) = h SF g s h SF u du dss
t
u
s
t
(( )( )) ( ) exp (( )( )) ( ) ( ) ( )λ λ λ σ σ
0
∫ ∫⋅ ′
.
By the inverse S-transform we obtain
( )( )DF gt = h F g s h F dM dss
t
u u
s
t
◊ ◊ ◊∫ ∫◊ ′
( ) ( ) exp (( ) ˆ ( )
0
σ ∈ ( )H −τ .
Remark 2.8. It is not difficult to understand that main results of this paper can be
reformulated “on the language of a so-called Q-system” under the biorthogonal appro-
ach to construction of a non-Gaussian infinite-dimensional analysis (see, e.g., [3, 6 –
10, 17, 18] and references therein) and, therefore, can be applied in a more general case
than the Meixner analysis. This follows from the fact that the Q-system is an orthogo-
nal basis in the spaces ( )H − −τ q and Q Fn
n( )( ) = 〈 〉−P U Fn n
n, ( )1 , see [15].
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758 N. A. KACHANOVSKY
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Received 28.12.06
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 6
|
| id | umjimathkievua-article-3193 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:37:58Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b4/330199decc42d2a1d29b75845d9ee5b4.pdf |
| spelling | umjimathkievua-article-31932020-03-18T19:48:06Z Generalized stochastic derivatives on spaces of nonregular generalized functions of Meixner white noise Узагальнені cтoxacтичнi похідні на пов'язаннх із білим шумом Майкснера просторах нерегулярннх узагальнених функцій Kachanovskii, N. A. Качановський, М. О. We introduce and study generalized stochastic derivatives on Kondratiev-type spaces of nonregular generalized functions of Meixner white noise. Properties of these derivatives are quite analogous to properties of stochastic derivatives in the Gaussian analysis. As an example, we calculate the generalized stochastic derivative of a solution of a stochastic equation with Wick-type nonlinearity. Вводяться та вивчаються узагальнені стохастичні похідні на пов'язаних із білим шумом Майкснера просторах типу Кондратьєва нерегулярних узагальнених функцій. Властивості цих похідних аналогічні властивостям стохастичних похідних у гауссівському аналізі. Як приклад обчислено узагальнену стохастичну похідну розв'язку певного стохастичного рівняння з нелінійністю типу Віка. Institute of Mathematics, NAS of Ukraine 2008-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3193 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 6 (2008); 737–758 Український математичний журнал; Том 60 № 6 (2008); 737–758 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3193/3132 https://umj.imath.kiev.ua/index.php/umj/article/view/3193/3133 Copyright (c) 2008 Kachanovskii N. A. |
| spellingShingle | Kachanovskii, N. A. Качановський, М. О. Generalized stochastic derivatives on spaces of nonregular generalized functions of Meixner white noise |
| title | Generalized stochastic derivatives on spaces of nonregular generalized functions of Meixner white noise |
| title_alt | Узагальнені cтoxacтичнi похідні на пов'язаннх із білим шумом Майкснера просторах нерегулярннх узагальнених функцій |
| title_full | Generalized stochastic derivatives on spaces of nonregular generalized functions of Meixner white noise |
| title_fullStr | Generalized stochastic derivatives on spaces of nonregular generalized functions of Meixner white noise |
| title_full_unstemmed | Generalized stochastic derivatives on spaces of nonregular generalized functions of Meixner white noise |
| title_short | Generalized stochastic derivatives on spaces of nonregular generalized functions of Meixner white noise |
| title_sort | generalized stochastic derivatives on spaces of nonregular generalized functions of meixner white noise |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3193 |
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