A Series Approach to Perturbed Stochastic Volterra Equations of Convolution Type ()
1. Introduction
Let
be a seperable Hilbert space and let
denote a probability space. We consider perturbed stochastic Volterra Equations in H of the form
(1)
where
is an H-valued
-measurable random variable, kernels a, k, b are real valued and locally inte- grable functions defined on
and A is a closed unbounded linear operator in H with a dense domain
.
The domain
is equipped with the graph norm
of A, i.e.
.
In our work, the Equation (1) is driven by series of scalar Wiener processes;
and
are appropriate processes defined below.
The goal of this paper is to formulate sufficient conditions for the existence and regularity of strong solutions to the perturbed Volterra Equation driven by series of scalar Wiener processes. Previously, in [1] - [4] , the stochastic integral for Hilbert-Schmidt operator-valued integrands and Wiener processes with values in Hilbert space has been constructed. Moreover, the particular series expansion of the Wiener process with respect to the eigenvectors of its covariance operator has been used. The stochastic integral used in this paper, originally introduced in [5] , bases on the construction directly in terms of the sequence of independent scalar processes. In consequence, the stochastic integral is independent of any covariance operator usually connected with a noise process.
In the paper, we use the resolvent approach to the Equation (1). This means that a deterministic counterpart of the Equation (1), that is, the Equation
(2)
admits a resolvent family. In (2), the operator A and the kernel functions are the same as previously in (1) and f is a H-valued function.
By
, we shall denote the family of resolvent operators corresponding to the Volterra Equation (2), which is defined as follows.
Definition 1 A family
of bounded linear operators in H is called resolvent for (2), if the following conditions are satisfied:
1)
is strongly continuous on
and
;
2)
commutes with the operator A, that is,
and
for all
,
;
3) the following resolvent equation holds
(3)
for all
.
In this paper, the following result concerning convergence of resolvents for the Equation (1) will play the key role.
As in [6] , we shall assume the following hypotheses:
The solution of the Equation
![]()
is nonnegative, nonincreasing and convex.
The solution of the Equation
is differentiable.
Theorem 1 ( [6] , Th.~3.5) Assume that A is the generator of bounded analytic semigroup of H. Suppose that the hypotheses
and
are satisfied. Then the Equation (2) admits a resolvent
. Addi- tionally, there exists bounded operators
and corresponding resolvent families
satisfying
for all
, such that
(4)
for all
. Moreover, the convergence (4) is uniform in t on every compact subset of
.
Below we give an example illustrating conditions
and
.
Example 1 Consider in the Equation (1) the following kernel functions
![]()
Then the functions
![]()
fulfil conditions
and
.
The paper is organized as follows. Section 2 constains the construction of the stochastic integral due to O. van Gaans [5] . In Section 3, we compare mild and weak solutions and then we provide sufficient conditions for stochastic convolution to be a strong solution to the Equation (1). Section 4 gives regularity of stochastic con- volution arising in perturbed Volterra Equation while in Section 5 we derive the analogue of Itô formula to the perturbed Volterra Equation.
2. The Stochastic Integral
In this section we recall the construction of the stochastic integral due to O. van Gaans [5] .
Definition 2 A function
is called piecewise uniformly continuous (PUC) if there are
such that f is uniformly continuous on
for each
.
Definition 3 A function
is called piecewise uniformly continuous (PUC), if
is uniformly continuous for all
.
Theorem 2 ( [5] ) Assume that
is a series of independent standard scalar Wiener processes with res-
pect to the filtration
in
. Let
be a series of piecewise uniformly continuous functions (PUC)
acting from
into
, adapted with respect to the filtration
. Then the following results hold.
1) For any
, the integral
is well-defined as the limit of Riemann sums
of the form
![]()
where
.
2) For each
, the Itô isometry holds
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3) For any
, such that
we have
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Definition 4 By
we shall denote the space of series
of piecewise uni- formly continuous functions (PUC) acting from
into
, adapted with respect to the filtration
, such that
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Theorem 3 ( [5] ) Assume that
is a series of independent standard scalar Wiener processes with respect to the filtration
in
. Let
. Then the integral
![]()
exists in
and
![]()
3. The Main Results
We begin this section with definitions of solutions to the Equation (1).
Definition 5 An h-valued predictable process
, is said to be a strong solution to (1), if
has a version such that
for almost all
; for any ![]()
(5)
(6)
and for any
the Equation (1) holds P − a.s.
Let
denote the adjoint of A with a dense domain
and the graph norm
.
Definition 6 An H-valued predictable process
, is said to be a weak solution to (1), if
![]()
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and if for all
and all
the following equation holds
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As we have already written, in the paper we assume that (2) admits a resolvent family
,
. So, we can introduce the following idea.
Definition 7 An H-valued predictable process
, is said to be a mild solution to the pertur- bed stochastic Volterra Equation (1), if
![]()
and, for all
,
(7)
where
is the resolvent for the deterministic perturbed Volterra Equation (2).
We introduce the stochastic convolution
(8)
where
and the resolvent operators
, are the same as above.
Let us formulate some auxiliary results concerning the convolution
.
Proposition 4 For arbitrary process
, the process
,
, given by (8) has a predictable version.
Proposition 5 Assume that
. Then the process
,
, defined by (8) has square integrable trajectories.
For the idea of proofs of Propositions 4 and 5, we refer to [2] or [3] .
In some cases, weak solutions of Equation (1) coincides with mild solutions of (1), see e.g. [2] or [3] . In con- sequence, having results for the convolution (8) we obtain results for weak solutions.
Proposition 6 Assume that
. Then the stochastic convolution
,
, is a weak solution to the Equation (1).
Hence, we are able to conclude the following result.
Corollary 1 Let A be a linear bounded operator in H. If
then
(9)
The formula (9) says that the convolution
is a strong solution to (1) if the operator A is bounded.
Here we provide sufficient conditions under which the stochastic convolution
,
, defined by (8) is a strong solution to the Equation (1).
Lemma 1 Assume that
and
. Then
(10)
and
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Proof Because
is a Hilbert space, then the integral
![]()
exists in
by Theorem 3.
Denote by
division of the interval
, that
,
. From the definition of the integral and closedness of the operator A we have
□
Theorem 7 Let A be a closed linear unbounded operator with the dense domain
equipped with the
graph norm
. Suppose that assumptions of Theorem 1 hold. If
and
, then the stochastic convolution
is a strong solution to the perturbed sto- chastic Volterra Equation (1).
Proof Since closed unbounded linear operator A becomes bounded in
, we have
Then from the properties of stochastic convolution we obtain integrability of
and
. Therefore, conditions (5) and (6) from the definition of strong solution to Equation (1) hold.
It remains to show that the Equation (1) holds P − a.s., i.e.
(11)
Because the formula (9) holds for any bounded operator, then it holds for the Yosida approximation
of the operator A, too. Then we have
![]()
where
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and
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To prove that (11) holds, we need to show the following convergences
(12)
and
(13)
By assumption
. Because the operators
are
deterministic and bounded for any
,
, then
belong to
, too. Hence, the difference
(14)
belongs to
for every
and
. This means that
(15)
From the definition of stochastic integral (Theorem 3), for
, we have
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By Theorem 1, the convergence of the resolvent families is uniform with respect to t on every closed intervals, particularly on
. Then we have
(16)
Summing up the above considerations, we obtain
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as
. Then, by the dominated convergence theorem the convergence (12) holds.
From the fact that
and
we have
. Then, by Lemma 1,
.
For any
,
, we can write
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Then
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To prove that the convergence (13) holds, we need to show that
(17)
and
(18)
We shall study the term
first. Because the operator A generates a semigroup, we can use the following property of the Yosida approximation
(19)
where
for any
,
.
Moreover
(20)
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For any big enough n and any
, we have
.
Next, by Lemma 1 and closedness of the operator A
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Analogously, we have
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Using (19), we receive
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From assumption
, so the term
may be treated like the difference
defined by (14).
Then, using (19) and (12), we obtain (17).
For the term
we can repeat the proof of the convergence (12).
![]()
By assumption
. Because
and
, are bounded, so
.
Analogously,
.
Since
, this term can be treated like the difference
de- fined by (14). Hence, for any
, we may write
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Using the convergence (20), we have
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Therefore the convergence (18) holds. □
4. Continuity of Trajectories
In this section, we give sufficient conditions for the continuity of trajectories of the stochastic convolution when the kernel function
. Thus, we study the stochastic convolution corresponding to the equation
(21)
where
.
Theorem 8 Let the operator A be the generator of strongly continuous bounded analytic semigroup
,
. Assume that the functions
,
are the skalar kernel functions and con-
dition
holds. If
, then the following formula holds
(22)
where
,
is a constant and
![]()
like in Equation (1).
Proof Since formula (9) holds for any bounded operator, then it holds for the Yosida approximation
of operator A as well, that is
(23)
where
![]()
Denoting
(24)
and using the Leibniz rule twice we obtain
(25)
From (23) and (24), we can write
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If
, then from (25) we have
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and next
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For simplicity, we introduce the following term
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Then
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Since
, we can write
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From (23) and (24) we obtain
![]()
where
.
Then
![]()
Basing on Theorem 1, properties of Yosida approximation
of the operator A, and dominated convergence theorem, we have
![]()
![]()
![]()
and
![]()
Since the operator A is closed, we can conclude that
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Hence, passing to the limit with
, we obtain
![]()
where
□
Lemma 2 Let the assumptions of Theorem 8 hold and
. Then for
,
(26)
where
(27)
and
![]()
Moreover,
, P − a.s. and
(28)
Proof The formula (26) results from (22) and the definition (27) of the process
. Moreover, from properties of convolution
, P − a.s.
Using the Leibniz rule and property of semigroup we obtain
![]()
where
. □
To conclude continuity of trajectories of stochastic convolution, we use regularity of solutions for the non- homogeneous Cauchy problem [7] to the formula (26).
Theorem 9 Suppose that the assumptions of Theorem 8 hold. If both processes Y and
,
, have continuous trajectories in the space
, then the stochastic convolution
,
, has continuous trajectories in
.
In the previous theorem, the space
,
, is defined as follows. For any
, we set
![]()
We denote by
the Banach space of all
such that
, endowed with the norm
. By the interpolation theory,
is an invariant space of
,
, and the restriction of
to
generates a
-semigroup in
.
5. Analogue of the Itô Formula
In this section, we derive the analogue of the Itô formula to the perturbed Volterra Equation (1).
Proposition 10 Let the process X be a strong solution to the Equation (1) and
. Suppose that the function
and its partial derivatives
,
,
are uniformly continu- ous on
. Then, for any
,
![]()
The following proposition is an example of application of the above analogue of the It formula.
Proposition 11 Let the operator A be the generator of bounded analytic semigroup in H. Suppose that
, the conditions
and
hold and
.
Assume that the function
satisfies the following conditions:
1) function v and its partial derivatives
,
are uniformly continuous on bounded subsets of H,
2) for any
and the constant ![]()
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3) for any
and the constant ![]()
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Then the stochastic convolution
satisfies the following inequality
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The idea of the proof bases on Ichikawa’s scheme, see ( [8] , Theorem 3.1), and on Theorem 1 and Proposition 10. It seems to be a good starting point in the study of stability of mild solution to perturbed Volterra Equation (1).