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

Authors: Stefan Tappe
Journal: International Journal of Stochastic Analysis
 (2012)

---

## 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.


---

## Body

## 1. Introduction

Semilinear stochastic partial differential equations (SPDEs) on Hilbert spaces, being of the type(1.1)Zt=(AZt+α(t,Zt))dt+σ(t,Zt)dWt,Z0=z0,
have widely been studied in the literature, see, for example, [1–4]. In (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 an SPDE of the type
(1.2)dZt=(AZt+α(t,Zt))dt+σ(t,Zt)dWt+∫E‍γ(t,Zt-,x)(μ(dt,dx)-F(dx)dt),Z0=z0,
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 [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(1.3)dZt=(AZt+α(t,Zt))dt+σ(t,Zt)dWt+∫B‍γ(t,Zt-,x)(μ(dt,dx)-F(dx)dt)+∫Bc‍γ(t,Zt-,x)μ(dt,dx),Z0=z0,
where B∈ℰ is a set with F(Bc)<∞. Then, the integral ∫B‍ represents the small jumps, and ∫Bc‍ 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 Theorem4.5) concerning existence and uniqueness of local and global mild solutions to (1.3):
(1)
if(α,σ,γ|B) are locally Lipschitz and of linear growth, then existence and uniqueness of global mild solutions to (1.3) hold;(2)
if(α,σ,γ|B) are locally Lipschitz and locally bounded, then existence and uniqueness of local mild solutions to (1.3) hold;(3)
if(α,σ,γ|B) 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 A 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
(1.4)dYt=a(t,Yt)dt+b(t,Yt)dWt+∫B‍c(t,Yt-,x)(μ(dt,dx)-F(dx)dt)+∫Bc‍c(t,Yt-,x)μ(dt,dx),Y0=y0.
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
(1.5)dYt=a(t,Yt)dt+b(t,Yt)dWt+∫B‍c(t,Yt-,x)(μ(dt,dx)-F(dx)dt),Y0=y0,
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 Section2, 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(Ω,ℱ,𝔽,ℙ) with 𝔽=(ℱt)t≥0 be a filtered probability space satisfying the usual conditions.LetU 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 (ej)j∈ℕ of U and a sequence (λj)j∈ℕ⊂(0,∞) with ∑j∈ℕ‍λj<∞ such that
(2.1)Qej=λjej∀j∈ℕ,
namely, the λj are the eigenvalues of Q, and each ej is an eigenvector corresponding to λj. The space U0∶=Q1/2(U), equipped with the inner product
(2.2)〈u,v〉U0∶=〈Q-1/2u,Q-1/2v〉U,
is another separable Hilbert space, and (λjej)j∈ℕ is an orthonormal basis. Let W be a U-valued Q-Wiener process, see [1, page 86, 87]. For another separable Hilbert space H, we denote by L20(H)∶=L2(U0,H) the space of Hilbert-Schmidt operators from U0 into H, which, endowed with the Hilbert-Schmidt norm
(2.3)∥Φ∥L20(H)∶=(∑j∈ℕ∥Φ(λjej)∥2‍)1/2,Φ∈L20(H),
itself is a separable Hilbert space.Let(E,ℰ) be a measurable space which we assume to be a Blackwell space (see [16, 17]). 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 [13, Definition II.1.20]. Then its compensator is of the form dt⊗F(dx), where F is a σ-finite measure on (E,ℰ).For the following definitions, letτ be a finite stopping time.(i)
We define the new filtration𝔽(τ)=(ℱt(τ))t≥0 by
(2.4)ℱt(τ)∶=ℱτ+t,t≥0.(ii)
We define the newU-valued process W(τ) by
(2.5)Wt(τ)∶=Wτ+t-Wτ,t≥0.(iii)
We define the new random measureμ(τ) on ℝ+×E by
(2.6)μ(τ)(ω;B)∶=μ(ω;Bτ(ω)),ω∈Ω,B∈ℬ(ℝ+)⊗ℰ,
where we use the notation
(2.7)Bτ∶={(t+τ,x)∈ℝ+×E:(t,x)∈B}.Then,W(τ) is an 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. [18, Lemma 4.6].Lemma 2.1.
Letϱ be another stopping time. Then, the mapping (ϱ-τ)+ is an 𝔽(τ)-stopping time.Proof.
For everyt∈ℝ+, we have
(2.8){(ϱ-τ)+≤t}={ϱ-τ≤t}={ϱ≤τ+t}∈ℱτ+t=ℱt(τ),
showing that (ϱ-τ)+ is an 𝔽(τ)-stopping time.Denoting by𝒫(τ) the predictable σ-algebra relative to the filtration 𝔽(τ), we have the following auxiliary result.Lemma 2.2.
The following statements are true:(1)
the mapping(2.9)θτ:Ω×ℝ+→Ω×ℝ+,θτ(ω,t)∶=(ω,τ(ω)+t)
is 𝒫(τ)-𝒫-measurable.(2)
the mapping(2.10)ϑτ:Ω→Ω×ℝ+,ϑτ(ω)∶=(ω,τ(ω))
is ℱτ-𝒫-measurable.Proof.
According to [13, Theorem I.2.2], the system of sets
(2.11){A×{0}:A∈ℱ0}∪{⟦0,ϱ⟧:ϱisastoppingtime}
is a generating system of the predictable σ-algebra 𝒫. For any set A∈ℱ0, we have
(2.12)θτ-1(A×{0})=(A∩{τ=0})×{0}∈𝒫(τ).
Furthermore, for any 𝔽-stopping time ϱ, we have
(2.13)θτ-1(⟦0,ϱ⟧)=θτ-1({(ω,t)∈Ω×ℝ+:0≤t≤ϱ(ω)})={(ω,t)∈Ω×ℝ+:0≤τ(ω)+t≤ϱ(ω)}={(ω,t)∈Ω×ℝ+:0≤t≤ϱ(ω)-τ(ω)}=⟦0,ϱ-τ⟧=⟦0,(ϱ-τ)+⟧∖({τ>ϱ}×{0})∈𝒫(τ),
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
(2.14){A×{0}:A∈ℱ0}∪{A×(s,t]:s<t,A∈ℱs}
is a generating system of the predictable σ-algebra 𝒫. For any set A∈ℱ0, we have
(2.15)ϑτ-1(A×{0})=A∩{τ=0}∈ℱ0⊂ℱτ.
Furthermore, for all s,t∈ℝ+ with s<t and A∈ℱs, we have
(2.16)ϑτ-1(A×(s,t])=A∩{s<τ}∩{τ≤t}∈ℱτ,
establishing the second statement.Let us further investigate the Poisson random measureμ. According to [13, Proposition II.1.14], there exist a sequence (κn)n∈ℕ of finite stopping times with ⟦κn⟧∩⟦κm⟧=∅ 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
(2.17)ℙ(∫tu∫E∥γ(s,x)∥‍μ(ds,dx)<∞‍)=1,
we have
(2.18)∫tu∫E‍γ(s,x)μ(ds,dx)=∑n∈ℕ‍γ(κn,ξκn)1{t<κn≤u}.
Let B∈ℰ be a set with F(Bc)<∞. We define the mappings ϱk:Ω→ℝ¯+, k∈ℕ0 as
(2.19)ϱk∶=inf{t≥0:μ([0,t]×Bc)=k},k∈ℕ0.Lemma 2.3.
The following statements are true:(1)
for eachk∈ℕ, the mapping ϱk is a finite stopping time,(2)
one hasϱ0=0 and ℙ(ϱk<ϱk+1)=1 for all k∈ℕ0,(3)
one hasℙ(ϱk→∞)=1.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).Letℋ be a separable Hilbert space, and let B∈ℰ be a set with F(Bc)<∞. Furthermore, let a:Ω×ℝ+×ℋ→ℋ and b:Ω×ℝ+×ℋ→L20(ℋ) be 𝒫⊗ℬ(ℋ)-measurable mappings, and let c:Ω×ℝ+×ℋ×E→ℋ be a 𝒫⊗ℬ(ℋ)⊗ℰ-measurable mapping.Definition 3.1.
One says that existence of (local) strong solutions to (1.4) holds, if for each ℱ0-measurable random variable y0:Ω→ℋ, there exists a (local) strong solution to (1.4) with initial condition y0 (and some strictly positive lifetime τ>0).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 y0 and y0′ (and lifetimes τ and τ') we have up to indistinguishability
(3.1)Y1{y0=y0′}=Y′1{y0=y0′},(Yτ∧τ′1{y0=y0′}=(Y′)τ∧τ′1{y0=y0′}).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 3.3.
One says that the mappings(a,b,c|B) are locally Lipschitz if ℙ-almost surely
(3.2)(∫B∥c(t,y,x)∥2‍F(dx))1/2<∞,∀t∈ℝ+andally∈ℋ,
and for each n∈ℕ, there is a nondecreasing function Ln:ℝ+→ℝ+ such that ℙ-almost surely
(3.3)∥a(t,y1)-a(t,y2)∥≤Ln(t)∥y1-y2∥,∥b(t,y1)-b(t,y2)∥L20(ℋ)≤Ln(t)∥y1-y2∥,(∫B∥c(t,y1,x)-c(t,y2,x)∥2‍F(dx))1/2≤Ln(t)∥y1-y2∥,
for all t∈ℝ+ and all y1,y2∈ℋ with ∥y1∥,∥y2∥≤n.Definition 3.4.
One says that the mappings(a,b,c|B) satisfy the linear growth condition if there exists a nondecreasing function K:ℝ+→ℝ+ such that ℙ-almost surely
(3.4)∥a(t,y)∥≤K(t)(1+∥y∥),∥b(t,y)∥L20(ℋ)≤K(t)(1+∥y∥),(∫B‍∥c(t,y,x)∥2F(dx))1/2≤K(t)(1+∥y∥),
for all t∈ℝ+ and all y∈ℋ.Definition 3.5.
One says that the mappings(a,b,c|B) are locally bounded if for each n∈ℕ, there is a nondecreasing function Mn:ℝ+→ℝ+ such that ℙ-almost surely
(3.5)∥a(t,y)∥≤Mn(t),∥b(t,y)∥L20(ℋ)≤Mn(t),(∫B‍∥c(t,y,x)∥2F(dx))1/2≤Mn(t),
for all t∈ℝ+ and all y∈ℋ with ∥y∥≤n.For a finite stopping timeτ and a set Γ∈ℱτ, we define the mappings a(τ,Γ):Ω×ℝ+×ℋ→ℋ, b(τ,Γ):Ω×ℝ+×ℋ→L20(ℋ), and c(τ,Γ):Ω×ℝ+×ℋ×E→ℋ as
(3.6)a(τ,Γ)(t,y)∶=a(τ+t,y)1Γ,(3.7)b(τ,Γ)(t,y)∶=b(τ+t,y)1Γ,(3.8)c(τ,Γ)(t,y,x)∶=c(τ+t,y,x)1Γ.
By Lemma 2.2, the mappings a(τ,Γ) and b(τ,Γ) are 𝒫(τ)⊗ℬ(ℋ)-measurable, and c(τ,Γ) is 𝒫(τ)⊗ℬ(ℋ)⊗ℰ-measurable. We will also use the notation
(3.9)a(τ)∶=a(τ,Ω),b(τ)∶=b(τ,Ω),c(τ)∶=c(τ,Ω).Lemma 3.6.
Suppose thatτ1Γ is bounded. Then, the following statements are true:(1)
if(a,b,c|B) are locally Lipschitz, then (a(τ,Γ),b(τ,Γ),c(τ,Γ)|B) are locally Lipschitz, too;(2)
if(a,b,c|B) satisfy the linear growth condition, then (a(τ,Γ),b(τ,Γ),c(τ,Γ)|B) satisfy the linear growth condition, too.Proof.
Suppose that(a,b,c|B) satisfy the linear growth condition. Since τ1Γ is bounded, there exists a constant T≥0 such that τ1Γ≤T. The mapping K~∶=K(•+T):ℝ+→ℝ+ is nondecreasing, and we have ℙ-almost surely
(3.10)∥a(τ,Γ)(t,y)∥=∥a(t+τ,y)1Γ∥≤K(t+τ)1Γ(1+∥y∥)≤K~(t)(1+∥y∥),
for all t∈ℝ+ and y∈ℋ. Analogous estimates for b(τ,Γ) and c(τ,Γ) prove that (a(τ,Γ),b(τ,Γ),c(τ,Γ)|B) satisfy the linear growth condition, too. The remaining statement is proven analogously.Lemma 3.7.
Letτ and ϱ be two finite stopping times, and let Γ∈ℱτ be a set with Γ⊂{τ≤ϱ}. If Y is an 𝔽-adapted local strong solution to (1.4) with lifetime ϱ, then
(3.11)Y(τ,Γ)∶=Yτ+•1Γ
is an 𝔽(τ)-adapted local strong solution to (1.4) with parameters
(3.12)a=a(τ,Γ),b=b(τ,Γ),c=c(τ,Γ),W=W(τ),μ=μ(τ),
initial condition Yτ1Γ, and lifetime (ϱ-τ)+.Proof.
The processY(τ,Γ) given by (3.11) is 𝔽(τ) adapted, and we have
(3.13)Yt(τ,Γ)1⟦0,(ϱ-τ)+⟧(t)=Yτ+t1Γ1⟦0,(ϱ-τ)+⟧(t)=[Yτ+(Yτ+t-Yτ)]1Γ1⟦0,(ϱ-τ)+⟧(t)=[Yτ+∫ττ+t‍a(s,Ys)ds+∫ττ+t‍b(s,Ys)dWsYτ+∫ττ+t‍∫B‍c(s,Ys-,x)(μ(ds,dx)-F(dx)ds)Yτ+∫ττ+t‍∫Bc‍c(s,Ys-,x)μ(ds,dx)]1Γ1⟦0,(ϱ-τ)+⟧(t).
Therefore, we obtain
(3.14)Yt(τ,Γ)1⟦0,(ϱ-τ)+⟧(t)=[Yτ1Γ+∫0t‍a(τ+s,Yτ+s)1Γds+∫0t‍b(τ+s,Yτ+s)1ΓdWs(τ)Yτ+∫0t‍∫B‍c(τ+s,Y(τ+s)-,x)1Γ(μ(τ)(ds,dx)-F(dx)ds)Yτ+∫0t‍∫Bc‍c(τ+s,Y(τ+s)-,x)1Γμ(τ)(ds,dx)]1⟦0,(ϱ-τ)+⟧(t).
Taking into account the Definitions (3.6)–(3.8) of a(τ,Γ), b(τ,Γ), c(τ,Γ) and the Definition (3.11) of Y(τ,Γ), it follows that
(3.15)Yt(τ,Γ)1⟦0,(ϱ-τ)+⟧(t)=[Yτ1Γ+∫0t‍a(τ,Γ)(s,Ys(τ,Γ))ds+∫0t‍b(τ,Γ)(s,Ys(τ,Γ))dWs(τ)Yτ+∫0t‍∫B‍c(τ)(s,Ys-(τ,Γ),x)(μ(τ)(ds,dx)-F(dx)ds)Yτ+∫0t‍∫Bc‍c(τ)(s,Ys-(τ,Γ),x)μ(τ)(ds,dx)]1⟦0,(ϱ-τ)+⟧(t).
Consequently, Y(τ,Γ) is a local strong solution to (1.4) with parameters (3.12), initial condition Yτ1Γ, and lifetime (ϱ-τ)+.Lemma 3.8.
Letτ≤ϱ be two finite stopping times. If Y(0) is an 𝔽-adapted local strong solution to (1.4) with lifetime τ, and Y(τ) is an 𝔽(τ)-adapted local strong solution to (1.4) with parameters
(3.16)a=a(τ),b=b(τ),c=c(τ),W=W(τ),μ=μ(τ),
initial condition Yτ(0), and lifetime ϱ-τ, then
(3.17)Y∶=Y(0)1⟦0,τ⟧+Y•-τ(τ)1⟧τ,ϱ⟧
is an 𝔽-adapted local strong solution to (1.4) with lifetime ϱ.Proof.
Lett∈ℝ+ be arbitrary. Then, the random variable Yt(0)1{τ≥t} is ℱt-measurable. Let C∈ℬ(ℋ) be an arbitrary Borel set. We define DC∈ℱt as
(3.18)DC∶={({τ<t}∩{t≤ϱ})cif0∈C,∅if0∉C.
According to Lemma 2.1, the mapping (t-τ)+ is an 𝔽(τ)-stopping time. Therefore, we get
(3.19){Y(t-τ)+(τ)∈C}∈ℱ(t-τ)+(τ)=ℱτ+(t-τ)+,
and hence, we obtain
(3.20){Yt-τ(τ)1{τ<t≤ϱ}∈C}={Y(t-τ)+(τ)1{τ<t≤ϱ}∈C}=({τ<t}∩{t≤ϱ}∩{Y(t-τ)+(τ)∈C})∪DC=({ϱ≥t}∩{τ≠t}∩{τ≤t}∩{Y(t-τ)+(τ)∈C})∪DC=({ϱ≥t}∩{τ≠t}∩{τ+(t-τ)+=t}∩{Y(t-τ)+(τ)∈C})∪DC∈ℱt,
showing that the process Y defined in (3.17) is 𝔽 adapted. Moreover, since Y(τ) is local strong solution to (1.4) with initial condition Yτ(0) and lifetime ϱ-τ, we have
(3.21)Yt-τ(τ)1⟧τ,ϱ⟧(t)=[Yτ(0)+∫0t-τ‍a(τ)(s,Ys(τ))ds+∫0t-τ‍b(τ)(s,Ys(τ))dWs(τ)ds+∫0t-τ‍∫B‍c(τ)(s,Ys-(τ),x)(μ(τ)(ds,dx)-F(dx)ds)ds+∫0t-τ‍∫Bc‍c(τ)(s,Ys-(τ),x)μ(τ)(ds,dx)]1⟧τ,ϱ⟧(t).
By the Definitions (3.6)–(3.9) of a(τ), b(τ), c(τ), we obtain
(3.22)Yt-τ(τ)1⟧τ,ϱ⟧(t)=[Yτ(0)+∫0t-τ‍a(τ+s,Ys(τ))ds+∫0t-τ‍b(τ+s,Ys(τ))dWs(τ)ds+∫0t-τ‍∫B‍c(τ+s,Ys-(τ),x)(μ(τ)(ds,dx)-F(dx)ds)ds+∫0t-τ‍∫Bc‍c(τ+s,Ys-(τ),x)μ(τ)(ds,dx)]1⟧τ,ϱ⟧(t).
Therefore, we get
(3.23)Yt-τ(τ)1⟧τ,ϱ⟧(t)=[Yτ(0)+∫τt‍a(s,Ys-τ(τ))ds+∫τt‍b(s,Ys-τ(τ))dWsds+∫τt‍∫B‍c(s,Y(s-τ)-(τ),x)(μ(ds,dx)-F(dx)ds)ds+∫τt‍∫Bc‍c(s,Y(s-τ)-(τ),x)μ(ds,dx)]1⟧τ,ϱ⟧(t).
By the Definition (3.17) of Y, we obtain
(3.24)Yt-τ(τ)1⟧τ,ϱ⟧(t)=[Yτ(0)+∫τt‍a(s,Ys)ds+∫τt‍b(s,Ys)dWsds+∫τt‍∫B‍c(s,Ys-,x)(μ(ds,dx)-F(dx)ds)ds+∫τt‍∫Bc‍c(s,Ys-,x)μ(ds,dx)]1⟧τ,ϱ⟧(t).
Since Y(0) is a local strong solution to (1.4) with lifetime τ, we deduce that the process Y given by (3.17) is a local strong solution to (1.4) with lifetime ϱ.Letk∈ℕ0 be arbitrary. By Lemmas 2.1 and 2.3, the mapping ϱk+1-ϱk is a strictly positive 𝔽(ϱk)-stopping time. Furthermore, let Γ∈ℱϱk be arbitrary, and let y0(ϱk):Ω→ℋ be an arbitrary ℱ0(ϱk)-measurable random variable.Lemma 3.9.
IfY(ϱk,Γ) is an 𝔽(ϱk)-adapted local strong solution to (1.4) with parameters
(3.25)a=a(ϱk,Γ),b=b(ϱk,Γ),c=c(ϱk,Γ),W=W(ϱk),μ=μ(ϱk),
initial condition y0(ϱk)1Γ, and lifetime τ, then
(3.26)Y(ϱk,Γ)-∶=Y(ϱk,Γ)-c(ϱk+1,Y(ϱk+1-ϱk)-(ϱk,Γ),ξϱk+1)1⟦ϱk+1-ϱk⟧1{ϱk+1-ϱk≤τ}1Γ
is a 𝔽(ϱk)-adapted local strong solution to (1.5) with parameters (3.25), initial condition y0(ϱk)1Γ, and lifetime τ∧(ϱk+1-ϱk).Proof.
We defineJ:Ω→ℋ as
(3.27)J∶=c(ϱk+1,Y(ϱk+1-ϱk)-(ϱk,Γ),ξϱk+1)1{ϱk+1-ϱk≤τ}1Γ,
and the stochastic process (Jt)t≥0 as Jt∶=J1⟦ϱk+1-ϱk⟧(t). By Lemma 2.2, the mapping J is ℱϱk+1-measurable. Let C∈ℬ(ℋ) be an arbitrary Borel set. We define DC∈ℱt as
(3.28)DC∶={{ϱk+1-ϱk≠t}if0∈C,∅if0∉C.
Then, for each t∈ℝ+, we have
(3.29){Jt∈C}={J1⟦ϱk+1-ϱk⟧(t)∈C}=({J∈C}∩{ϱk+1-ϱk=t})∪DC=({J∈C}∩{ϱk+1=ϱk+t})∪DC∈ℱϱk+t=ℱt(ϱk).
Consequently, the process Y(ϱk,Γ)- defined in (3.26) is 𝔽(ϱk)-adapted. Furthermore, by the Definition (3.26), we have
(3.30)Y-(ϱk,Γ)1⟦0,τ∧(ϱk+1-ϱk)⟧=Y-(ϱk,Γ)-1⟦0,τ∧(ϱk+1-ϱk)⟧,
and, by the Definition (3.8) of c(ϱk,Γ) and identity (2.18), we obtain
(3.31)(∫0t‍∫Bc‍c(ϱk,Γ)(s,Ys-(ϱk,Γ),x)μ(ϱk)(ds,dx))1⟦0,τ∧(ϱk+1-ϱk)⟧(t)=(∫0t‍∫Bc‍c(ϱk+s,Ys-(ϱk,Γ),x)1Γμ(ϱk)(ds,dx))1⟦0,τ∧(ϱk+1-ϱk)⟧(t)=(∫ϱkϱk+t‍∫Bc‍c(s,Y(s-ϱk)-(ϱk,Γ),x)1Γμ(ds,dx))1⟦0,τ∧(ϱk+1-ϱk)⟧(t)=(∑n∈ℕ‍c(κn,Y(κn-ϱk)-(ϱk,Γ),ξκn)1{ξκn∉B}1{ϱk<κn≤ϱk+t})1⟦0,τ∧(ϱk+1-ϱk)⟧(t)1Γ=c(ϱk+1,Y(ϱk+1-ϱk)-(ϱk,Γ),ξϱk+1)1⟦ϱk+1-ϱk⟧(t)1{ϱk+1-ϱk≤τ}1Γ,
showing that Y(ϱk,Γ)- is a local strong solution to (1.5) with parameters (3.25) and lifetime τ∧(ϱk+1-ϱk).Lemma 3.10.
IfY(ϱk,Γ)- is an 𝔽(ϱk)-adapted local strong solution to (1.5) with parameters (3.25), initial condition y0(ϱk)1Γ, and lifetime τ, then
(3.32)Y(ϱk,Γ)∶=Y(ϱk,Γ)-+c(ϱk+1,Y(ϱk+1-ϱk)-(ϱk,Γ)-,ξϱk+1)1⟦ϱk+1-ϱk⟧1{ϱk+1-ϱk≤τ}1Γ
is a 𝔽(ϱk)-adapted local strong solution to (1.4) with parameters (3.25), initial condition y0(ϱk)1Γ, and lifetime τ∧(ϱk+1-ϱk).Proof.
The proof is analogous to that of Lemma3.9.


### 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(a,b,c|B) 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(a,b,c|B) are locally Lipschitz. Then, uniqueness of local strong solutions to (1.4) holds.Proof.
LetY and Y' be two local strong solutions to (1.5) with initial conditions y0 and y0′, and lifetimes τ and τ'. By induction, we will prove that up to indistinguishability
(3.33)Y1⟦0,τ∧τ'∧ϱk⟧1{y0=y0′}=Y′1⟦0,τ∧τ′∧ϱk⟧1{y0=y0′}∀k∈ℕ0.
The identity (3.33) holds true for k=0, because by Lemma 2.3, we have ϱ0=0.
For the induction stepk→k+1, we suppose that identity (3.33) is satisfied. We define the stopping time τk∶=τ∧τ'∧ϱk+1 and the set Γk∶={ϱk≤τk}∩{y0=y0′}∈ℱϱk. By Lemma 3.7, the processes Y(ϱk,Γk)∶=Yϱk+•1Γk and Y'(ϱk,Γk):=Yϱk+•′1Γk defined according to (3.11) are 𝔽(ϱk)-adapted local strong solutions to (1.4) with parameters (3.12), where τ=ϱk and Γ=Γk, initial conditions Yϱk1Γk and Yϱk′1Γk, and lifetime (τk-ϱk)+.
Letn∈ℕ be arbitrary, and set Γkn∶=Γk∩{ϱk≤n}∈ℱϱk. The processes Y(ϱk,Γkn)∶=Y(ϱk,Γk)1Γkn and Y'(ϱk,Γkn)∶=Y'(ϱk,Γk)1Γkn are 𝔽(ϱk)-adapted local strong solutions to (1.4) with parameters (3.25), where Γ=Γkn, initial conditions Yϱk1Γkn and Yϱk′1Γkn, and lifetime (τk-ϱk)+. By Lemma 3.9, the processes Y(ϱk,Γkn)- and Y'(ϱk,Γkn)- defined according to (3.26) are 𝔽(ϱk)-adapted local strong solutions to (1.5) with parameters (3.25), where Γ=Γkn, initial conditions Yϱk1Γkn and Yϱk′1Γkn, and lifetime (τk-ϱk)+. According to Lemma 3.6, the mappings (a(ϱk,Γkn),b(ϱk,Γkn),c(ϱk,Γkn)|B) are locally Lipschitz, too. Therefore, by Proposition 3.11, we have up to indistinguishability
(3.34)Y(ϱk,Γkn)-1⟦0,(τk-ϱk)+⟧=Y'(ϱk,Γkn)-1⟦0,(τk-ϱk)+⟧∀n∈ℕ.
By the Definition (3.26), we deduce that up to indistinguishability
(3.35)Y(ϱk,Γkn)1⟦0,(τk-ϱk)+⟧=Y'(ϱk,Γkn)1⟦0,(τk-ϱk)+⟧∀n∈ℕ,
and hence, we have up to indistinguishability
(3.36)Y(ϱk,Γk)1{ϱk≤n}1⟦0,(τk-ϱk)+⟧=Y'(ϱk,Γk)1{ϱk≤n}1⟦0,(τk-ϱk)+⟧∀n∈ℕ.
By Lemma 2.3, we have ℙ(ϱk<∞)=1, and hence, we get up to indistinguishability
(3.37)Y(ϱk,Γk)1⟦0,(τk-ϱk)+⟧=Y'(ϱk,Γk)1⟦0,(τk-ϱk)+⟧.
Therefore, we have up to indistinguishability
(3.38)Yϱk+•1{ϱk≤τk}1⟦0,(τk-ϱk)+⟧1{y0=y0′}=Yϱk+•′1{ϱk≤τk}1⟦0,(τk-ϱk)+⟧1{y0=y0′}.
Consequently, we have up to indistinguishability
(3.39)Y1{ϱk≤τk}1⟦ϱk,τk⟧1{y0=y0′}=Y'1{ϱk≤τk}1⟦ϱk,τk⟧1{y0=y0′}.
Together with the induction hypothesis, it follows that
(3.40)Y1⟦0,τk⟧1{y0=y0′}=Y'1⟦0,τk⟧1{y0=y0′},
which establishes (3.33). Since by Lemma 2.3, that we have ℙ(ϱk→∞)=1, we deduce
(3.41)Yτ∧τ'1{y0=y0′}=(Y′)τ∧τ'1{y0=y0′},
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(a,b,c|B) are locally Lipschitz and satisfy the linear growth condition. Then, existence of strong solutions to (1.5) holds.Proof.
If the mappings(a,b,c|B) are Lipschitz continuous, then we have existence and uniqueness of strong solutions to (1.5) for every initial condition y0∈ℒ2(ℱ0;ℋ), see, for example, [6, Corollary 10.3].
For(a,b,c|B) being locally Lipschitz and satisfying the linear growth condition, for any initial condition y0∈ℒ2(ℱ0;ℋ), we adopt the technique from the proof of [21, Theorem 4.11]. For k∈ℕ, we define the retraction
(3.42)Rk:ℋ→ℋ,Rk(y)∶={yif∥y∥≤k,ky∥y∥if∥y∥>k
and the mappings ak:Ω×ℝ+×ℋ→ℋ, bk:Ω×ℝ+×ℋ→L20(ℋ), and ck:Ω×ℝ+×ℋ×E→ℋ as
(3.43)ak∶=a∘Rk,bk∶=b∘Rk,ck(•,x)∶=c(•,x)∘Rk.
These mappings are Lipschitz continuous, and hence, there exists a strong solution Y(k) to the SDE (1.5) with parameters a=ak, b=bk, and c=ck, and initial condition y0. Using the linear growth condition, Gronwall’s lemma, and Doob’s martingale inequality, we can show that ℙ(τk→∞)=1, where
(3.44)τk∶=inf{t≥0:∥Yt(k)∥>k},k∈ℕ0,
that is, the solutions do not explode. Consequently, the process
(3.45)Y∶=y01⟦τ0⟧+∑k∈ℕ‍Y(k)1⟧τk-1,τk⟧
is a strong solution to (1.5) with initial condition y0.
Finally, for a generalℱ0-measurable initial condition y0:Ω→ℋ, the process Y∶=∑k∈ℕ‍Y(k)1Ωk is a strong solution to (1.5) with initial condition y0, where (Ωk)k∈ℕ⊂ℱ0 denotes the partition of Ω given by Ωk∶={∥y0∥∈[k-1,k)}, and where for each k∈ℕ the process Y(k) denotes a strong solution to (1.5) with initial condition y01Ωk.Theorem 3.14.
One supposes 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.
Lety0:Ω→ℋ be an arbitrary ℱ0-measurable random variable. By induction, we will prove that for each k∈ℕ0 there exists a local strong solution Y(k) to (1.4) with initial condition y0 and lifetime ϱk. By Lemma 2.3, we have ϱ0=0, providing the assertion for k=0.
For the induction stepk→k+1, let Y(k) be a local strong solution to (1.4) with initial condition y0 and lifetime ϱk. Let n∈ℕ be arbitrary, and set Γkn∶={ϱk∈[n-1,n)}∈ℱϱk. By Lemma 3.6, the mappings (a(ϱk,Γkn),b(ϱk,Γkn),c(ϱk,Γkn)|B) are locally Lipschitz, too. Therefore, by Proposition 3.13, there exists an 𝔽(ϱk)-adapted strong solution Y(ϱk,Γkn)- to (1.5) with parameters (3.25), where Γ=Γkn, and initial condition Yϱk(k)1Γkn. By Lemma 3.10, the process Y(ϱk,Γkn) defined according to (3.32) is an 𝔽(ϱk)-adapted local strong solution to (1.4) with parameters (3.25), where Γ=Γkn, initial condition Yϱk(k)1Γkn, and lifetime ϱk+1-ϱk. Noting that (Γkn)n∈ℕ is a partition of Ω, it follows that Y(ϱk)∶=∑n∈ℕ‍Y(ϱk,Γkn) is an 𝔽(ϱk)-adapted local strong solution to (1.4) with initial condition Yϱk(k) and lifetime ϱk+1-ϱk. By Lemma 3.8, the process
(3.46)Y(k+1)∶=Y(k)1⟦0,ϱk⟧+Y•-ϱk(ϱk)1⟧ϱk,ϱk+1⟧
defined according to (3.17) is an 𝔽-adapted local strong solution to (1.4) with initial condition y0 and lifetime ϱk+1.
Consequently, for eachk∈ℕ0, there exists a local strong solution Y(k) to (1.4) with initial condition y0 and lifetime ϱk. By Lemma 2.3, we have ℙ(ϱk→∞)=1. Hence, it follows that
(3.47)Y∶=y01⟦ϱ0⟧+∑k∈ℕ‍Y(k)1⟧ϱk-1,ϱk⟧
is an 𝔽-adapted strong solution to (1.4) with initial condition y0.Theorem 3.15.
One supposes that the mappings(a,b,c|B) are locally Lipschitz and locally bounded. Then, existence of local strong solutions to (1.4) holds.Proof.
Lety0:Ω→ℋ be an arbitrary ℱ0-measurable random variable. We define the partition (Ωk)k∈ℕ⊂ℱ0 of Ω by Ωk∶={∥y0∥∈[k-1,k)}. Furthermore, for each k∈ℕ, we define the mappings ak:Ω×ℝ+×ℋ→ℋ, bk:Ω×ℝ+×ℋ→L20(ℋ), and ck: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=ak, b=bk, and c=ck, and initial condition y01Ωk. The stopping time
(3.48)τk∶=inf{t≥0:∥Yt(k)∥>k}
is strictly positive, and Y(k) is a local strong solution to (1.4) with initial condition y01Ωk and lifetime τk. The stopping time τ∶=∑k∈ℕ‍τk1Ωk is strictly positive, and the process Y∶=∑k∈ℕ‍Y(k)1Ωk is a local strong solution to (1.4) with initial condition y0 and lifetime τ.


### 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(a,b,c|B) 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 Theorems3.12 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 p≥2, we need the estimate
(3.49)∥a(y1)-a(y2)∥p+∥b(y1)-b(y2)∥L20(ℋ)p+∫B‍∥c(y1,x)-c(y2,y)∥pF(dx)+(∫B∥c(y1,x)-c(y2,x)∥2‍F(dx))p/2≤κ(∥y1-y2∥p)∀y1,y2∈ℋ,
where κ:ℝ+→ℝ+ denotes a continuous, nondecreasing function with κ(0)=0, and further conditions, which are precisely stated in [14], must be fulfilled. These conditions are satisfied if κ is a continuous, nondecreasing, and concave function such that
(3.50)∫0ϵ‍1κ(u)du=∞foreachϵ>0.
In particular, we may choose κ(u)=u for u∈ℝ+, 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 ℋ=ℝ and b≡c≡0.Example 3.17.
We fix an arbitrary constant0<δ<exp(-1) and define the functions κ,ρ:ℝ+→ℝ+ by
(3.51)κ(u)∶={0,u=0,-ulnu,0<u<δ,-δlnδ-(1+lnδ)(u-δ)u≥δ,
as well as
(3.52)ρ(u)∶={0,u=0,u-ln(u2),0<u<δ,-δlnδ-(1+lnδ)(u2-δ)u≥δ,
compare with [22, Remark 1]. Let a:ℝ→ℝ be a mapping such that
(3.53)|a(y1)-a(y2)|≤ρ(|y1-y2|)∀y1,y2∈ℝ.
Then we have the estimate
(3.54)|a(y1)-a(y2)|2≤κ(|y1-y2|2)∀y1,y2∈ℝ,
showing that condition (3.49) with p=2 is satisfied. Moreover, κ is a continuous, nondecreasing, concave function, and condition (3.50) is satisfied, because for each 0<ϵ<δ, we have
(3.55)∫0ϵ‍1κ(u)du=-∫0ϵ‍1ulnudu=-ln|lnu||u=0u=ϵ=-ln|lnϵ|+limu→0ln|lnu|=∞.
Consequently, [14, Theorem 2.1] applies. However, we have
(3.56)ρ′(u)=-ln(u2)-1-ln(u2)foru∈(0,δ),
and thus, limu→0ρ'(u)=∞. Therefore, the mapping a:ℝ→ℝ might fail to be locally Lipschitz, and hence, Theorem 3.16 does not apply.Example 3.18.
Let us define the mappinga:ℝ→ℝ as follows. For n∈ℕ0, we define a on the interval [n,n+1] by
(3.57)a(y)∶={n,y∈[n,n+1-1n+1],n+(n+1)(y-(n+1-1n+1)),y∈[n+1-1n+1,n+1].
This defines the mapping a:ℝ+→ℝ, which we extend to a mapping a:ℝ→ℝ by symmetry
(3.58)a(y)∶=a(-y),y∈ℝ-.
Then, a is locally Lipschitz and satisfies the linear growth condition, and hence, Theorem 3.16 applies. However, there are no constant p≥2 and no continuous, nondecreasing function κ:ℝ+→ℝ+ with κ(0)=0 such that
(3.59)|a(y1)-a(y2)|p≤κ(|y1-y2|p)∀y1,y2∈ℝ.
Suppose, on the contrary, that there exists a continuous, nondecreasing function κ:ℝ+→ℝ+ with κ(0)=0 fulfilling (3.59). Then we have
(3.60)|κ(u)|≥1∀u∈(0,1].
Indeed, let u∈(0,1] be arbitrary. Then, there exists n∈ℕ with 1/n≤u. Moreover, by the definition of the mapping a:ℝ→ℝ, there are y1,y2∈ℝ such that
(3.61)|y1-y2|≤(1n)1/p,|a(y1)-a(y2)|=1.
Therefore, using the monotonicity of κ and (3.59), we obtain
(3.62)κ(u)≥κ(1n)≥κ(|y1-y2|p)≥|a(y1)-a(y2)|p=1,
showing (3.60). Now, the continuity of κ yields the contradiction κ(0)≥1. Consequently, condition (3.49) is not satisfied, and thus, we cannot use [14, Theorem 2.1] in this case.


## 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(a,b,c|B) 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(a,b,c|B) are locally Lipschitz. Then, uniqueness of local strong solutions to (1.4) holds.Proof.
LetY and Y' be two local strong solutions to (1.5) with initial conditions y0 and y0′, and lifetimes τ and τ'. By induction, we will prove that up to indistinguishability
(3.33)Y1⟦0,τ∧τ'∧ϱk⟧1{y0=y0′}=Y′1⟦0,τ∧τ′∧ϱk⟧1{y0=y0′}∀k∈ℕ0.
The identity (3.33) holds true for k=0, because by Lemma 2.3, we have ϱ0=0.
For the induction stepk→k+1, we suppose that identity (3.33) is satisfied. We define the stopping time τk∶=τ∧τ'∧ϱk+1 and the set Γk∶={ϱk≤τk}∩{y0=y0′}∈ℱϱk. By Lemma 3.7, the processes Y(ϱk,Γk)∶=Yϱk+•1Γk and Y'(ϱk,Γk):=Yϱk+•′1Γk defined according to (3.11) are 𝔽(ϱk)-adapted local strong solutions to (1.4) with parameters (3.12), where τ=ϱk and Γ=Γk, initial conditions Yϱk1Γk and Yϱk′1Γk, and lifetime (τk-ϱk)+.
Letn∈ℕ be arbitrary, and set Γkn∶=Γk∩{ϱk≤n}∈ℱϱk. The processes Y(ϱk,Γkn)∶=Y(ϱk,Γk)1Γkn and Y'(ϱk,Γkn)∶=Y'(ϱk,Γk)1Γkn are 𝔽(ϱk)-adapted local strong solutions to (1.4) with parameters (3.25), where Γ=Γkn, initial conditions Yϱk1Γkn and Yϱk′1Γkn, and lifetime (τk-ϱk)+. By Lemma 3.9, the processes Y(ϱk,Γkn)- and Y'(ϱk,Γkn)- defined according to (3.26) are 𝔽(ϱk)-adapted local strong solutions to (1.5) with parameters (3.25), where Γ=Γkn, initial conditions Yϱk1Γkn and Yϱk′1Γkn, and lifetime (τk-ϱk)+. According to Lemma 3.6, the mappings (a(ϱk,Γkn),b(ϱk,Γkn),c(ϱk,Γkn)|B) are locally Lipschitz, too. Therefore, by Proposition 3.11, we have up to indistinguishability
(3.34)Y(ϱk,Γkn)-1⟦0,(τk-ϱk)+⟧=Y'(ϱk,Γkn)-1⟦0,(τk-ϱk)+⟧∀n∈ℕ.
By the Definition (3.26), we deduce that up to indistinguishability
(3.35)Y(ϱk,Γkn)1⟦0,(τk-ϱk)+⟧=Y'(ϱk,Γkn)1⟦0,(τk-ϱk)+⟧∀n∈ℕ,
and hence, we have up to indistinguishability
(3.36)Y(ϱk,Γk)1{ϱk≤n}1⟦0,(τk-ϱk)+⟧=Y'(ϱk,Γk)1{ϱk≤n}1⟦0,(τk-ϱk)+⟧∀n∈ℕ.
By Lemma 2.3, we have ℙ(ϱk<∞)=1, and hence, we get up to indistinguishability
(3.37)Y(ϱk,Γk)1⟦0,(τk-ϱk)+⟧=Y'(ϱk,Γk)1⟦0,(τk-ϱk)+⟧.
Therefore, we have up to indistinguishability
(3.38)Yϱk+•1{ϱk≤τk}1⟦0,(τk-ϱk)+⟧1{y0=y0′}=Yϱk+•′1{ϱk≤τk}1⟦0,(τk-ϱk)+⟧1{y0=y0′}.
Consequently, we have up to indistinguishability
(3.39)Y1{ϱk≤τk}1⟦ϱk,τk⟧1{y0=y0′}=Y'1{ϱk≤τk}1⟦ϱk,τk⟧1{y0=y0′}.
Together with the induction hypothesis, it follows that
(3.40)Y1⟦0,τk⟧1{y0=y0′}=Y'1⟦0,τk⟧1{y0=y0′},
which establishes (3.33). Since by Lemma 2.3, that we have ℙ(ϱk→∞)=1, we deduce
(3.41)Yτ∧τ'1{y0=y0′}=(Y′)τ∧τ'1{y0=y0′},
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(a,b,c|B) are locally Lipschitz and satisfy the linear growth condition. Then, existence of strong solutions to (1.5) holds.Proof.
If the mappings(a,b,c|B) are Lipschitz continuous, then we have existence and uniqueness of strong solutions to (1.5) for every initial condition y0∈ℒ2(ℱ0;ℋ), see, for example, [6, Corollary 10.3].
For(a,b,c|B) being locally Lipschitz and satisfying the linear growth condition, for any initial condition y0∈ℒ2(ℱ0;ℋ), we adopt the technique from the proof of [21, Theorem 4.11]. For k∈ℕ, we define the retraction
(3.42)Rk:ℋ→ℋ,Rk(y)∶={yif∥y∥≤k,ky∥y∥if∥y∥>k
and the mappings ak:Ω×ℝ+×ℋ→ℋ, bk:Ω×ℝ+×ℋ→L20(ℋ), and ck:Ω×ℝ+×ℋ×E→ℋ as
(3.43)ak∶=a∘Rk,bk∶=b∘Rk,ck(•,x)∶=c(•,x)∘Rk.
These mappings are Lipschitz continuous, and hence, there exists a strong solution Y(k) to the SDE (1.5) with parameters a=ak, b=bk, and c=ck, and initial condition y0. Using the linear growth condition, Gronwall’s lemma, and Doob’s martingale inequality, we can show that ℙ(τk→∞)=1, where
(3.44)τk∶=inf{t≥0:∥Yt(k)∥>k},k∈ℕ0,
that is, the solutions do not explode. Consequently, the process
(3.45)Y∶=y01⟦τ0⟧+∑k∈ℕ‍Y(k)1⟧τk-1,τk⟧
is a strong solution to (1.5) with initial condition y0.
Finally, for a generalℱ0-measurable initial condition y0:Ω→ℋ, the process Y∶=∑k∈ℕ‍Y(k)1Ωk is a strong solution to (1.5) with initial condition y0, where (Ωk)k∈ℕ⊂ℱ0 denotes the partition of Ω given by Ωk∶={∥y0∥∈[k-1,k)}, and where for each k∈ℕ the process Y(k) denotes a strong solution to (1.5) with initial condition y01Ωk.Theorem 3.14.
One supposes 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.
Lety0:Ω→ℋ be an arbitrary ℱ0-measurable random variable. By induction, we will prove that for each k∈ℕ0 there exists a local strong solution Y(k) to (1.4) with initial condition y0 and lifetime ϱk. By Lemma 2.3, we have ϱ0=0, providing the assertion for k=0.
For the induction stepk→k+1, let Y(k) be a local strong solution to (1.4) with initial condition y0 and lifetime ϱk. Let n∈ℕ be arbitrary, and set Γkn∶={ϱk∈[n-1,n)}∈ℱϱk. By Lemma 3.6, the mappings (a(ϱk,Γkn),b(ϱk,Γkn),c(ϱk,Γkn)|B) are locally Lipschitz, too. Therefore, by Proposition 3.13, there exists an 𝔽(ϱk)-adapted strong solution Y(ϱk,Γkn)- to (1.5) with parameters (3.25), where Γ=Γkn, and initial condition Yϱk(k)1Γkn. By Lemma 3.10, the process Y(ϱk,Γkn) defined according to (3.32) is an 𝔽(ϱk)-adapted local strong solution to (1.4) with parameters (3.25), where Γ=Γkn, initial condition Yϱk(k)1Γkn, and lifetime ϱk+1-ϱk. Noting that (Γkn)n∈ℕ is a partition of Ω, it follows that Y(ϱk)∶=∑n∈ℕ‍Y(ϱk,Γkn) is an 𝔽(ϱk)-adapted local strong solution to (1.4) with initial condition Yϱk(k) and lifetime ϱk+1-ϱk. By Lemma 3.8, the process
(3.46)Y(k+1)∶=Y(k)1⟦0,ϱk⟧+Y•-ϱk(ϱk)1⟧ϱk,ϱk+1⟧
defined according to (3.17) is an 𝔽-adapted local strong solution to (1.4) with initial condition y0 and lifetime ϱk+1.
Consequently, for eachk∈ℕ0, there exists a local strong solution Y(k) to (1.4) with initial condition y0 and lifetime ϱk. By Lemma 2.3, we have ℙ(ϱk→∞)=1. Hence, it follows that
(3.47)Y∶=y01⟦ϱ0⟧+∑k∈ℕ‍Y(k)1⟧ϱk-1,ϱk⟧
is an 𝔽-adapted strong solution to (1.4) with initial condition y0.Theorem 3.15.
One supposes that the mappings(a,b,c|B) are locally Lipschitz and locally bounded. Then, existence of local strong solutions to (1.4) holds.Proof.
Lety0:Ω→ℋ be an arbitrary ℱ0-measurable random variable. We define the partition (Ωk)k∈ℕ⊂ℱ0 of Ω by Ωk∶={∥y0∥∈[k-1,k)}. Furthermore, for each k∈ℕ, we define the mappings ak:Ω×ℝ+×ℋ→ℋ, bk:Ω×ℝ+×ℋ→L20(ℋ), and ck: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=ak, b=bk, and c=ck, and initial condition y01Ωk. The stopping time
(3.48)τk∶=inf{t≥0:∥Yt(k)∥>k}
is strictly positive, and Y(k) is a local strong solution to (1.4) with initial condition y01Ωk and lifetime τk. The stopping time τ∶=∑k∈ℕ‍τk1Ωk is strictly positive, and the process Y∶=∑k∈ℕ‍Y(k)1Ωk is a local strong solution to (1.4) with initial condition y0 and lifetime τ.


## 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(a,b,c|B) 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 Theorems3.12 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 p≥2, we need the estimate
(3.49)∥a(y1)-a(y2)∥p+∥b(y1)-b(y2)∥L20(ℋ)p+∫B‍∥c(y1,x)-c(y2,y)∥pF(dx)+(∫B∥c(y1,x)-c(y2,x)∥2‍F(dx))p/2≤κ(∥y1-y2∥p)∀y1,y2∈ℋ,
where κ:ℝ+→ℝ+ denotes a continuous, nondecreasing function with κ(0)=0, and further conditions, which are precisely stated in [14], must be fulfilled. These conditions are satisfied if κ is a continuous, nondecreasing, and concave function such that
(3.50)∫0ϵ‍1κ(u)du=∞foreachϵ>0.
In particular, we may choose κ(u)=u for u∈ℝ+, 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 ℋ=ℝ and b≡c≡0.Example 3.17.
We fix an arbitrary constant0<δ<exp(-1) and define the functions κ,ρ:ℝ+→ℝ+ by
(3.51)κ(u)∶={0,u=0,-ulnu,0<u<δ,-δlnδ-(1+lnδ)(u-δ)u≥δ,
as well as
(3.52)ρ(u)∶={0,u=0,u-ln(u2),0<u<δ,-δlnδ-(1+lnδ)(u2-δ)u≥δ,
compare with [22, Remark 1]. Let a:ℝ→ℝ be a mapping such that
(3.53)|a(y1)-a(y2)|≤ρ(|y1-y2|)∀y1,y2∈ℝ.
Then we have the estimate
(3.54)|a(y1)-a(y2)|2≤κ(|y1-y2|2)∀y1,y2∈ℝ,
showing that condition (3.49) with p=2 is satisfied. Moreover, κ is a continuous, nondecreasing, concave function, and condition (3.50) is satisfied, because for each 0<ϵ<δ, we have
(3.55)∫0ϵ‍1κ(u)du=-∫0ϵ‍1ulnudu=-ln|lnu||u=0u=ϵ=-ln|lnϵ|+limu→0ln|lnu|=∞.
Consequently, [14, Theorem 2.1] applies. However, we have
(3.56)ρ′(u)=-ln(u2)-1-ln(u2)foru∈(0,δ),
and thus, limu→0ρ'(u)=∞. Therefore, the mapping a:ℝ→ℝ might fail to be locally Lipschitz, and hence, Theorem 3.16 does not apply.Example 3.18.
Let us define the mappinga:ℝ→ℝ as follows. For n∈ℕ0, we define a on the interval [n,n+1] by
(3.57)a(y)∶={n,y∈[n,n+1-1n+1],n+(n+1)(y-(n+1-1n+1)),y∈[n+1-1n+1,n+1].
This defines the mapping a:ℝ+→ℝ, which we extend to a mapping a:ℝ→ℝ by symmetry
(3.58)a(y)∶=a(-y),y∈ℝ-.
Then, a is locally Lipschitz and satisfies the linear growth condition, and hence, Theorem 3.16 applies. However, there are no constant p≥2 and no continuous, nondecreasing function κ:ℝ+→ℝ+ with κ(0)=0 such that
(3.59)|a(y1)-a(y2)|p≤κ(|y1-y2|p)∀y1,y2∈ℝ.
Suppose, on the contrary, that there exists a continuous, nondecreasing function κ:ℝ+→ℝ+ with κ(0)=0 fulfilling (3.59). Then we have
(3.60)|κ(u)|≥1∀u∈(0,1].
Indeed, let u∈(0,1] be arbitrary. Then, there exists n∈ℕ with 1/n≤u. Moreover, by the definition of the mapping a:ℝ→ℝ, there are y1,y2∈ℝ such that
(3.61)|y1-y2|≤(1n)1/p,|a(y1)-a(y2)|=1.
Therefore, using the monotonicity of κ and (3.59), we obtain
(3.62)κ(u)≥κ(1n)≥κ(|y1-y2|p)≥|a(y1)-a(y2)|p=1,
showing (3.60). Now, the continuity of κ yields the contradiction κ(0)≥1. 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).LetH be a separable Hilbert space, let (St)t≥0 be a C0-semigroup on H with infinitesimal generator A:𝒟(A)⊂H→H, and let B∈ℰ be a set with F(Bc)<∞. Furthermore, let α:Ω×ℝ+×H→H and σ:Ω×ℝ+×H→L20(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 spaceℋ, a C0-group (Ut)t∈ℝ on ℋ, and continuous linear operators ℓ∈L(H,ℋ), π∈L(ℋ,H) such that the diagram
(4.1)
commutes for every t∈ℝ+, that is,
(4.2)πUtℓ=St∀t∈ℝ+.Remark 4.1.
According to [6, Proposition 8.7], this assumption is satisfied if the semigroup (St)t≥0 is pseudocontractive (one also uses the notion quasicontractive), that is, there is a constant ω∈ℝ such that
(4.3)∥St∥≤eωt∀t≥0.
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 (Ut)t∈ℝ is called a dilation of the semigroup (St)t≥0.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 mappingsa:Ω×ℝ+×ℋ→ℋ, b:Ω×ℝ+×ℋ→L20(ℋ), and c:Ω×ℝ+×ℋ×E→ℋ by
(4.4)a(t,y)∶=U-tℓα(t,πUty),b(t,y)∶=U-tℓσ(t,πUty),c(t,y,x)∶=U-tℓγ(t,πUty,x).
Note that a and b are 𝒫⊗ℬ(ℋ)-measurable and that c is 𝒫⊗ℬ(ℋ)⊗ℰ-measurable.Lemma 4.3.
The following statements are true:(1)
if(α,σ,γ|B) are locally Lipschitz, then (a,b,c|B) are locally Lipschitz, too;(2)
if(α,σ,γ|B) satisfy the linear growth condition, then (a,b,c|B) satisfy the linear growth condition, too;(3)
if(α,σ,γ|B) are locally bounded, then (a,b,c|B) are locally bounded, too.Proof.
All three statements are straightforward to check.Proposition 4.4.
Letz0:Ω→H be a ℱ0-measurable random variable, and let τ be a stopping time. Then, the following statements are true:(1)
ifY is a local strong solution to (1.4) with initial condition ℓz0 and lifetime τ, then Z∶=πUY is a local mild solution to (1.3) with initial condition z0 and lifetime τ;(2)
ifZ is a local mild solution to (1.3) with initial condition z0 and lifetime τ, then the process Y defined as
(4.5)Yt∶=ℓz0+∫0t∧τ‍U-sℓα(s,Zs)ds+∫0t∧τ‍U-sℓσ(s,Zs)dWs+∫0t∧τ‍∫B‍U-sℓγ(s,Zs-,x)(μ(ds,dx)-F(dx)ds)+∫0t∧τ‍∫Bc‍U-sℓγ(s,Zs-,x)μ(ds,dx),t≥0
is a local strong solution to (1.4) with initial condition ℓz0 and lifetime τ, and one has Zτ=πUYτ.Proof.
LetY be a local strong solution to (1.4) with initial condition ℓz0 and lifetime τ. Then we have
(4.6)Zt∧τ=πUt∧τYt∧τ=πUt∧τ(ℓz0+∫0t∧τ‍a(s,Ys)ds+∫0t∧τ‍b(s,Ys)dWsππUt∧τ+∫0t∧τ‍∫B‍c(s,Ys-,x)(μ(ds,dx)-F(dx)ds)+∫0t∧τ‍∫Bc‍c(s,Ys-,x)μ(ds,dx)).
By the Definitions (4.4) of a,b,c, we obtain
(4.7)Zt∧τ=πUt∧τ(ℓz0+∫0t∧τ‍U-sℓα(s,πUsYs)ds+∫0t∧τ‍U-sℓσ(s,πUsYs)dWsππUt∧τ+∫0t∧τ‍∫B‍U-sℓγ(s,πUsYs-,x)(μ(ds,dx)-F(dx)ds)ππUt∧τ+∫0t∧τ‍∫Bc‍U-sℓγ(s,πUsYs-,x)μ(ds,dx)).
Therefore, by (4.2), and since Z=πUY, we arrive at
(4.8)Zt∧τ=St∧τz0+∫0t∧τ‍S(t∧τ)-sα(s,Zs)ds+∫0t∧τ‍S(t∧τ)-sσ(s,Zs)dWs+∫0t∧τ‍∫B‍S(t∧τ)-sγ(s,Zs-,x)(μ(ds,dx)-F(dx)ds)+∫0t∧τ‍∫Bc‍S(t∧τ)-sγ(s,Zs-,x)μ(ds,dx),
showing that Z is a local mild solution to (1.3) with initial condition z0 and lifetime τ. This establishes the first statement. Now, let Z be a local mild solution to (1.3) with initial condition z0 and lifetime τ. Then we have (4.8), and therefore, by (4.2) and the Definition (4.5) of Y, we obtain
(4.9)Zt∧τ=πUt∧τ(ℓz0+∫0t∧τ‍U-sℓα(s,Zs)ds+∫0t∧τ‍U-sℓσ(s,Zs)dWsππUt∧τ+∫0t∧τ‍∫B‍U-sℓγ(s,Zs-,x)(μ(ds,dx)-F(dx)ds)ππUt∧τ+∫0t∧τ‍∫Bc‍U-sℓγ(s,Zs-,x)μ(ds,dx))=πUt∧τYt∧τ,
showing that Zτ=πUYτ. Therefore, by the Definition (4.5) of Y, we obtain
(4.10)Yt∧τ=ℓz0+∫0t∧τ‍U-sℓα(s,πUsYs)ds+∫0t∧τ‍U-sℓσ(s,πUsYs)dWs+∫0t∧τ‍∫B‍U-sℓγ(s,πUsYs-,x)(μ(ds,dx)-F(dx)ds)+∫0t∧τ‍∫Bc‍U-sℓγ(s,πUsYs-,x)μ(ds,dx).
Taking into account the Definitions (4.4) of a,b,c, we get
(4.11)Yt∧τ=ℓz0+∫0t∧τ‍a(s,Ys)ds+∫0t∧τ‍b(s,Ys)dWs+∫0t∧τ‍∫B‍c(s,Ys-,x)(μ(ds,dx)-F(dx)ds)+∫0t∧τ‍∫Bc‍c(s,Ys-,x)μ(ds,dx),
showing that Y is a local strong solution to (1.4) with initial condition ℓz0 and lifetime τ.Theorem 4.5.
The following statements are true:(1)
if(α,σ,γ|B) are locally Lipschitz and satisfy the linear growth condition, then existence and uniqueness of mild solutions to (1.3) hold;(2)
if(α,σ,γ|B) are locally Lipschitz and locally bounded, then existence and uniqueness of local mild solutions to (1.3) hold;(3)
if(α,σ,γ|B) are locally Lipschitz, then uniqueness of local mild solutions to (1.3) holds.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 z0 and z0′ 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 ℓz0 and ℓz0′ and lifetimes τ and τ', and we have Zτ=πUYτ 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
(4.12)Yτ∧τ'1{ℓz0=ℓz0′}=(Y′)τ∧τ'1{ℓz0=ℓz0′}.
Therefore, we have up to indistinguishability
(4.13)Zτ∧τ'1{z0=z0′}=πUYτ∧τ'1{z0=z0′}=πU(Y′)τ∧τ'1{z0=z0′}=(Z′)τ∧τ'1{z0=z0′},
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 z0:Ω→H be an arbitrary ℱ0-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 ℓz0. According to Proposition 4.4, the process Z:=πUY is a mild solution to (1.3) with initial condition z0, 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.Remark 4.6.
The structureZ=πUY shows that mild solutions to (1.3) obtained from Theorem 4.5 have càdlàg sample paths.Remark 4.7.
As pointed out in [5], the existence of weak solutions to (1.3) relies on a suitable stochastic Fubini theorem. Sufficient conditions can be found in [6].



---

*Source: 236327-2012-12-25.xml*