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In the paper, perturbed stochastic Volterra Equations with noise terms driven by series of independent scalar Wiener processes are considered. In the study, the resolvent approach to the equations under consideration is used. Sufficient conditions for the existence of strong solution to the class of perturbed stochastic Volterra Equations of convolution type are given. Regularity of stochastic convolution is supplied, as well.

Let

where

The domain

In our work, the Equation (1) is driven by series of scalar Wiener processes;

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 [

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

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

Definition 1 A family

1)

2)

3) the following resolvent equation holds

for all

In this paper, the following result concerning convergence of resolvents for the Equation (1) will play the key role.

As in [

is nonnegative, nonincreasing and convex.

Theorem 1 ( [

for all

Below we give an example illustrating conditions

Example 1 Consider in the Equation (1) the following kernel functions

Then the functions

fulfil conditions

The paper is organized as follows. Section 2 constains the construction of the stochastic integral due to O. van Gaans [

In this section we recall the construction of the stochastic integral due to O. van Gaans [

Definition 2 A function

Definition 3 A function

Theorem 2 ( [

pect to the filtration

acting from

1) For any

where

2) For each

3) For any

Definition 4 By

Theorem 3 ( [

exists in

We begin this section with definitions of solutions to the Equation (1).

Definition 5 An h-valued predictable process

and for any

Let

Definition 6 An H-valued predictable process

and if for all

As we have already written, in the paper we assume that (2) admits a resolvent family

Definition 7 An H-valued predictable process

and, for all

where

We introduce the stochastic convolution

where

Let us formulate some auxiliary results concerning the convolution

Proposition 4 For arbitrary process

Proposition 5 Assume that

For the idea of proofs of Propositions 4 and 5, we refer to [

In some cases, weak solutions of Equation (1) coincides with mild solutions of (1), see e.g. [

Proposition 6 Assume that

Hence, we are able to conclude the following result.

Corollary 1 Let A be a linear bounded operator in H. If

The formula (9) says that the convolution

Here we provide sufficient conditions under which the stochastic convolution

Lemma 1 Assume that

and

Proof Because

exists in

Denote by

Theorem 7 Let A be a closed linear unbounded operator with the dense domain

graph norm

Proof Since closed unbounded linear operator A becomes bounded in

It remains to show that the Equation (1) holds P − a.s., i.e.

Because the formula (9) holds for any bounded operator, then it holds for the Yosida approximation

where

and

To prove that (11) holds, we need to show the following convergences

and

By assumption

deterministic and bounded for any

belongs to

From the definition of stochastic integral (Theorem 3), for

By Theorem 1, the convergence of the resolvent families is uniform with respect to t on every closed intervals, particularly on

Summing up the above considerations, we obtain

as

From the fact that

For any

Then

To prove that the convergence (13) holds, we need to show that

and

We shall study the term

where

Moreover

For any big enough n and any

Next, by Lemma 1 and closedness of the operator A

Analogously, we have

Using (19), we receive

From assumption

Then, using (19) and (12), we obtain (17).

For the term

By assumption

Analogously,

Since

Using the convergence (20), we have

Therefore the convergence (18) holds. □

In this section, we give sufficient conditions for the continuity of trajectories of the stochastic convolution when the kernel function

where

Theorem 8 Let the operator A be the generator of strongly continuous bounded analytic semigroup

dition

where

like in Equation (1).

Proof Since formula (9) holds for any bounded operator, then it holds for the Yosida approximation

where

Denoting

and using the Leibniz rule twice we obtain

From (23) and (24), we can write

If

and next

For simplicity, we introduce the following term

Then

Since

From (23) and (24) we obtain

where

Then

Basing on Theorem 1, properties of Yosida approximation

and

Since the operator A is closed, we can conclude that

Hence, passing to the limit with

where

Lemma 2 Let the assumptions of Theorem 8 hold and

where

and

Moreover,

Proof The formula (26) results from (22) and the definition (27) of the process

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 [

Theorem 9 Suppose that the assumptions of Theorem 8 hold. If both processes Y and

In the previous theorem, the space

We denote by

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

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

Assume that the function

1) function v and its partial derivatives

2) for any

3) for any

Then the stochastic convolution

The idea of the proof bases on Ichikawa’s scheme, see ( [

Anna Karczewska,Bartosz Bandrowski, (2015) A Series Approach to Perturbed Stochastic Volterra Equations of Convolution Type. Advances in Pure Mathematics,05,660-671. doi: 10.4236/apm.2015.511060