Some refinements of existence results for SPDEs driven by Wiener processes and Poisson random measures

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.


Introduction
Semilinear stochastic partial differential equations (SPDEs) on Hilbert spaces, being of the type dZ t = (AZ t + α(t, Z t ))dt + σ(t, Z t )dW t Z 0 = z 0 , (1. 1) have widely been studied in the literature, see e.g. [4,23,27,10]. In equation (1.1), A denotes the generator of a strongly continuous semigroup, and W 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 a SPDE of the type      dZ t = (AZ t + α(t, Z t ))dt + σ(t, Z t )dW t + E γ(t, Z t− , x)(µ(dt, dx) − F (dx)dt) where µ denotes a Poisson random measure on some mark space (E, E) with dt ⊗ F (dx) being its compensator. SPDEs of this type have been investigated in [20,7], see also [17,18,16,21,1,22], 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: • We consider more general SPDEs of the form where B ∈ E is a set with F (B c ) < ∞. Then, the integral B represents the small jumps, and B c represents the large jumps of the solution process. Similar SDEs have been considered in finite dimension in [14,Sec. II.2.c] and in infinite dimension in [3].
• We will prove the following results (see Theorem 4.5) (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 the reference [27], where the author treats Wiener process driven SPDEs of the type (1.1), even on 2smooth Banach spaces, and provides existence and uniqueness under local Lipschitz and linear growth conditions. In [27], the crucial assumption on the operator A is that is generates an analytic semigroup, while our results hold true for every pseudo-contractive semigroup.
• 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 [7]. As a direct consequence, we obtain that any mild solution to (1.3) is càdlàg.
As just mentioned, we shall utilize the "method of the moving frame" from [7], 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 ( 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 the reference [3] 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 [3].
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".

Preliminaries and notation
In this section, we provide the required preliminary results and some basic notation.
Throughout this text, let (Ω, F, F, P) with F = (F t ) t≥0 be a filtered probability space satisfying the usual conditions. Let U be a separable Hilbert space and let Q ∈ L(U ) be a nuclear, self-adjoint, positive definite linear operator. Then, there exist an orthonormal basis (e j ) j∈N of U and a sequence (λ j ) j∈N ⊂ (0, ∞) with j∈N λ j < ∞ such that Qe j = λ j e j for all j ∈ N, namely, the λ j are the eigenvalues of Q, and each e j is an eigenvector corresponding to λ j . The space U 0 := Q 1/2 (U ), equipped with the inner product is another separable Hilbert space and ( λ j e j ) j∈N is an orthonormal basis. Let W be an U -valued Q-Wiener process, see [4, p. 86, 87]. For another separable Hilbert space H, we denote by L 0 2 (H) := L 2 (U 0 , H) the space of Hilbert-Schmidt operators from U 0 into H, which, endowed with the Hilbert-Schmidt norm itself is a separable Hilbert space.
Let (E, E) be a measurable space which we assume to be a Blackwell space (see [6,11]). We remark that every Polish space with its Borel σ-field is a Blackwell space. Furthermore, let µ be a time-homogeneous Poisson random measure on R + × E, see [14,Def. II.1.20]. Then its compensator is of the form For the following definitions, let τ be a finite stopping time.
• We define the new filtration • We define the new random measure µ (τ ) on R + × E by where we use the notation Then, W (τ ) is a F (τ ) -adapted Q-Wiener process and µ (τ ) is a time-homogeneous Poisson random measure relative to the filtration F (τ ) with compensator dt⊗F (dx), cf. [8, Lemma 4.6].
Proof. For every t ∈ R + we have showing that ( − τ ) + is a F (τ ) -stopping time.
Denoting by P (τ ) the predictable σ-algebra relative to the filtration F (τ ) , we have the following auxiliary result.
(2) The mapping Proof. According to [14, Thm. I.2.2], the system of sets : is a stopping time} is a generating system of the predictable σ-algebra P. For any set A ∈ F 0 we have . Furthermore, for any F-stopping time we have where, in the last step, we have used Lemma 2.1. This proves the first statement.
According to [14, Thm. I.2.2], the system of sets : s < t and A ∈ F s } is a generating system of the predictable σ-algebra P. For any set A ∈ F 0 we have Furthermore, for all s, t ∈ R + with s < t and A ∈ F s we have ]] = ∅ for n = m and an E-valued optional process ξ such that for every optional process γ : Ω × R + × E → H, where H denotes a separable Hilbert space, and all 0 ≤ t ≤ u with The following statements are true: (1) For each k ∈ N the mapping k is a finite stopping time.

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 H be a separable Hilbert space and let B ∈ E be a set with F (B c ) < ∞. Furthermore, let a : Ω × R + × H → H and b : Ω × R + × H → L 0 2 (H) be P ⊗ B(H)measurable mappings, and let c : Ω×R + ×H×E → H be a P ⊗B(H)⊗E-measurable mapping.
3.1. Definition. We say that existence of (local) strong solutions to (1.4) holds, if for each F 0 -measurable random variable y 0 : Ω → H there exists a (local) strong solution to (1.4) with initial condition y 0 (and some strictly positive lifetime τ > 0).

3.2.
Definition. We say that uniqueness of (local) strong solutions to (1.4) holds, if for two (local) strong solutions to (1.4) with initial conditions y 0 and y 0 (and lifetimes τ and τ ) 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 τ := ∞ and τ := ∞.

Definition. We say that the mappings
< ∞ for all t ∈ R + and all y ∈ H, and for each n ∈ N there is a non-decreasing function L n : R + → R + such that P-almost surely for all t ∈ R + and all y 1 , y 2 ∈ H with y 1 , y 2 ≤ n.

Definition.
We say that the mappings (a, b, c| B ) satisfy the linear growth condition, if there exists a non-decreasing function K : for all t ∈ R + and all y ∈ H.

3.5.
Definition. We say that the mappings (a, b, c| B ) are locally bounded, if for each n ∈ N there is a non-decreasing function M n : for all t ∈ R + and all y ∈ H with y ≤ n.
For a finite stopping time τ and a set Γ ∈ F τ we define the mappings a (τ,Γ) : We shall also use the notation 3.6. Lemma. Suppose that τ 1 Γ is bounded. Then, the following statements are true: satisfy the linear growth condition, too.
Proof. Let t ∈ R + be arbitrary. Then, the random variable Y be an arbitrary Borel set. We define D C ∈ F t as According to Lemma 2.1, the mapping (t − τ ) + is a F (τ ) -stopping time. Therefore, we get and hence, we obtain showing that the process Y defined in (3.14) is F-adapted. Moreover, since Y (τ ) is local strong solution to (1.4) .
Therefore, we get By the Definition (3.14) of Y we obtain Since Y (0) is a local strong solution to (1.4) with lifetime τ , we deduce that the process Y given by (3.14) is a local strong solution to (1.4) with lifetime .
Proof. We define J : Ω → H as . By Lemma 2.2, the mapping J is F k+1 -measurable. Let C ∈ B(H) be an arbitrary Borel set. We define D C ∈ F t as Then, for each t ∈ R + we have and, by the Definition (3.9) of c ( k ,Γ) and identity (2.1) we obtain showing that Y ( k ,Γ)− is a local strong solution to (1.5) with parameters (3.15) and lifetime τ ∧ ( k+1 − k ).

3.10.
Lemma. If Y ( k ,Γ)− is a F ( k ) -adapted local strong solution to (1.5) with parameters (3.15), initial condition y ( k ) 0 1 Γ , and lifetime τ , then is a F ( k ) -adapted local strong solution to (1.4) with parameters (3.15), initial condition y Proof. The proof is analogous to that of Lemma 3.9. Proof. Let Y and Y be two local strong solutions to (1.5) with initial conditions y 0 and y 0 , and lifetimes τ and τ . By induction, we will prove that up to indistinguishability The identity (3.18) holds true for k = 0, because by Lemma 2.3 we have 0 = 0. For the induction step k → k + 1 we suppose that identity (3.18) is satisfied. We define the stopping time τ k := τ ∧τ ∧ k+1 and the set Γ k := { k ≤ τ k }∩{y 0 = y 0 } ∈ F k . By Lemma 3.7, the processes (3.11) are F ( k ) -adapted local strong solutions to (1.4) with parameters (3.12), where τ = k and Γ = Γ k , initial conditions Y k 1 Γ k and Y k 1 Γ k , and lifetime (τ k − k ) + . Let n ∈ N be arbitrary and set Γ kn := Γ k ∩ { k ≤ n} ∈ F k . The processes cal strong solutions to (1.4) with parameters (3.15), where Γ = Γ kn , initial conditions Y k 1 Γ kn and Y k 1 Γ kn , and lifetime (τ k − k ) + . By Lemma 3.9, the processes Y ( k ,Γ kn )− and Y ( k ,Γ kn )− defined according to (3.16) are F ( k ) -adapted local strong solutions to (1.5) with parameters (3.15), where Γ = Γ kn , initial conditions Y k 1 Γ kn and Y k 1 Γ kn , and lifetime (τ k − k ) + . According to Lemma 3.6, the mappings (a ( k ,Γ kn ) , b ( k ,Γ kn ) , c ( k ,Γ kn ) | B ) are locally Lipschitz, too. Therefore, by Proposition 3.11 we have up to indistinguishability By the Definition (3.16), we deduce that up to indistinguishability and hence, we have up to indistinguishability By Lemma 2.3 we have P( k < ∞) = 1, 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  For (a, b, c| B ) being locally Lipschitz and satisfying the linear growth condition, for any initial condition y 0 ∈ L 2 (F 0 ; H) we adopt the technique from the proof of [19,Thm. 4.11]. For k ∈ N we define the retraction if y ≤ k, k y y , if y > k, and the mappings a k : Ω × R + × H → H, b k : Ω × R + × H → L 0 2 (H) and c k : These mappings are Lipschitz continuous, and hence there exists a strong solution Y (k) to the SDE (1.5) with parameters a = a k , b = b k and c = c k , and initial condition y 0 . Using the linear growth condition, Gronwall's lemma and Doob's martingale inequality, we can show that P(τ k → ∞) = 1, where i.e. the solutions do not explode. Consequently, the process is a strong solution to (1.5) with initial condition y 0 .
Finally, for a general F 0 -measurable initial condition y 0 : Ω → H, the process Y := k∈N Y (k) 1 Ω k is a strong solution to (1.5) with initial condition y 0 , where (Ω k ) k∈N ⊂ F 0 denotes the partition of Ω given by Ω k := { y 0 ∈ [k − 1, k)}, and where for each k ∈ N the process Y (k) denotes a strong solution to (1.5) with initial condition y 0 1 Ω k .
3.14. Theorem. We suppose that the mappings (a, b, c| B ) are locally Lipschitz and satisfy the linear growth condition. Then, existence of strong solutions to (1.4) holds.
Proof. Let y 0 : Ω → H be an arbitrary F 0 -measurable random variable. By induction, we will prove that for each k ∈ N 0 there exists a local strong solution Y (k) to (1.4) with initial condition y 0 and lifetime k . By Lemma 2.3 we have 0 = 0, providing the assertion for k = 0. For the induction step k → k + 1 let Y (k) be a local strong solution to (1.4) with initial condition y 0 and lifetime k . Let n ∈ N be arbitrary and set Γ kn := { k ∈ [n − 1, n)} ∈ F k . By Lemma 3.6, the mappings (a ( k ,Γ kn ) , b ( k ,Γ kn ) , c ( k ,Γ kn ) | B ) are locally Lipschitz, too. Therefore, by Proposition 3.13 there exists a F ( k ) -adapted strong a solution Y ( k ,Γ kn )− to (1.5) with parameters (3.15), where Γ = Γ kn , and initial condition Y defined according to (3.14) is a F-adapted local strong solution to (1.4) with initial condition y 0 and lifetime k+1 .
Consequently, for each k ∈ N 0 there exists a local strong solution Y (k) to (1.4) with initial condition y 0 and lifetime k . By Lemma 2.3 we have P( k → ∞) = 1. Hence, it follows that is a F-adapted strong solution to (1.4) with initial condition y 0 .
3.15. Theorem. We suppose that the mappings (a, b, c| B ) are locally Lipschitz and locally bounded. Then, existence of local strong solutions to (1.4) holds.
Proof. Let y 0 : Ω → H be an arbitrary F 0 -measurable random variable. We define the partition (Ω k ) k∈N ⊂ F 0 of Ω by Ω k := { y 0 ∈ [k −1, k)}. Furthermore, for each k ∈ N we define the mappings a k : Ω × R + × H → H, b k : Ω × R + × H → L 0 2 (H) and c k : Ω × R + × H × E → H 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 Y (k) to (1.4) with parameters a = a k , b = b k and c = c k , and initial condition y 0 1 Ω k . The stopping time is strictly positive, and Y (k) is a local strong solution to (1.4) with initial condition y 0 1 Ω k and lifetime τ k . The stopping time τ := k∈N τ k 1 Ω k is strictly positive, and the process Y := k∈N Y (k) 1 Ω k is a local strong solution to (1.4) with initial condition y 0 and lifetime τ . Proof. This is a direct consequence of Theorems 3.12 and 3.14. Now, we shall provide a comparison with reference [3], where the authors also study Hilbert space valued SDEs of the type (1.4). Their result [3, Theorem 2.1] is based on the method of successive approximations (see also [26,25]) 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 [3, Theorem 2.1], for some constant p ≥ 2 we need the estimate (3.19) where κ : R + → R + denotes a continuous, nondecreasing function with κ(0) = 0, and further conditions, which are precisely stated in [3], must be fulfilled. These conditions are satisfied if κ is a continuous, nondecreasing and concave function such that 3.17. Example. We fix an arbitrary constant 0 < δ < exp(−1) and define the functions κ, ρ : as well as cf. [26,Remark 1]. Let a : R → R be a mapping such that |a(y 1 ) − a(y 2 )| ≤ ρ(|y 1 − y 2 |) for all y 1 , y 2 ∈ R.
Then we have the estimate showing that condition (3.19) with p = 2 is satisfied. Moreover, κ is a continuous, nondecreasing, concave function and condition (3.20) is satisfied, because for each 0 < < δ we have Consequently, [3, Theorem 2.1] applies. However, we have and thus lim u→0 ρ (u) = ∞. Therefore, the mapping a : R → R might fail to be locally Lipschitz, and hence, Theorem 3.16 does not apply.

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 H be a separable Hilbert space, let (S t ) t≥0 be a C 0 -semigroup on H with infinitesimal generator A : D(A) ⊂ H → H, and let B ∈ E be a set with F (B c ) < ∞. Furthermore, let α : Ω×R + ×H → H and σ : Ω×R + ×H → L 0 2 (H) be P ⊗B(H)measurable mappings, and let γ : Ω×R + ×H ×E → H be a P ⊗B(H)⊗E-measurable mapping.
Throughout this section, we suppose that there exist another separable Hilbert space H, a C 0 -group (U t ) t∈R on H and continuous linear operators ∈ L(H, H), π ∈ L(H, H) such that the diagram Remark. According to [7,Prop. 8.7], this assumption is satisfied if the semigroup (S t ) t≥0 is pseudo-contractive (one also uses the notion quasi-contractive), that is, there is a constant ω ∈ R such that S t ≤ e ωt for all t ≥ 0.
(2) If (α, σ, γ| B ) satisfy the linear growth condition, then (a, b, c| B ) satisfy the linear growth condition, too. Proof. All three statements are straightforward to check.

4.4.
Proposition. Let z 0 : Ω → H be a F 0 -measurable random variable, and let τ be a stopping time. Then, the following statements are true: (1) If Y is a local strong solution to (1.4) with initial condition z 0 and lifetime τ , then Z := πU Y is a local mild solution to (1.3) with initial condition z 0 and lifetime τ . (2) If Z is a local mild solution to (1.3) with initial condition z 0 and lifetime τ , then the process Y defined as is a local strong solution to (1.4) with initial condition z 0 and lifetime τ , and we have Z τ = πU Y τ .
Proof. Let Y be a local strong solution to (1.4) with initial condition z 0 and lifetime τ . Then we have By the Definitions (4.2)-(4.4) of a, b, c we obtain Therefore, by (4.1), and since Z = πU Y , we arrive at Proof. Suppose that (α, σ, γ| B ) are locally Lipschitz. Let Z and Z be two local mild solutions to (1.3) with initial conditions z 0 and z 0 , and lifetimes τ and τ . We define the H-valued processes Y and Y according to (4.5). By Proposition 4.4, the processes Y and Y are local strong solutions to (1.4) with initial conditions z 0 and z 0 , and lifetimes τ and τ , and we have Z τ = πU Y τ and (Z ) τ = πU (Y ) τ . By Lemma 4.3, the mappings (a, b, c| B ) are also locally Lipschitz, and hence, Theorem 3.12 yields that up to indistinguishability Therefore, we have up to indistinguishability proving uniqueness of local mild solutions to (1.3). Now, we suppose that (α, σ, γ| B ) are locally Lipschitz and satisfy the linear growth condition. Let z 0 : Ω → H be an arbitrary F 0 -measurable random variable. By Lemma 4.3, the mappings (a, b, c| B ) are also locally Lipschitz and satisfy the linear growth condition. Thus, by Theorem 3.14 there exists a strong solution Y to (1.4) with initial condition z 0 . According to Proposition 4.4, the process Z := πU Y is a mild solution to (1.3) with initial condition z 0 , proving the existence of mild solutions to (1.3).
If (α, σ, γ| B ) are locally Lipschitz and locally bounded, then a similar proof, which uses Theorem 3.15, shows that existence of local mild solutions to (1.3) holds. 4.6. Remark. The structure Z = πU Y shows that mild solutions to (1.3) obtained from Theorem 4.5 have càdlàg sample paths.

4.7.
Remark. As pointed out in [20], the existence of weak solutions to (1.3) relies on a suitable stochastic Fubini theorem. Sufficient conditions can be found in [7].