Invariant manifolds with boundary for jump-diffusions

We provide necessary and sufficient conditions for stochastic invariance of finite dimensional submanifolds with boundary in Hilbert spaces for stochastic partial differential equations driven by Wiener processes and Poisson random measures.


Introduction
Consider a stochastic partial differential equation (SPDE) of the form dr t = (Ar t + α(r t ))dt + σ(r t )dW t + E γ(r t− , x)(µ(dt, dx) − F (dx)dt) on a separable Hilbert space H driven by some trace class Wiener process W on a separable Hilbert space H and a compensated Poisson random measure µ on some mark space E with dt ⊗ F (dx) being its compensator. Throughout this paper, we assume that A is the generator of a C 0 -semigroup on H and that the mappings α, σ = (σ j ) j∈N and γ satisfy appropriate regularity conditions. Given a finite dimensional C 3 -submanifold M with boundary of H, we study the stochastic viability and invariance problem related to the SPDE (1.1). In particular, we provide necessary and sufficient conditions such that for each h 0 ∈ M there is a (local) mild solution r to (1.1) with r 0 = h 0 which stays (locally) on the submanifold M.
Any finite dimensional invariant submanifold M for the SPDE (1.1) gives rise to a finite dimensional Markovian realization of the respective particular solution processes r with initial values in M, i.e. a deterministic C 3 -function G and a finite dimensional Markov process X such that r t = G(X t ) up to some stopping time. This proves to be very useful in applications, since it renders the stochastic evolution model (1.1) analytically and numerically tractable for initial values in M. An important example is the so called Heath-Jarrow-Morton (HJM) SPDE that describes the evolution of the interest rate curve. Stochastic invariance for the HJM SPDE has been discussed in detail in [3,4,5,10,11,16,17,23] for the diffusion case. The present paper completes the results from [12,16,17] by providing explicit stochastic invariance conditions for the general case of an SPDE with jumps.
Stochastic invariance has been extensively studied also for other sets than manifolds. In finite dimension the general stochastic invariance problem for closed sets has been treated, e.g., in [7] in the diffusion case, and in [27] in the case of jump-diffusions. In infinite dimension we mention, e.g., the works of [22,28], where stochastic invariance has been established by means of support theorems for diffusion-type SPDEs.
We shall now present and explain the invariance conditions which we derive in this paper. Let us first consider the situation where the jumps in (1.1) are of finite variation. Then, the conditions are necessary and sufficient for stochastic invariance of M for (1.1).
Condition (1.2) says that the submanifold M lies in the domain of the infinitesimal generator A. This ensures that the mapping in (1.5) is well-defined. Condition (1.3) means that the volatilities h → σ j (h) must be tangential to M in its interior and tangential to the boundary ∂M at boundary points. Condition (1.4) says that the functions h → h + γ(h, x) map the submanifold M into its closure M. Condition (1.5) means that the adjusted drift must be tangential to M in its interior and additionally inward pointing at boundary points.
In the general situation, where the jumps in (1.1) may be of infinite variation, condition (1.5) is replaced by the three conditions E | η h , γ(h, x) |F (dx) < ∞, h ∈ ∂M (1.6) where η h denotes the inward pointing normal (tangent) vector to ∂M at boundary points h ∈ ∂M.
Condition (1.6) concerns the small jumps of r at the boundary of the submanifold and means that the discontinuous part of the solution must be of finite variation, unless it is parallel to the boundary ∂M. Denoting by Π K the orthogonal projection on a closed subspace K ⊂ H, we decompose As we will show, condition (1.4) implies The essential idea is to perform a second order Taylor expansion for a parametrization φ around h to obtain for some constant C ≥ 0. By virtue of (1.9), the integral in (1.7) exists, and hence, conditions (1.7), (1.8) correspond to (1.5).
As in previous papers on this subject we are dealing with mild solutions of SPDEs, i.e. stochastic processes taking values in a Hilbert space whose drift characteristic is quite irregular (e.g., not continuous with respect to the state variables). Therefore, the arguments to translate stochastic invariance into conditions on the characteristics are not straightforward. The arguments to prove our stochastic invariance results can be structured as follows: First, we show that we can (pre-)localize the problem by separating big and small jumps. Second, prelocal invariance of parametrized submanifolds can be pulled back to R m by a linear projection argument tracing back to [13]. Both steps require a careful analysis of jump structures, which leads to the involved invariance conditions. The remainder of this paper is organized as follows: We state our main results in Section 2. In Section 3 we prove invariance results for submanifolds with one chart, and in Section 4 we prove the main results. In Section 5 we apply our results to the particular situation, where the Poisson random measure is generated by finitely many independent Lévy processes, and in Section 6 we present several examples illustrating our results. For the convenience of the reader, we provide the prerequisites on SPDEs in Appendix A and on finite dimensional submanifolds with boundary in Appendix B. For the sake of lucidity, we postpone the proofs of some auxiliary results to Appendix C.

Statement of the main results
In this section we introduce the necessary terminology and state our main results. We fix a filtered probability space (Ω, F, (F t ) t≥0 , P) satisfying the usual conditions. In Appendix A below we review some basic facts about SPDEs of the type (1.1) and we recall the concepts of (local) strong, weak and mild solutions. In particular, in view of (A.2), equation (1.1) can be rewritten equivalently      dr t = (Ar t + α(r t ))dt + j∈N σ j (r t )dβ j t + E γ(r t− , x)(µ(dt, dx) − F (dx)dt) where (β j ) j∈N is a sequence of real-valued independent standard Wiener processes. We next formulate the concept of stochastic invariance.

Definition.
A non-empty Borel set B ⊂ H is called prelocally (locally) invariant for (2.1), if for all h 0 ∈ B there exists a local mild solution r = r (h0) to (2.1) with lifetime τ > 0 such that up to an evanescent set 1 (r τ ) − ∈ B and r τ ∈ B r τ ∈ B .
The following standing assumptions prevail throughout this paper: • A generates a C 0 -semigroup (S t ) t≥0 on H.
• For each n ∈ N there exists a sequence (κ j n ) j∈N ⊂ R + with j∈N (κ j n ) 2 < ∞ such that for all j ∈ N the mapping σ j : H → H satisfies Consequently, for each j ∈ N the mapping σ j is locally Lipschitz continuous.
• The mapping γ : H × E → H is measurable, and for each n ∈ N there exists a measurable function ρ n : such that for all x ∈ E the mapping γ(•, x) : H → H satisfies Consequently, for each x ∈ E the mapping γ(•, x) is locally Lipschitz continuous.
• We assume that for each j ∈ N the mapping σ j : H → H is continuously differentiable, that is The first four conditions ensure that we may apply the results about SPDEs from Appendix A. We furthermore assume that: • M is a finite-dimensional C 3 -submanifold with boundary of H, see Appendix B.
Our first main result now reads as follows.
In either case, A and the mapping in (1.7) are continuous on M, and for each h 0 ∈ M there is a local strong solution r = r (h0) to (2.1). Moreover, if instead of (1.4) we even have h + γ(h, x) ∈ M for F -almost all x ∈ E, for all h ∈ M, (2.9) then M is locally invariant for (2.1).

2.3.
Remark. It follows from Theorem 2.2 that (pre-)local invariance of M is a property which only depends on the parameters {α, σ j , γ, F } -that is, on the law of the solution to (2.1). It does not depend on the actual stochastic basis {(Ω, F, (F t ) t≥0 , P), W, µ}.
Then we have (r (h0) ) τ ∈ M up to an evanescent set, because showing that M is locally invariant for (2.10). However, the jump condition (2.9) is not satisfied, because for h = 0 we have Nevertheless, we see that condition (1.4) holds true, because 1 ∈ M.
If M is a closed subset of H and global Lipschitz conditions are satisfied, then we obtain global invariance. This is the content of our second main result. Recall that the semigroup (S t ) t≥0 is called pseudo-contractive, if S t ≤ e ωt , t ≥ 0 (2.11) for some ω ∈ R.
2.5. Theorem. Assume that the semigroup (S t ) t≥0 is pseudo-contractive and that conditions (2.2)-(2.7) hold globally, i.e. the coefficients L n , (κ j n ) j∈N , ρ n do not depend on n ∈ N, and with the right-hand sides of (2.4), (2.7) multiplied by (1 + h ). If M is a closed subset of H, then (1.2)-(1.4) and (1.6)-(1.8) imply that for any h 0 ∈ M there exists a unique strong solution r = r (h0) to (2.1) and r ∈ M up to an evanescent set.
The above two theorems simplify in the case of jumps with finite variation: 2.6. Theorem. Suppose for each n ∈ N there exists a measurable function θ n : E → R + with E θ n (x)F (dx) < ∞ such that γ(h, x) ≤ θ n (x) for all h ∈ M with h ≤ n and all x ∈ E. (2.12) Then, Theorems 2.2 and 2.5 remain true with (1.6)-(1.8) being replaced by (1.5) and the mapping in (1.7) being replaced by the mapping in (1.5).

2.7.
Remark. We shall briefly comment on our assumptions: • The assumption that the submanifold M is of class C 3 is a technical assumption, which we require for the proof of Lemma 3.5. Theorems 2.2, 2.5 and 2.6 also hold true for C 2 -submanifolds, but in this case the proof of Proposition 3.15 below is more involved, because we only obtain the existence of martingale solutions to (3.26). • In Theorem 2.5, the assumption that the semigroup (S t ) t≥0 is pseudocontractive is also a technical assumption, which allows us to apply Theorem A.6, ensuring existence of mild solutions to (2.1) with càdlàg paths. Theorem 2.5 holds true for every C 0 -semigroup, but then the proof becomes more involved, because we only obtain the existence of mild solutions to (2.1) which might not have a càdlàg version.
The following results supplement Theorems 2.2, 2.5 and 2.6 by providing necessary conditions for (pre-)local stochastic invariance. If the mark space E is a Banach space, then the support of the measure F is defined as 2.8. Proposition. Suppose condition (1.4) is satisfied and the mark space E is a Banach space. Let h ∈ M and x ∈ E be such that γ(h, •) : E → H is continuous in a neighborhood of x and differentiable in x with γ(h, x) = 0. Then, for every direction v ∈ E, v = 0 the following statements are true: (1) Suppose there exists a sequence (t n ) n∈N ⊂ (0, ∞) with t n → 0 such that Then we have Then we have Then we have Further examples illustrating our main results from this section are provided in Sections 5 and 6 below.

Invariance results for submanifolds with one chart
In this section, we shall prove invariance results for submanifolds with one chart. First, we provide a stronger invariance property than Definition 2.1. Let τ 0 be a bounded stopping time and consider the time-shifted version of (2.1), for some F τ0 -measurable random variable h 0 : Ω → H. In (3.1), the sequence (β (τ0),j t ) j∈N is a sequence of real-valued independent standard Wiener processes and µ (τ0) is a time-homogeneous Poisson random measure, both relative to the filtration (F τ0+t ) t≥0 . We refer to Appendix A for further details.
3.1. Definition. A non-empty Borel set B ⊂ H is called locally strong invariant for (3.1), if for each bounded stopping time τ 0 and each bounded F τ0 -measurable random variable h 0 : Ω → H with P(h 0 ∈ B) = 1 there exists a local mild solution r = r (τ0,h0) to (3.1) with lifetime τ > 0 such that r τ ∈ B up to an evanescent set.
For technical reasons, we will also need the following concepts of prelocal (strong) invariance: 3.2. Definition. Let B 1 ⊂ B 2 ⊂ H be two nonempty Borel sets.
(1) B 1 is called prelocally invariant in B 2 for (2.1), if for all h 0 ∈ B 1 there exists a local mild solution r = r (h0) to (2.1) with lifetime τ > 0 such that (r τ ) − ∈ B 1 and r τ ∈ B 2 up to an evanescent set.
Note that any non-empty Borel set B ⊂ H is prelocally (strong) invariant for (2.1) in the sense of Definition 2.1 (Definition 3.1) if and only if B is prelocally (strong) invariant in B for (2.1) in the sense of Definition 3.2. Now, let G be another separable Hilbert space. For any k ∈ N we denote by . . , k − 1 are Lipschitz continuous. We do not demand that f itself is bounded, as this would exclude continuous linear operators f ∈ L(G, H).
and let f : G → H and g ∈ C 2 b (H; G) be mappings. We define the mappings where h = f (z).
For our subsequent analysis, the following technical definitions will be useful.  The following two auxiliary results are direct consequences of the previous definitions.
3.9. Lemma. Suppose O M is prelocally invariant in C M for (2.1) with solutions given by (3.9) and f . Then, the following statements are true: If the generator A is a continuous, i.e. (2.1) is rather an SDE than an SPDE, then the just introduced invariance concept transfers to the sets O N and C N : 3.11. Lemma. Suppose A ∈ L(H). Then, the following statements are equivalent: (1) O M is prelocally invariant in C M for (2.1) with solutions given by (3.9) and f . (2) O N is prelocally invariant in C N for (3.9) with solutions given by (2.1) and g. Proof.
(1) ⇒ (2): Let z 0 ∈ O N be arbitrary and set h 0 := f (z 0 ) ∈ O M . There exists a local strong solution Z = Z (g(h0)) = Z (z0) to (3.9) with lifetime τ > 0 such that (Z τ ) − ∈ O N and Z τ ∈ C N up to an evanescent set and, since A ∈ L(H), the process r = f (Z) is a local strong solution to (2.1) with initial condition h 0 = f (z 0 ). Therefore, we have (r τ ) − ∈ O M and r τ ∈ C M up to an evanescent set and g(r) is a local strong solution to (2.1) with initial condition z 0 and lifetime τ . Proof. The proof is analogous to that of Lemma 3.11.
3.13. Proposition. The following statements are equivalent: (1) O M is prelocally invariant in C M for (2.1) with solutions given by (3.9) and f . Proof. (1) ⇒ (2): By Lemma 3.9, the set O N is prelocally invariant in C N for (3.9). Let h ∈ O M be arbitrary. Since O M is prelocally invariant in C M for (2.1) with solutions given by (3.9) and f , there exists a local strong solution Z = Z (g(h)) to (3.9) with lifetime τ > 0 such that (Z τ ) − ∈ O N and Z τ ∈ C N up to an evanescent set and r := f (Z) is a local mild solution to (2.1) with initial condition h 0 and lifetime τ . By Itô's formula (Theorem A.16) we obtain P-almost surely Let ζ ∈ D(A * ) be arbitrary. Since r is also a local weak solution to (2.1) with lifetime τ , we have P-almost surely Therefore, we get up to an evanescent set where the processes B, M c , M d are given by The process B is a finite variation process which is continuous, and hence predictable, M c is a continuous square-integrable martingale and M d is a purely discontinuous square-integrable martingale. Therefore B + M c + M d is a special semimartingale. Since the decomposition B + M of a special semimartingale into a finite variation process B and a local martingale M is unique (see [ Since the process r is càdlàg, by Lemma 3.5 and Lebesgue's dominated convergence theorem (applied to the sum j∈N and to the integral E ) the integrands appearing in (3.15)-(3.17) are continuous in s = 0, and hence, we get Identity (3.18) shows that ζ → A * ζ, h is continuous on D(A * ), proving h ∈ D(A * * ). Since A = A * * , see [25,Thm. 13.12], we obtain h ∈ D(A), which yields (3.11). Using the identity A * ζ, h = ζ, Ah , we obtain ζ, Ah + α(h) − (g a)(h) = 0 for all ζ ∈ D(A * ), and hence (3.12). For an arbitrary j ∈ N we obtain, by using (3.19), showing (3.13). By (3.20), for all ζ ∈ D(A * ) we have This proves (3.14).
(2) ⇒ (1): Let h 0 ∈ O M be arbitrary. Since O N is prelocally invariant in C N for (3.9), there exists a local strong solution Z = Z (g(h0)) to (3.9) with lifetime τ > 0 such that (Z τ ) − ∈ O N and Z τ ∈ C N up to an evanescent set. By Itô's formula (Theorem A.16), conditions (3.11)-(3.14) and taking into account the Itô isometry, the process r := f (Z) satisfies P-almost surely Therefore, the process r is a local strong solution to (2.1) with initial condition h 0 and lifetime τ , showing that O M is prelocally strong invariant in C M for (3.1) with solutions given by (3.9) and f .
If condition (3.11), (3.12) are satisfied, then the continuity of A on O M follows from Lemma 3.5, proving the additional statement.
3.14. Proposition. The following statements are equivalent: (1) O M is prelocally strong invariant in C M for (3.1) with solutions given by (3.10) and f . (2) O N is prelocally strong invariant in C N for (3.10) and we have (3.11)-(3.14).
In either case, A is continuous on O M .
Proof. The proof is analogous to that of Proposition 3.13.
For the rest of this section, let G = R m , where m ∈ N denotes the dimension of the submanifold M. We assume there exist elements ζ 1 , . . . , ζ m ∈ D(A * ) such that the mapping f : N → M has the inverse This is illustrated by the following diagram: where A * ζ, f := ( A * ζ 1 , f , . . . , A * ζ m , f ). Then, for each h ∈ O M we have where z = ζ, h ∈ O N . Furthermore, we define the mappings Θ := (ψ, Ψ) a : R m → R m , as well as the time-shifted version (3.27) According to Lemma 3.5, the mappings a, (b j ) j∈N , c as well as Θ, (Σ j ) j∈N , Γ satisfy the regularity conditions (2.2)-(2.4) and (2.6)-(2.8). Note that The following result provides necessary and sufficient conditions regarding prelocal invariance of C V in O V for (3.26). Note that V is also a C 3 -submanifold with boundary of R m , and that for y ∈ ∂V the inward pointing normal (tangent) vector to ∂V at y is given by the first unit vector e 1 = (1, 0, . . . , 0).

3.15.
Proposition. The following statements are equivalent:   From now on, we assume that y ∈ ∂O := O ∩ ∂V . Let (Φ j ) j∈N ⊂ R be a sequence with Φ j = 0 for only finitely many j ∈ N, and let Ψ : E → R be a measurable function of the form Ψ = c1 B with c > −1 and B ∈ E satisfying F (B) < ∞. Let Z be the Doléans-Dade exponential By [19,Thm. I.4.61] the process Z is a solution of and, since Ψ > −1, the process Z is a strictly positive local martingale. There exists a strictly positive stopping time τ 1 such that Z τ1 is a martingale. Integration by parts (see [19,Thm. I.4.52]) yields Taking into account the dynamics (3.26), we have Incorporating (3.26), (3.37) and (3.38) into (3.36), we obtain where M is a local martingale with M 0 = 0. There exists a strictly positive stopping time τ 2 such that M τ2 is a martingale.
. . , y m ) = ((y 1 ) + , y 2 , . . . , y m ), (3.41) and therefore, it satisfies Consequently, the mappings Θ Π : • Π also satisfy the regularity conditions (2.2)-(2.4) and (2.6)-(2.8), which, due to Proposition A.8, ensures existence and uniqueness of strong solutions to the SDE by taking into account that the metric projection Π on R n + is given by (3.41), we have we have the identities and defining the sequence we have the inclusions Now, let τ 0 be a bounded stopping time and let y 0 : Ω → R m be a bounded F τ0 -measurable random variable. Defining the F τ0 -measurable sets According to Lemma A.2 and recalling the notation (A.19), the mappings are strictly positive stopping times. By Proposition A.9, the mapping is a strictly positive stopping time and is a local strong solution to (3.42) with initial condition y 0 and lifetime τ . We obtain on {y 0 ∈ Int O} up to an evanescent set By (3.41) and (3.32), for all y ∈ P ∩ R m + we have Furthermore, by (3.31)-(3.34) and (3.43), for all y ∈ P ∩ R m − we have The function φ : R → R, φ(y) := (−y 3 ) + is of class C 2 (R) and we have φ (y) < 0 for y < 0 and φ (y) = φ (y) = 0 for y ≥ 0. By (3.44)-(3.48) and Lemma A.22, we obtain Let k ∈ N be arbitrary. Applying Itô's formula (Theorem A.16) yields P-almost Since Ω ∂ k ⊂ {y 0 ∈ ∂O}, by (3.47) and Taylor's theorem we obtain P-almost surely By (3.49)-(3.52) and Lemmas A.14, A.15, we deduce that φ( e 1 , (Y ∂,k ) τ ∂ k ) ≤ 0 on Ω ∂ k up to an evanescent set. Therefore, we obtain on {y 0 ∈ ∂O} up to an evanescent Consequently, we get up to an evanescent set Using (3.32) and Corollary A.25 we obtain Y τ ∈ C up to an evanescent set. Since Proof. This is a direct consequence of Lemma C.1.
there exists a local mild solution r = r (h0) to (2.1) with lifetime τ > 0 such that (r τ ) − ∈ O M and r τ ∈ C M up to an evanescent set. Since ζ 1 , . . . , ζ m ∈ D(A * ) and r is also a local weak solution to (2.1), setting Z := ζ, r we have, by taking into account (3.23)-(3.25), P-almost surely Therefore, the process Z is a local strong solution to (3.9) with initial condition ζ, h 0 and lifetime τ such that (Z τ ) − ∈ O N and Z τ ∈ C N up to an evanescent set. By (3.21), we have f (Z τ ) = r τ , and hence, the process f (Z) is a local mild solution to (2.1) with initial condition h 0 and lifetime τ .
3.18. Proposition. The following statements are equivalent: (1) O M is prelocally strong invariant in C M for (3.1) with solutions given by (3.10) and f .
1) with solutions given by (3.9) and f .
In either case, A and the mapping in    • Proposition 3.13 yields (3.53) and • By Lemma 3.9, the set O N is prelocally invariant in C N for (3.9). Hence, by (3.28)-(3.30) and Proposition 3.13, the set O V is prelocally invariant in C V for (3.26) with solutions given by (3.9) and Ψ.
The latter statement has two further consequences: • By Lemma 3.11, the set O N is prelocally invariant in C N for (3.9) with solutions given by (3.26) and ψ. Thus, Proposition 3.13 yields where ξ z denotes the inward pointing normal (tangent) vector to ∂N at z. Taking into account (3.59)-(3.61), applying Proposition C.12 we arrive at (3.54)-(3.58).

Proof of the main results
In this section, we shall prove our main results by using Proposition 3.18. Let B ∈ E be a set with F (B c ) < ∞ and define the mappings α B : H → H and By Lemma A.18, the mappings α B , (σ j ) j∈N , γ B are well-defined and satisfy the regularity conditions (2.2)-(2.4) and (2.6)-(2.8) from Section 2. We shall consider the SPDE According to Lemma A.20, the mapping is a strictly positive stopping time.
Then, the following statements are true: (2) The mapping in (3.57) Taking into account Lemma A.2, the mapping is a strictly positive stopping time. We obtain up to an evanescent set Furthermore, using (4.6) and Corollary A. 25 Since we have the coverings • a sequence (M k ) k∈N of submanifolds with one chart as in Section 3, i.e. as the submanifold M in Diagram (3.22), and M k ⊂ M for all k ∈ N, and the submanifold M has the countable coverings Let k ∈ N be arbitrary. By the submanifold M has the disjoint covering M = k∈N P k , and defining the F τ0measurable sets such that on Ω k up to an evanescent set According to Proposition A.21, there exists a local mild solution r k to (3.1) with initial condition h 0 1 Ω k and lifetime τ k such that (r Proposition A.9, the mapping τ := k∈N τ k 1 Ω k is a strictly positive stopping time and r := k∈N r k 1 Ω k is a local mild solution to (3.1) with initial condition h 0 . We obtain up to an evanescent set Using Proposition A.24, by (1.4) we obtain r τ ∈ M up to an evanescent set, proving that M is prelocally invariant for (3.1). If even condition (2.9) is satisfied, then by Proposition A.24 we obtain r τ ∈ M up to an evanescent set, and hence, M is locally strong invariant for (3.1). This concludes the proof of Theorem 2.2.
We shall now prove Theorem 2.5. Let h 0 ∈ M be arbitrary. According to Theorem A.6, there exists a unique mild and weak solution r = r (h0) to (2.1). By Lemma A.2 the mapping is a stopping time. We claim that Suppose, on the contrary, that (4.9) is not satisfied. Then, there exists N ∈ N such that P(τ ≤ N ) > 0. We define the bounded stopping time τ 0 := τ ∧ N . By the closedness of M in H, we have (r τ0 ) − ∈ M up to an evanescent set. Therefore, by relation (1.4) and Corollary A.25 we obtain r τ0 ∈ M up to an evanescent set. The process r τ0+• is a weak solution to the time-shifted SPDE (3.1) with initial condition r τ0 . Indeed, for each ζ ∈ D(A * ) we have P-almost surely ζ, r τ0+t = ζ, r τ0 + ζ, r τ0+t − r τ0 According to Theorem A.6, there exists a unique mild solution r K to (3.1) with initial condition being the bounded F τ0 -measurable random variable r τ0 1 { rτ 0 ≤K} .
By the second part of the proof of Theorem 2.2, the submanifold M is prelocally strong invariant for (3.1), and hence, there exists an (F τ0+t )-stopping time > 0 such that (r K ) ∈ M up to an evanescent set. Noting that {τ ≤ N } = {τ = τ 0 }, by Proposition A.10 we obtain up to an evanescent set which contradicts the Definition (4.8) of τ . Therefore, relation (4.9) is valid and we obtain r ∈ M up to an evanescent set.

4.4.
Remarks on the existence of invariant manifolds. The existence of (locally) invariant submanifolds (with boundary) for a given jump-diffusion is an (even more) involved question. As in [16], Frobenius-type theorems have to be applied to construct (locally) invariant submanifolds for given drift, volatility and jump mappings. Assuming Notice that in this case the manifolds M constructed by Frobenius-type methods from σ j the mapping in (1.5) have to satisfy an additional condition.
If (4.13) is not fulfilled, then for h ∈ H we have to determine the set Let M be a submanifold such that (1.4) is satisfied. Then we have (1.9), and hence Therefore we have to construct via the methods of [16] submanifolds M such that σ j and N ⊥ h are tangent. Then we have to check among these submanifolds M those, which are left invariant by the maps h → h + γ(h, x) for F -almost all x ∈ E, and which are tangent to the mapping in (1.7). Notice that in this case the actual construction of M already involved the jump structure via the distribution h → N ⊥ h .

Invariant manifolds with boundary for Lévy driven jump-diffusions
In this section, we investigate the invariance problem for submanifolds with boundary for the particular situation, where the Poisson random measure µ in the SPDE (2.1) is generated by finitely many independent Lévy processes. The following additional assumptions prevail throughout this section: • The mark space is (E, E) = (R e , B(R e )) for some e ∈ N.
• Concerning the compensator dt ⊗ F (dx) of the Poisson random measure µ, we assume that F is given by where F 1 , . . . , F e are Lévy measures on (R, B(R)) such that • For each n ∈ N there exists a measurable mapping θ n : R e → R + with for any nonnegative measurable function g : R e → R. We define the index set K := {1, . . . , e} and the disjoint subsets which constitute a decomposition of K. This corresponds to the three types A,B,C of Lévy processes from [26,Def. 11.9]. In terms of Lévy processes, K A means "finite activity", K B means "infinite activity" but "finite variation", and K C means "infinite variation". Let us further introduce By symmetry, we may assume, without loss of generality, that we can decompose the set The following statements are true: (1) For each k ∈ K B ∪K C there exists a sequence (t n ) n∈N ⊂ (0, ∞) with t n → 0 such that {t n e k : n ∈ N} ⊂ supp(F ) or {−t n e k : n ∈ N} ⊂ supp(F ). We start with the proof of the second statement. Let k ∈ K + B be arbitrary. By the Definition (5.10) of K + B we have F k (R + ) = ∞. Therefore, for all n ∈ N we have (0, 1 n ) ∩ supp(F k ) = ∅. Indeed, suppose, on the contrary, that (0, 1 n ) ∩ supp(F k ) = ∅ for some n ∈ N. Then, for all x ∈ (0, 1 n ) there exists x > 0 such that F k (B x (x)) = 0. Using Lindelöf's Lemma [1, Lemma 1.1.6], for some countable subset I ⊂ (0, 1 n ) we obtain which contradicts F k ((0, 1 n )) = ∞. Hence, there exists a sequence (t n ) n∈N ⊂ (0, ∞) with t n → 0 such that {t n : n ∈ N} ⊂ supp(F k ). In view of (5.15), we obtain (5.13), proving the second statement. Analogous argumentations provide the first and the third statement.

5.2.
Theorem. The statements of Theorems 2.2 and 2.5 remain true with (1.6)-(1.8) being replaced by and the mapping in (1.7) being replaced by the mapping in (5.17). Furthermore, in either case we have h ∈ ∂M and k ∈ K +− B .

Thus, we have (1.7) if and only if
Ah

8) if and only if
Ah Consequently, conditions (1.6)-(1.8) are equivalent to (5.16), (5.17). By our assumptions stated at the beginning of the section, for each x ∈ E the mapping ∆(•, x) is continuous on M and for each k ∈ K A ∪ K B the mapping δ k is continuous on M. Since, by Theorem 2.2, the mapping in (1.7) is continuous on M, identity (5.21) together with relation (5.5) and Lebesgue's dominated convergence theorem shows that the mapping in (5.17) is continuous M.

Examples of invariant manifolds with boundary
In order to demonstrate our results from the previous sections, we provide four examples in this section. In the first two examples, we treat the unit circle and the closed unit ball, both in the Euclidean plane R 2 . In our third example, we consider Ornstein-Uhlenbeck processes on closed, convex cones in separable Hilbert spaces, and in the fourth example, we deal with the existence of finite dimensional realizations for HJM interest rate models with jumps.
6.1. Stochastic invariance of the unit circle. Let the state space H = R 2 be the Euclidean plane and let be the unit circle. Then M is a one-dimensional submanifold without boundary, i.e. ∂M = ∅, which is a closed subset of R 2 . We choose H = R, i.e. the Wiener process W is one-dimensional, and the mark space (E, E) = (R, B(R)). Concerning the compensator dt ⊗ F (dx) of the Poisson random measure µ, we assume that F is a Lévy measure satisfying where the mappings α : R 2 → R 2 , σ : R 2 → R 2 and γ : R 2 × R → R 2 are given by In order to apply Theorem 2.5, it suffices to show that The unit circle M has the tangent spaces Therefore, condition (6.2) satisfied. Furthermore, for all (h, x) ∈ M × [−π, π] we have, by noting that h = 1, and for (h, x) ∈ M × R we have, by noting that h = 1, showing that (6.4) is fulfilled. Now, Theorem 2.6 applies and yields that for each h 0 ∈ M there exists a unique strong solution r = r (h0) to (6.1) and r ∈ M up to an evanescent set.
6.1. Remark. Note that due to (6.6) we have showing that relation (1.9) is indeed true.
An alternative way to prove the previous invariance result is established by applying Theorem 5.2. We shall consider the case π −π |x|F (dx) = ∞, i.e. K A = K B = ∅ and K C = {1}. Note that we have the decomposition where the mappings δ : M → R 2 and ∆ : M × R → R 2 are given by In order to apply Theorem 5.2, it suffices to show that Taking into account (6.5), we obtain showing (6.9). Now, Theorem 5.2 applies and yields that for each h 0 ∈ M there exists a unique strong solution r = r (h0) to (6.1) and r ∈ M up to an evanescent set.
6.2. Remark. Note that by the Definition (6.7) of δ we have showing that relation (5.18) is indeed true.

6.2.
Stochastic invariance of the closed unit ball. As in Section 6.1, let the state space H = R 2 be the Euclidean plane. Let the submanifold be the closed unit ball Then M is a two-dimensional submanifold with boundary of R 2 , which is a closed subset of R 2 , and its boundary is the unit circle ∂M = {h ∈ R 2 : h = 1}.
As before, we choose H = R, i.e. the Wiener process W is one-dimensional, and the mark space (E, E) = (R, B(R)). Concerning the compensator dt ⊗ F (dx) of the Poisson random measure µ, we assume that F is a Lévy measure satisfying We define the mappings α : R 2 → R 2 , σ : R 2 → R 2 and γ : R 2 × R → R 2 of the two-dimensional SDE (6.1) as where a ≥ 0 is a constant and f : R 2 → R is a function satisfying the boundary condition f (h) = 2, h ∈ ∂M. (6.10) In order to apply Theorem 2.5, it suffices to show that The inward pointing normal (tangent) vectors to ∂M at boundary points are given by Therefore, condition (6.11) is satisfied. Moreover, for all (h, x) ∈ M×[0, 1] we have, by noting that h ≤ 1, providing (6.12). By the boundary condition (6.10), for all h ∈ ∂M we have, by noting that h = 1, showing (6.13). Moreover, by (6.5) and (6.15), for each boundary point h ∈ ∂M we have, by noting that h = 1, which yields (6.14). Now, Theorem 2.6 applies and yields that for each h 0 ∈ M there exists a unique strong solution r = r (h0) to (6.1) and r ∈ M up to an evanescent set.

6.3.
Ornstein-Uhlenbeck processes on closed, convex cones. In our third example, let H be a separable Hilbert space and let A be the generator of a pseudocontractive semigroup (S t ) t≥0 on H. We consider an SPDE of Ornstein-Uhlenbeck type with α ∈ H, a sequence (σ j ) j∈N ⊂ H satisfying j∈N σ j 2 < ∞ and a measurable mapping γ : E → H such that E γ(x) 2 F (dx) < ∞. Note that for each h 0 ∈ H there exists a unique mild solution r = r (h0) to (6.16), which is given by Let C ⊂ H be a closed, convex cone, i.e. C is a nonempty, closed subset of H such that h + g ∈ C for all h, g ∈ C and λh ∈ C for all λ ≥ 0 and h ∈ C.

Proposition. Suppose we have
Then, for each h 0 ∈ C the mild solution r = r (h0) to (6.16) satisfies r ∈ C up to an evanescent set.
Proof. Let h 0 ∈ C be arbitrary. The mild solution r = r (h0) to (6.16) is given by (6.17), and hence, by (6.18) and (6.20) we can write it as Since (6.19), (6.21) and (6.22) are satisfied, using Lemmas A.14 and A.15 we deduce that r ∈ C up to an evanescent set. Now, we deal with the necessity of conditions (6.18)-(6.22). For the sake of simplicity, we shall assume that C is a polyhedral cone generated by linearly independent vectors, that is where m ∈ N denotes a positive integer and v 1 , . . . , v m ∈ H are linearly independent vectors. We also define the interior of the cone  (1) For each h 0 ∈ C there exists a unique strong solution r = r (h0) to (6.16) and r ∈ C up to an evanescent set. where (e tA ) t≥0 denotes the norm-continuous semigroup on V generated by the linear operator A| V : V → V .
Proof. (2) ⇒ (1): Conditions (6.18)-(6.21) imply that α ∈ V , σ j ∈ V for all j ∈ N and γ(x) ∈ V for F -almost all x ∈ E. Thus, in view of (6.23), (6.24), the SPDE (6.16) is an V -valued SDE. By (6.25) and Proposition 6.3, for each h 0 ∈ C there exists a unique strong solution r = r (h0) to (6.16) and r ∈ C up to an evanescent set. where η h denotes the inward pointing normal (tangent) vector to ∂M at h. The vectors w 1 , . . . , w m ∈ V are linearly independent. Indeed, let c ∈ R m with m j=1 c j w j = 0 be arbitrary. Then we have Bc = 0, where B ∈ R m×m denotes the matrix given by B ij := v i , w j . By Lemma B.7 we have B ii > 0 for i = 1, . . . , n and B ij = 0 for i = j. Therefore, we obtain det B > 0, which implies c = 0.
We can apply Theorem 6.4 in order to derive a well-known invariance result about Lévy processes. Let X = (X (x0) ) x0∈R be a family of one-dimensional Lévy processes with Lévy-Itô decomposition where α ∈ R denotes the drift, σ ≥ 0 the diffusion part, W a one-dimensional Wiener process and F the Lévy measure, which we assume to be square-integrable, that is R x 2 F (dx) < ∞. Then, by Theorem 6.4, the half space R + is invariant for (6.32) if and only if we have Indeed, these conditions are also known to be necessary and sufficient for a Lévy processes to be a subordinator, i.e., to have non-decreasing sample paths, see [26,Thm. 21.5].
6.4. HJM interest rate models from finance. In this section, we shall deal with the existence of finite dimensional realizations for the HJMM (Heath-Jarrow-Morton-Musiela) equation from interest rate theory. The HJMM equation (6.33) describes the evolution of interest rates in a bond market In order to ensure the absence of arbitrage, we consider the HJMM equation (6.33) under a martingale measure Q. Then, the drift is given by The state space H = H β for the forward curve evolution (6.33) is the space of all absolutely continuous functions h : R + → R such that The parameter β > 0 is an arbitrary positive constant, and (S t ) t≥0 denotes the shift semigroup defined by S t h := h(t + ·) for t ≥ 0, which is generated by the differential operator d/dξ. We refer to [15] for further details on this topic.
Proceeding as in Section 4.4, we construct a finite dimensional submanifold M, which is (locally) invariant for the HJMM equation (6.33) if and only if for F -almost all x ∈ E the mapping h → h + γ(h, x) leaves M invariant. Under appropriate conditions on the mappings, the results from [16] yield that we have an affine term structure, that is, the submanifold M has an affine parametrization of form

Appendix A. SPDEs driven by Wiener processes and Poisson measures
For convenience of the reader, we provide the crucial results on SPDEs driven by Wiener processes and Poisson measures in this appendix. References on this topic are, e.g., [2,21,14].
In the sequel, (Ω, F, (F t ) t≥0 , P) denotes a filtered probability space satisfying the usual conditions. Let H be a separable Hilbert space and let (S t ) t≥0 be a C 0semigroup on H with infinitesimal generator A : D(A) ⊂ H → H. We denote by A * : D(A * ) ⊂ H → H the adjoint operator of A. Recall that the domains D(A) and D(A * ) are dense in H, see, e.g., [25,Thm. 13.35.c,Thm. 13.12].
Let H be another separable Hilbert space and let Q ∈ L(H) be a nuclear, selfadjoint, positive definite linear operator. Then, there exist an orthonormal basis (e j ) j∈N of H and a sequence (λ j ) j∈N ⊂ (0, ∞) with j∈N λ j < ∞ such that Qu = j∈N λ j u, e j H e j , u ∈ H namely, the λ j are the eigenvalues of Q, and each e j is an eigenvector corresponding to λ j . The space H 0 := Q 1/2 (H), 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 H-valued Q-Wiener process, see [8, p. 86,87]. We denote by L 0 2 (H) := L 2 (H 0 , H) the space of Hilbert-Schmidt operators from H 0 into H, which, endowed with the Hilbert-Schmidt norm itself is a separable Hilbert space. According to [8,Prop. 4.1], the sequence of stochastic processes (β j ) j∈N defined as is a sequence of real-valued independent standard Wiener processes and we have the expansion is an isometric isomorphism. According to [8,Thm. 4.3], for every predictable process Φ : Let (E, E) be a measurable space which we assume to be a Blackwell space (see [9,18]). 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 [19,Def. II.1.20]. Then its compensator is of the form dt ⊗ F (dx), where F is a σ-finite measure on (E, E).
A.1. Lemma. Let τ 0 be a bounded stopping time. We define: • The sequence (β (τ0),j ) j∈N of real-valued processes by • The new random measure µ (τ0) on R + × E by where we use the notation Then, W (τ0) is a Q-Wiener process adapted to (F (τ0) t ) t≥0 , the sequence (β (τ0),j ) j∈N is a sequence of real-valued independent standard Wiener processes, adapted to (F Proof. We claim that the process r is optional, that is, measurable with respect to the optional σ-algebra O, which is generated by all real-valued càdlàg adapted processes. Indeed, according to [8,Prop. 1

.3] we have
But for any h ∈ H and C ∈ B(R) we have because h, r is a real-valued càdlàg adapted process. Therefore, the set A = {r ∈ B} is optional, and hence τ (ω) = inf{t ≥ 0 : (ω, t) ∈ A} is a stopping time due to [19,Thm. I.1.27].
A.3. Definition. Let h 0 : Ω → H be an F 0 -measurable random variable. Furthermore, let r = r (h0) be an H-valued càdlàg adapted process and let τ > 0 a stopping time such that for all t ≥ 0 we have Then, the process r is called • a local weak solution to (1.1), if for all ζ ∈ D(A * ) we have P-almost surely We call τ the lifetime of r. If τ = ∞, then we call r a strong, weak or mild solution to (1.1), respectively.
A.4. Remark. Since the process r is càdlàg, we have r t = r t− for almost all t ∈ R + , P-almost surely, and hence, relation (A.3) implies r (t∧τ )− ∈ D(A) for almost all t ∈ R + , P-almost surely.
According to [13,Lemma 2.4.2], the process f defined by is predictable. By slight abuse of notation, we have written Ar instead of f in (A.4) and (A.5).
A.5. Remark. The following results are well-known: • Every (local) strong solution to (1.1) is also a (local) weak solution to (1.1).
• If A is bounded, i.e. generates a norm-continuous semigroup (S t ) t≥0 , then the concepts of (local) strong, weak and mild solutions to (1.1) are equivalent.
For our upcoming results, we introduce the abbreviation L 2 (F ) := L 2 (E, E, F ; H).
continuous. Then, for each random variable h 0 ∈ L 2 (F 0 ; H) there exists a mild and weak solution r = r (h0) to (1.1), which is unique up to indistinguishability.
A.7. Remark. Recall that the semigroup (S t ) t≥0 is called pseudo-contractive if (2.11) is satisfied for some ω ∈ R. The pseudo-contractive property is needed to ensure that mild solutions to the SPDE (1.1) have càdlàg sample paths, which we demand in Definition A.3. If A is the generator of a general C 0 -semigroup, then, under appropriate regularity conditions we also have existence and uniqueness of mild solutions, see [21], but the mild solution to (1.1) might not have a càdlàg version, see the counterexample in [24, Prop. 9.25].
Lipschitz continuous. Then, for every bounded F 0 -measurable random variable h 0 : Ω → H there exists a local mild and weak solution r = r (h0) to (1.1).
Proof. Let h 0 : Ω → H be a bounded F 0 -measurable random variable. Then, there exists N ∈ N such that h 0 < N . We define the retraction These mappings satisfy the Lipschitz conditions from Theorem A.6, and hence, there exists a unique mild and weak solution r = r (h0) to the SPDE By Lemma A.2, the mapping τ := inf{t ≥ 0 : r t ≥ N } is a strictly positive stopping time. Since (r τ ) − ∈ C up to an evanescent set, α| C = α R | C , σ| C = σ R | C and γ(•, x)| C = γ(•, x) R | C for all x ∈ E, we deduce that r is a local mild and weak solution to (1.1) with lifetime τ .
A.9. Proposition. For each k ∈ N let h k 0 : Ω → H be an F 0 -measurable random variable and let r k be a local mild solution to (1.1) with initial condition h k 0 and lifetime τ k > 0. Let (Ω k ) k∈N ⊂ F 0 be a decomposition of Ω. Then, τ := k∈N τ k 1 Ω k is a strictly positive stopping time and r := k∈N r k 1 Ω k is a local mild solution to (1.1) with initial condition h 0 := k∈N h k 0 1 Ω k and lifetime τ .
Proof. For each k ∈ N the mapping τ k 1 Ω k is a stopping time, because, taking into account that Ω k ∈ F 0 we have Consequently, τ = k∈N τ k 1 Ω k is a strictly positive stopping time by [19,Thm. I. 1.18]. For each k ∈ N, the process r k be a local mild solution to (1.1) with initial condition h k 0 and lifetime τ k , and hence, we have P-almost surely Since Ω k ∈ F 0 for each k ∈ N, this implies P-almost surely showing that r is a local mild solution to (1.1) with initial condition h 0 and lifetime τ .
A.10. Proposition. We suppose that: • α : H → H is locally Lipschitz continuous. Then, for all F 0 -measurable random variables h 0 , g 0 : Ω → H and any two local mild solutions r (h0) , r (g0) to (1.1) with initial conditions h 0 , g 0 and lifetimes τ (h0) , τ (g0) > 0 we have up to indistinguishability Proof. By Lemma A.2, the increasing sequence (τ n ) n∈N given by is a sequence of stopping times. Since the sample paths of r (h0) and r (g0) are càdlàg, we have P-almost surely Therefore, we have P(τ n → τ (h0) ∧ τ (g0) ) = 1. Set Γ := {h 0 = g 0 } ∈ F 0 and let n ∈ N and T ≥ 0 be arbitrary. We define the process X n as Since Γ ∈ F 0 , for all t ∈ [0, T ] we obtain P-almost surely There are constants M ≥ 1 and ω ∈ R such that S t ≤ M e ωt for all t ≥ 0, see, e.g., [25,Thm. 13.35.a]. Furthermore, by the assumed local Lipschitz continuity of the mappings, there exists a constant L n ≥ 0 such that By the Cauchy-Schwarz inequality and the Itô isometry, for all 0 ≤ s ≤ t ≤ T we obtain where the constant C n > 0 is given by Note that for all h, g ∈ H we have Therefore, estimate (A.7) and the Cauchy-Schwarz inequality yield Hence, the nonnegative function is continuous. In view of (A.7), the Gronwall lemma applies and yields Since T ≥ 0 was arbitrary and the sample paths of r (h0) and r (g0) are càdlàg, recalling that P(τ n → τ (h0) ∧ τ (g0) ) = 1 we deduce that up to an evanescent set completing the proof. Since D(A * ) is dense in H, we get P-almost surely showing that r is a local strong solution to (1.1) with lifetime τ . Furthermore, for every predictable process γ : the jumps of the integral process are given by A.13. Lemma. Let r = r (h0) be a local weak solution to (1.1) with lifetime τ > 0 for some F 0 -measurable random variable h 0 : Ω → H. Then, the following statements are true: (1) The jumps of the stopped process r τ are given by (2) For each n ∈ N we have Proof. Let X be the process Since r is a local weak solution to (1.1), for every ζ ∈ D(A * ) we have, by using (A.10), ζ, ∆r t∧τ = ∆ ζ, r t∧τ = ∆X t∧τ = ζ, γ(r (t∧τ )− , ξ t∧τ ) Recall that a closed, convex cone C is a nonempty, closed subset C ⊂ H such that h + g ∈ C for all h, g ∈ C and λh ∈ C for all λ ≥ 0 and h ∈ C.
A.14. Lemma. Let (G, G, ν) be a σ-finite measure space, let C ⊂ H be a closed, convex cone and let f ∈ L 1 (G; H) be such that f (x) ∈ C for ν-almost all x ∈ G.
Then we have Proof. First, we assume that f ∈ L 1 (G; H) is a simple function of the form with c k ∈ C and A k ∈ G satisfying ν(A k ) < ∞ for k = 1, . . . , m. Then we have Now, let f ∈ L 1 (G; H) be an arbitrary function such that f (x) ∈ C for ν-almost all x ∈ G. Arguing as in the proof of [8, Lemma 1.1], there exists a a sequence (f n ) n∈N of simple functions of the form (A.11) such that f n → f in L 1 (G; H). Therefore, we get finishing the proof.
A.15. Lemma. Let C ⊂ H be a closed, convex cone and let γ : Ω × R + × E → H be an optional process satisfying (A.8) such that γ(•, x) ∈ C up to an evanescent set, for F -almost all x ∈ E.
Then we have X ∈ C up to an evanescent set, where X denotes the integral process Proof. By assumption, there is an F -nullset N such that γ(•, x) ∈ C up to an evanescent set, for all x ∈ N c .
In this text, we apply the following version of Itô's formula.
A. 16. Theorem. Let α : Ω×R + → R m , σ : Ω×R + → L 0 2 (R m ) and γ : Ω×R + ×E → R m be predictable processes such that for all t ≥ 0 we have Furthermore, let Y 0 : Ω → R m be an F 0 -measurable random variable, let Y be the R m -valued Itô process and let φ ∈ C 2 (R m ; H) be arbitrary. Then we have P-almost surely where σ j := λ j σe j for each j ∈ N.
Proof. This follows from applying Itô's formula for finite dimensional semimartingales (see [19,Thm.I.4.57]) to h, φ(Y ) for each h ∈ H. Proof. Let j ∈ N be arbitrary. Furthermore, let h ∈ H be arbitrary. There exists n ∈ N such that h ≤ n. By estimates (2.3), (2.4) we have Since j∈N (κ j n ) 2 < ∞, we have (A. 12), showing that the first statement holds true. For each j ∈ N the mapping Denoting by ν the counting measure on (N, P(N)) given by ν({j}) = 1 for all j ∈ N, we can express the mapping (A.13) as Taking into account estimate (A.14), Lebesgue's dominated convergence theorem yields the continuity of the mapping (A.13).
Proof. Let h ∈ H be arbitrary. There exists n ∈ N with h ≤ n. By the Cauchy-Schwarz inequality and (2.7), (3.4) we have showing (A.15). Now, let n ∈ N and h 1 , h 2 ∈ H with h 1 , h 2 ≤ n be arbitrary. By the Cauchy-Schwarz inequality and (2.6) we obtain which, in view of (3.4), proves that α B also satisfies (2.2). Furthermore, the mapping γ B also satisfies (2.6), (2.7), which directly follows from its Definition (A.17).
A.19. Lemma. For every set B ∈ E with F (B c ) < ∞ the process is a càdlàg, adapted process with N 0 = 0, N ∈ N 0 and ∆N ∈ {0, 1} up to an evanescent set, and we have the representation which provides the representation (A.18) and shows that N ∈ N 0 . Since < ∞ for all t ≥ 0, we deduce that P(N t < ∞) = 1 for all t ≥ 0. Therefore, the representation (A. 18) shows that the process N is càdlàg, adapted with N ∈ N 0 up to an evanescent set. Since µ(ω; {t} × E) ≤ 1 for all (ω, t) ∈ Ω × R + by the definition of an integer-valued random measure, see [19,Def. II.1.13], we obtain ∆N ∈ {0, 1}.
For any set B ∈ E we define the mapping B : Ω → R + as For the representation (A.20) below we recall that for any stopping time τ and any set A ∈ F τ the mapping τ A : Ω → R + given by is also a stopping time. Proof. This is a direct consequence of Lemma A.19.
We shall now consider the SPDE Proof. Let r be a local mild solution to (1.1) with lifetime τ . We define the process r B by Then, r B is càdlàg and adapted, because γ(r B B − , ξ B ) is F B -measurable, and, since τ ≤ B , we have Therefore, we have (A. 22), and hence (r τ ) − = ((r B ) τ ) − . Since r is a local mild solution to (1.1) with lifetime τ , we have P-almost surely Hence, by the Definitions (A.16), (A.17) of α B , γ B we get P-almost surely Therefore, we obtain P-almost surely showing that r B is a local mild solution to (A.21) with lifetime τ . This proves the first statement. Now, let r B be a local mild solution to (A.21) with lifetime τ . We define the process r by Then, r is càdlàg and adapted, because γ(r B B − , ξ B ) is F B -measurable, and, since τ ≤ B , we have Therefore, we have (A. 22), and hence (r τ ) − = ((r B ) τ ) − . Since r B is a local mild solution to (A.21) with lifetime τ , we have P-almost surely Hence, by the Definitions (A.16), (A.17) of α B , γ B we get P-almost surely Arguing as in (A.23), we have P-almost surely showing that r is a local mild solution to (1.1) with lifetime τ . This proves the second statement.
A.22. Lemma. Let G 1 , G 2 be metric spaces such that G 1 is separable. Let B ⊂ G 1 be a Borel set, let C ⊂ G 2 be a closed set and let δ : Then, we even have Proof. By separability of G 1 there exists a countable set D, which is dense in B. By (A.24), for each h ∈ D there exists an F -nullset N h such that δ(h, x) ∈ C for all x ∈ N c h . The set N := h∈D N h is also an F -nullset. Now, let h ∈ B be arbitrary. Then, there exists a sequence (h n ) n∈N ⊂ D with h n → h, and hence δ(h n , x) ∈ C for all n ∈ N and x ∈ N c .
Since δ(•, x) is continuous for all x ∈ E and the set C is closed in G 2 , we deduce Proof. We denote by by the linear growth condition (2.7), for all n ∈ N, all h ∈ B 2 with h ≤ n and all x ∈ E we have By (2.6) and Lebesgue's dominated convergence theorem, the mapping is continuous. Now, let h ∈ B 1 be arbitrary. Since B 1 is prelocally invariant in B 2 for (1.1), there exists a local mild solution r = r (h) to (1.1) with lifetime τ > 0 such that (r τ ) − ∈ B 1 and r τ ∈ B 2 up to an evanescent set. Taking into account [19, Thm. II.1.8], identity (A.9) and Lemma A.13, we obtain Therefore, we have P-almost surely Since the process r is càdlàg with (r τ ) − ∈ B 1 up to an evanescent set and the mapping (A.27) is continuous, the integrand appearing in (A.28) is continuous in s = 0. Thus, we deduce that Therefore, by Lemma A.13 we obtain P-almost surely proving (A.30).
A.25. Corollary. Let B ⊂ C ⊂ H be two nonempty Borel sets such that C is closed in H and h + γ(h, x) ∈ C for F -almost all x ∈ E, for all h ∈ B.
Let h 0 : Ω → H be an F 0 -measurable random variable and let r = r (h0) be a local mild solution to (1.1) with lifetime τ > 0 such that (r τ ) − ∈ B up to an evanescent set. Then we have r τ ∈ C up to an evanescent set.
Proof. By the closedness of C in H, we have r τ 1 [[0,τ [[ ∈ C up to an evanescent set.
Thus, the statement follows from Proposition A.24.

Appendix B. Finite dimensional submanifolds with boundary in Hilbert spaces
For convenience of the reader, we provide the crucial properties of finite dimensional submanifolds with boundary in Hilbert spaces. For more details, we refer to any textbook about manifolds, e.g., [1], [20] or [29].
Let H be a Hilbert space and let m ∈ N be a positive integer. We denote by R m + the set of m-tuples y ∈ R m with non-negative first coordinate y 1 ≥ 0, that is Let k ∈ N be arbitrary.
For a C k -map φ : V ⊂ R m + → H and y ∈ V we define the derivative Dφ(y) := Dφ(y). Note that this definition does not depend on the choice ofφ.
The following lemma is a standard result, whence we omit the proof.
(2) For each y ∈ ∂V we have Dφ(y)R m + ⊂ R m + . Hence, boundary points of V are mapped to boundary points of W under a Since parametrizations of ∂M are provided by restricting parametrizations φ : In particular, we see that For a subset A ⊂ H we define A ⊥ := {h ∈ H : h, g = 0 for all g ∈ A}, A + := {h ∈ H : h, g ≥ 0 for all g ∈ A}.
In order to introduce the inward pointing normal (tangent) vectors at boundary points of the submanifold M, we require the following auxiliary result. The proof is elementary and therefore omitted.
Moreover, for each h ∈ ∂M we have B.8. Definition. For each h ∈ ∂M we call η h the inward pointing normal (tangent) vector to ∂M at h.
Then, for every h ∈ U ∩ ∂M there exists a unique number λ > 0 such that Proof. Let h ∈ U ∩ ∂M be arbitrary. We define the continuous linear functional There is a unique z ∈ R m such that In order to complete the proof, we shall show that z = λe 1 for some λ > 0. By identity (B.6) from Lemma B.7, for any v ∈ R m we have (v) = 0 if and only if Dφ(y)v ∈ T h ∂M, which, in view of (B.3), means that v ∈ ∂R m + . This shows ker( ) = ∂R m + , and hence, there exists a unique λ ∈ R such that z = λe 1 . Consequently, identity (B.
Proof. Let h 0 ∈ M be arbitrary, let φ : V ⊂ R m + → U ∩ M be a parametrization around h 0 and set y 0 is open in U ∩ M and φ(K) is compact. Therefore, and since U is an open neighborhood of h 0 , there exists 0 > 0 such that Let 0 ≤ ≤ 0 be arbitrary. Since φ(X) ⊂ φ(K) ⊂ U ∩ M, we have the identity Proof. Taking  We set φ :=φ| V , N :=f −1 (M), f :=f | N and ψ := f −1 • φ. Then, φ : V ⊂ R m + → M is a parametrization, N is an m-dimensional C k -submanifold with boundary of R m and ψ : V ⊂ R m + → N is a parametrization. By the inverse mapping theorem, see [1,Thm. 2.5.2], the parametrization ψ is a local diffeomorphism. Hence, arguing as in [13, Remark 6.1.1], we may assume that the mappings φ, ψ, Φ := φ −1 , Ψ := ψ −1 (after restricting to smaller neighborhoods, if necessary) have the desired extensions.
C.2. Definition. We introduce the following notions: (1) Let h 0 ∈ M be arbitrary. We say that γ satisfies the -δ-jump condition in h 0 , if there exists 0 > 0 such that for every 0 < ≤ 0 the set B (h 0 ) ∩ M is compact, and there are 0 < δ < and a set B ∈ E with F (B c ) < ∞ such that h + γ(h, x) ∈ B (h 0 ) ∩ M for F -almost all x ∈ B, for all h ∈ B δ (h 0 ) ∩ M.
For a closed subspace K ⊂ H we denote by Π K : H → K the orthogonal projection on K, that is, for each h ∈ H the vector Π K h is the unique element from K such that C.5. Lemma. Suppose that γ satisfies the -δ-jump condition on M. Then, the following statements are true: (1) For each h ∈ M we have E Π (T h M) ⊥ γ(h, x) F (dx) < ∞. (C.11) (2) The mapping is continuous.
Let G be another separable Hilbert space and let N an m-dimensional C 3submanifold with boundary of G. We assume there exist parametrizations φ : V ⊂ R m + → M and ψ : V ⊂ R m + → N . Defining f := φ • ψ −1 : N → M and g := ψ • φ −1 : M → N , the situation is illustrated by the following diagram: In the sequel, for z ∈ ∂N the vector ξ z denotes the inward pointing normal (tangent) vector to ∂N at z. and that the following conditions are satisfied: Then, the following conditions also hold true: (1) For each w ∈ T h M we have Dg(h)w ∈ T z N .
(3) For each w ∈ T h ∂M we have Dg(h)w ∈ T z ∂N .
Proof. Let w ∈ T h M be arbitrary and set y := φ −1 (h) ∈ V . By Lemma B.