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International Journal of Stochastic Analysis

Volume 2012 (2012), Article ID 236327, 24 pages

http://dx.doi.org/10.1155/2012/236327

## Some Refinements of Existence Results for SPDEs Driven by Wiener Processes and Poisson Random Measures

Institut für Mathematische Stochastik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany

Received 4 June 2012; Accepted 6 August 2012

Academic Editor: Hari Mohan Srivástava

Copyright © 2012 Stefan Tappe. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

Semilinear stochastic partial differential equations (SPDEs) on Hilbert spaces, being of the type have widely been studied in the literature, see, for example, [1–4]. In (1.1), denotes the generator of a strongly continuous semigroup, and is a trace class Wiener process. In view of applications, this framework has been extended by adding jumps to the SPDE (1.1). More precisely, consider an SPDE of the type where denotes a Poisson random measure on some mark space with being its compensator. SPDEs of this type have been investigated in [5, 6], see also [7–12], where SPDEs with jump noises have been studied.

The goal of the present paper is to extend results and methods for SPDEs of the type (1.2) in the following directions.(i) We consider more general SPDEs of the form where is a set with . Then, the integral represents the small jumps, and represents the large jumps of the solution process. Similar SDEs have been considered in finite dimension in [13, Section II.2.c] and in infinite dimension in [14].(ii) We will prove the following results (see Theorem 4.5) concerning existence and uniqueness of local and global mild solutions to (1.3):(1)if are locally Lipschitz and of linear growth, then existence and uniqueness of global mild solutions to (1.3) hold;(2)if are locally Lipschitz and locally bounded, then existence and uniqueness of local mild solutions to (1.3) hold;(3)if are locally Lipschitz, then uniqueness of mild solutions to (1.3) holds.In particular, the result that local Lipschitz and linear growth conditions ensure existence and uniqueness of global mild solutions does not seem to be well known for SPDEs, as most of the mentioned references impose global Lipschitz conditions. An exception is [3], where the author treats Wiener process-driven SPDEs of the type (1.1), even on 2-smooth Banach spaces, and provides existence and uniqueness under local Lipschitz and linear growth conditions. In [3], the crucial assumption on the operator is that it generates an analytic semigroup, while our results hold true for every pseudocontractive semigroup.(iii) 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 [6]. As a direct consequence, we obtain that any mild solution to (1.3) is càdlàg. As just mentioned, we will utilize the “method of the moving frame” from [6], which allows us to reduce the SPDE problems to SDE problems. Therefore, we will be concerned with SDEs in Hilbert spaces being of the type By using the technique of interlacing solutions at jump times (which, in particular cases has been applied, e.g., in [15, Section 6.2] and [10, Section 9.7]), we can reduce the SDE (1.4) to SDEs of the form without large jumps, and for those SDEs, suitable techniques and results are available in the literature. This allows us to derive existence and uniqueness results for the SDE (1.4), which are subject to the regularity conditions described above. We point out that [14] also studies Hilbert space-valued SDEs of the type (1.4) and provides an existence and uniqueness result considerably going beyond the classical results which impose global Lipschitz conditions. In Section 3.3, we provide a comparison of our existence and uniqueness result for SDEs of the type (1.4) with that from [14].

The remainder of this paper is organized as follows: in Section 2, we provide the required preliminaries and notation. In Section 3, we prove existence and uniqueness results for (local) strong solutions to SDEs of the form (1.4), and in Section 4, we prove existence and uniqueness results for (local) mild solutions to SPDEs of the form (1.3) by using the “method of the moving frame.”

#### 2. Preliminaries and Notation

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

Throughout this text, let with be a filtered probability space satisfying the usual conditions.

Let be a separable Hilbert space, and let be a nuclear, self-adjoint, positive definite linear operator. Then, there exist an orthonormal basis of and a sequence with such that namely, the are the eigenvalues of , and each is an eigenvector corresponding to . The space , equipped with the inner product is another separable Hilbert space, and is an orthonormal basis. Let be a -valued -Wiener process, see [1, page 86, 87]. For another separable Hilbert space , we denote by the space of Hilbert-Schmidt operators from into , which, endowed with the Hilbert-Schmidt norm itself is a separable Hilbert space.

Let be a measurable space which we assume to be a Blackwell space (see [16, 17]). We remark that every Polish space with its Borel -field is a Blackwell space. Furthermore, let be a time-homogeneous Poisson random measure on , see [13, Definition II.1.20]. Then its compensator is of the form , where is a -finite measure on .

For the following definitions, let be a finite stopping time.(i)We define the new filtration by (ii)We define the new -valued process by (iii)We define the new random measure on by where we use the notation

Then, is an -adapted -Wiener process, and is a time-homogeneous Poisson random measure relative to the filtration with compensator , cf. [18, Lemma 4.6].

Lemma 2.1. *Let be another stopping time. Then, the mapping is an -stopping time.*

*Proof. *For every , we have
showing that is an -stopping time.

Denoting by the predictable -algebra relative to the filtration , we have the following auxiliary result.

Lemma 2.2. *The following statements are true:*(1)*the mapping
is --measurable.*(2)*the mapping
is --measurable.*

*Proof. *According to [13, Theorem I.2.2], the system of sets
is a generating system of the predictable -algebra . For any set , we have
Furthermore, for any -stopping time , we have
where, in the last step, we have used Lemma 2.1. This proves the first statement.

According to [13, Theorem I.2.2], the system of sets
is a generating system of the predictable -algebra . For any set , we have
Furthermore, for all with and , we have
establishing the second statement.

Let us further investigate the Poisson random measure . According to [13, Proposition II.1.14], there exist a sequence of finite stopping times with for and an -valued optional process such that for every optional process , where denotes a separable Hilbert space, and all with we have Let be a set with . We define the mappings , as

Lemma 2.3. *The following statements are true:*(1)*for each , the mapping is a finite stopping time,*(2)*one has and for all ,*(3)*one has .*

*Proof. *This follows from [19, Lemma A.19].

#### 3. Existence and Uniqueness of Strong Solutions to Hilbert Space-Valued SDEs

In this section, we establish existence and uniqueness of (local) strong solutions to Hilbert space-valued SDEs of the type (1.4).

Let be a separable Hilbert space, and let be a set with . Furthermore, let and be -measurable mappings, and let be a -measurable mapping.

*Definition 3.1. *One says that existence of (local) strong solutions to (1.4) holds, if for each -measurable random variable , there exists a (local) strong solution to (1.4) with initial condition (and some strictly positive lifetime ).

*Definition 3.2. *One says that uniqueness of (local) strong solutions to (1.4) holds, if for two (local) strong solutions to (1.4) with initial conditions and (and lifetimes and ) we have up to indistinguishability

Note that uniqueness of local strong solutions to (1.4) implies uniqueness of strong solutions to (1.4). This is seen by setting and .

*Definition 3.3. *One says that the mappings are locally Lipschitz if -almost surely
and for each , there is a nondecreasing function such that -almost surely
for all and all with .

*Definition 3.4. *One says that the mappings satisfy the linear growth condition if there exists a nondecreasing function such that -almost surely
for all and all .

*Definition 3.5. *One says that the mappings are locally bounded if for each , there is a nondecreasing function such that -almost surely
for all and all with .

For a finite stopping time and a set , we define the mappings , , and as By Lemma 2.2, the mappings and are -measurable, and is -measurable. We will also use the notation

Lemma 3.6. *Suppose that is bounded. Then, the following statements are true:*(1)*if are locally Lipschitz, then are locally Lipschitz, too;*(2)*if satisfy the linear growth condition, then satisfy the linear growth condition, too.*

*Proof. *Suppose that satisfy the linear growth condition. Since is bounded, there exists a constant such that . The mapping is nondecreasing, and we have -almost surely
for all and . Analogous estimates for and prove that satisfy the linear growth condition, too. The remaining statement is proven analogously.

Lemma 3.7. *Let and be two finite stopping times, and let be a set with . If is an -adapted local strong solution to (1.4) with lifetime , then
**
is an -adapted local strong solution to (1.4) with parameters
**
initial condition , and lifetime .*

*Proof. *The process given by (3.11) is adapted, and we have
Therefore, we obtain
Taking into account the Definitions (3.6)–(3.8) of , , and the Definition (3.11) of , it follows that
Consequently, is a local strong solution to (1.4) with parameters (3.12), initial condition , and lifetime .

Lemma 3.8. *Let be two finite stopping times. If is an -adapted local strong solution to (1.4) with lifetime , and is an -adapted local strong solution to (1.4) with parameters
**
initial condition , and lifetime , then
**
is an -adapted local strong solution to (1.4) with lifetime .*

*Proof. *Let be arbitrary. Then, the random variable is -measurable. Let be an arbitrary Borel set. We define as
According to Lemma 2.1, the mapping is an -stopping time. Therefore, we get
and hence, we obtain
showing that the process defined in (3.17) is adapted. Moreover, since is local strong solution to (1.4) with initial condition and lifetime , we have
By the Definitions (3.6)–(3.9) of , , , we obtain
Therefore, we get
By the Definition (3.17) of , we obtain
Since is a local strong solution to (1.4) with lifetime , we deduce that the process given by (3.17) is a local strong solution to (1.4) with lifetime .

Let be arbitrary. By Lemmas 2.1 and 2.3, the mapping is a strictly positive -stopping time. Furthermore, let be arbitrary, and let be an arbitrary -measurable random variable.

Lemma 3.9. *If is an -adapted local strong solution to (1.4) with parameters
**
initial condition , and lifetime , then
**
is a -adapted local strong solution to (1.5) with parameters (3.25), initial condition , and lifetime .*

*Proof. *We define as
and the stochastic process as . By Lemma 2.2, the mapping is -measurable. Let be an arbitrary Borel set. We define as
Then, for each , we have
Consequently, the process defined in (3.26) is -adapted. Furthermore, by the Definition (3.26), we have
and, by the Definition (3.8) of and identity (2.18), we obtain
showing that is a local strong solution to (1.5) with parameters (3.25) and lifetime .

Lemma 3.10. *If is an -adapted local strong solution to (1.5) with parameters (3.25), initial condition , and lifetime , then
**
is a -adapted local strong solution to (1.4) with parameters (3.25), initial condition , and lifetime .*

*Proof. *The proof is analogous to that of Lemma 3.9.

##### 3.1. Uniqueness of Strong Solutions to Hilbert Space-Valued SDEs

Now, we will deal with the uniqueness of strong solutions to the SDE (1.4).

Proposition 3.11. *One supposes that the mappings are locally Lipschitz. Then, uniqueness of local strong solutions to (1.5) holds.*

*Proof. *We can adopt a standard technique (see, e.g., the proof of Theorem 5.2.5 in [20]), where we apply the Itô isometry and Gronwall’s lemma.

Theorem 3.12. *One supposes that the mappings are locally Lipschitz. Then, uniqueness of local strong solutions to (1.4) holds.*

*Proof. *Let and be two local strong solutions to (1.5) with initial conditions and , and lifetimes and . By induction, we will prove that up to indistinguishability
The identity (3.33) holds true for , because by Lemma 2.3, we have .

For the induction step , we suppose that identity (3.33) is satisfied. We define the stopping time and the set . By Lemma 3.7, the processes and defined according to (3.11) are -adapted local strong solutions to (1.4) with parameters (3.12), where and , initial conditions and , and lifetime .

Let be arbitrary, and set . The processes and are -adapted local strong solutions to (1.4) with parameters (3.25), where , initial conditions and , and lifetime . By Lemma 3.9, the processes and defined according to (3.26) are -adapted local strong solutions to (1.5) with parameters (3.25), where , initial conditions and , and lifetime . According to Lemma 3.6, the mappings are locally Lipschitz, too. Therefore, by Proposition 3.11, we have up to indistinguishability
By the Definition (3.26), we deduce that up to indistinguishability
and hence, we have up to indistinguishability
By Lemma 2.3, we have , and hence, we get up to indistinguishability
Therefore, we have up to indistinguishability
Consequently, we have up to indistinguishability
Together with the induction hypothesis, it follows that
which establishes (3.33). Since by Lemma 2.3, that we have , we deduce
completing the proof.

##### 3.2. Existence of Strong Solutions to Hilbert Space-Valued SDEs

Now, we will deal with the existence of strong solutions to the SDE (1.4).

Proposition 3.13. *One supposes that the mappings are locally Lipschitz and satisfy the linear growth condition. Then, existence of strong solutions to (1.5) holds.*

*Proof. *If the mappings are Lipschitz continuous, then we have existence and uniqueness of strong solutions to (1.5) for every initial condition , see, for example, [6, Corollary 10.3].

For being locally Lipschitz and satisfying the linear growth condition, for any initial condition , we adopt the technique from the proof of [21, Theorem 4.11]. For , we define the retraction
and the mappings , , and as
These mappings are Lipschitz continuous, and hence, there exists a strong solution to the SDE (1.5) with parameters , , and , and initial condition . Using the linear growth condition, Gronwall’s lemma, and Doob’s martingale inequality, we can show that , where
that is, the solutions do not explode. Consequently, the process
is a strong solution to (1.5) with initial condition .

Finally, for a general -measurable initial condition , the process is a strong solution to (1.5) with initial condition , where denotes the partition of given by , and where for each the process denotes a strong solution to (1.5) with initial condition .

Theorem 3.14. *One supposes that the mappings are locally Lipschitz and satisfy the linear growth condition. Then, existence of strong solutions to (1.4) holds.*

*Proof. *Let be an arbitrary -measurable random variable. By induction, we will prove that for each there exists a local strong solution to (1.4) with initial condition and lifetime . By Lemma 2.3, we have , providing the assertion for .

For the induction step , let be a local strong solution to (1.4) with initial condition and lifetime . Let be arbitrary, and set . By Lemma 3.6, the mappings are locally Lipschitz, too. Therefore, by Proposition 3.13, there exists an -adapted strong solution to (1.5) with parameters (3.25), where , and initial condition . By Lemma 3.10, the process defined according to (3.32) is an -adapted local strong solution to (1.4) with parameters (3.25), where , initial condition , and lifetime . Noting that is a partition of , it follows that is an -adapted local strong solution to (1.4) with initial condition and lifetime . By Lemma 3.8, the process
defined according to (3.17) is an -adapted local strong solution to (1.4) with initial condition and lifetime .

Consequently, for each , there exists a local strong solution to (1.4) with initial condition and lifetime . By Lemma 2.3, we have . Hence, it follows that
is an -adapted strong solution to (1.4) with initial condition .

Theorem 3.15. *One supposes that the mappings are locally Lipschitz and locally bounded. Then, existence of local strong solutions to (1.4) holds.*

*Proof. *Let be an arbitrary -measurable random variable. We define the partition of by . Furthermore, for each , we define the mappings , , and as in the proof of Proposition 3.13. These mappings are locally Lipschitz and satisfy the linear growth condition. By Theorem 3.14, there exists a strong solution to (1.4) with parameters , , and , and initial condition . The stopping time
is strictly positive, and is a local strong solution to (1.4) with initial condition and lifetime . The stopping time is strictly positive, and the process is a local strong solution to (1.4) with initial condition and lifetime .

##### 3.3. Comparison with the Method of Successive Approximations

So far, our investigations provide the following result concerning existence and uniqueness of global strong solutions to the SDE (1.4).

Theorem 3.16. *If are locally Lipschitz and satisfy the linear growth condition, then existence and uniqueness of strong solutions to (1.4) hold.*

*Proof. *This is a direct consequence of Theorems 3.12 and 3.14.

Now, we will provide a comparison with [14], where the authors also study Hilbert space-valued SDEs of the type (1.4). Their result [14, Theorem 2.1] is based on the method of successive approximations (see also [22, 23]) and considerably goes beyond the classical global Lipschitz conditions. For the sake of simplicity, let us recall the required assumptions in the time-homogeneous Markovian framework. In order to apply [14, Theorem 2.1], for some constant , we need the estimate where denotes a continuous, nondecreasing function with , and further conditions, which are precisely stated in [14], must be fulfilled. These conditions are satisfied if is a continuous, nondecreasing, and concave function such that In particular, we may choose for , and consequently, both results, Theorem 3.16 and [14, Theorem 2.1], cover the classical situation, where global Lipschitz conditions are imposed.

However, there are situations where [14, Theorem 2.1] can be applied, while Theorem 3.16 does not apply, and vice versa. For the sake of simplicity, in the following two examples, we assume that and .

*Example 3.17. *We fix an arbitrary constant and define the functions by
as well as
compare with [22, Remark 1]. Let be a mapping such that
Then we have the estimate
showing that condition (3.49) with is satisfied. Moreover, is a continuous, nondecreasing, concave function, and condition (3.50) is satisfied, because for each , we have
Consequently, [14, Theorem 2.1] applies. However, we have
and thus, . Therefore, the mapping might fail to be locally Lipschitz, and hence, Theorem 3.16 does not apply.

*Example 3.18. *Let us define the mapping as follows. For , we define on the interval by
This defines the mapping , which we extend to a mapping by symmetry
Then, is locally Lipschitz and satisfies the linear growth condition, and hence, Theorem 3.16 applies. However, there are no constant and no continuous, nondecreasing function with such that
Suppose, on the contrary, that there exists a continuous, nondecreasing function with fulfilling (3.59). Then we have
Indeed, let be arbitrary. Then, there exists with . Moreover, by the definition of the mapping , there are such that
Therefore, using the monotonicity of and (3.59), we obtain
showing (3.60). Now, the continuity of yields the contradiction . Consequently, condition (3.49) is not satisfied, and thus, we cannot use [14, Theorem 2.1] in this case.

#### 4. Existence and Uniqueness of Mild Solutions to Hilbert Space-Valued SPDEs

In this section, we establish existence and uniqueness of (local) mild solutions to Hilbert space-valued SPDEs of the type (1.3).

Let be a separable Hilbert space, let be a -semigroup on with infinitesimal generator , and let be a set with . Furthermore, let and be -measurable mappings, and let be a -measurable mapping.

Throughout this section, we suppose that there exist another separable Hilbert space , a -group on , and continuous linear operators , such that the diagram (4.1) commutes for every , that is,

*Remark 4.1. *According to [6, Proposition 8.7], this assumption is satisfied if the semigroup is pseudocontractive (one also uses the notion quasicontractive), that is, there is a constant such that
This result relies on the Szökefalvi-Nagy theorem on unitary dilations (see, e.g., [24, Theorem I.8.1] or [25, Section 7.2]). In the spirit of [24], the group is called a *dilation* of the semigroup .

*Remark 4.2. *The Szökefalvi-Nagy theorem was also utilized in [26, 27] in order to establish results concerning stochastic convolution integrals.

Now, we define the mappings , , and by Note that and are -measurable and that is -measurable.

Lemma 4.3. *The following statements are true:*(1)*if are locally Lipschitz, then are locally Lipschitz, too;*(2)*if satisfy the linear growth condition, then satisfy the linear growth condition, too;*(3)*if are locally bounded, then are locally bounded, too.*

*Proof. *All three statements are straightforward to check.

Proposition 4.4. *Let be a -measurable random variable, and let be a stopping time. Then, the following statements are true:*(1)if is a local strong solution to (1.4) with initial condition and lifetime , then is a local mild solution to (1.3) with initial condition and lifetime ;(2)if is a local mild solution to (1.3) with initial condition and lifetime , then the process defined as
is a local strong solution to (1.4) with initial condition and lifetime , and one has .

*Proof. *Let be a local strong solution to (1.4) with initial condition and lifetime . Then we have
By the Definitions (4.4) of , we obtain
Therefore, by (4.2), and since , we arrive at
showing that is a local mild solution to (1.3) with initial condition and lifetime . This establishes the first statement. Now, let be a local mild solution to (1.3) with initial condition and lifetime . Then we have (4.8), and therefore, by (4.2) and the Definition (4.5) of