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We provide existence and uniqueness of global (and local) mild solutions for a general class of semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures under local Lipschitz and linear growth (or local boundedness, resp.) conditions. The so-called “method of the moving frame” allows us to reduce the SPDE problems to SDE problems.

Semilinear stochastic partial differential equations (SPDEs) on Hilbert spaces, being of the type

The goal of the present paper is to extend results and methods for SPDEs of the type (

We consider more general SPDEs of the form

We will prove the following results (see Theorem

if

if

if

We reduce the proofs of these SPDE results to the analysis of SDE problems. This is due to the “method of the moving frame”, which has been presented in [

The remainder of this paper is organized as follows: in Section

In this section, we provide the required preliminary results and some basic notation.

Throughout this text, let

Let

Let

For the following definitions, let

We define the new filtration

We define the new

We define the new random measure

Then,

Let

For every

Denoting by

The following statements are true:

the mapping

the mapping

According to [

According to [

Let us further investigate the Poisson random measure

The following statements are true:

for each

one has

one has

This follows from [

In this section, we establish existence and uniqueness of (local) strong solutions to Hilbert space-valued SDEs of the type (

Let

One says that existence of (local) strong solutions to (

One says that uniqueness of (local) strong solutions to (

Note that uniqueness of local strong solutions to (

One says that the mappings

One says that the mappings

One says that the mappings

For a finite stopping time

Suppose that

if

if

Suppose that

Let

The process

Let

Let

Let

If

We define

If

The proof is analogous to that of Lemma

Now, we will deal with the uniqueness of strong solutions to the SDE (

One supposes that the mappings

We can adopt a standard technique (see, e.g., the proof of Theorem 5.2.5 in [

One supposes that the mappings

Let

For the induction step

Let

Now, we will deal with the existence of strong solutions to the SDE (

One supposes that the mappings

If the mappings

For

Finally, for a general

One supposes that the mappings

Let

For the induction step

Consequently, for each

One supposes that the mappings

Let

So far, our investigations provide the following result concerning existence and uniqueness of global strong solutions to the SDE (

If

This is a direct consequence of Theorems

Now, we will provide a comparison with [

However, there are situations where [

We fix an arbitrary constant

Let us define the mapping

In this section, we establish existence and uniqueness of (local) mild solutions to Hilbert space-valued SPDEs of the type (

Let

Throughout this section, we suppose that there exist another separable Hilbert space

According to [

The Szökefalvi-Nagy theorem was also utilized in [

Now, we define the mappings

The following statements are true:

if

if

if

All three statements are straightforward to check.

Let

if

if

Let

The following statements are true:

if

if

if

Suppose that

Now, we suppose that

If

The structure

As pointed out in [

The author is grateful to an anonymous referee for valuable comments and suggestions.