Global well posedness of a class of stochastic equations with jumps

R E S E A R C H

Open Access

Global well-posedness of a class of stochastic

equations with jumps

Guoli Zhou

*_{Correspondence:} [email protected] School of Mathematics and Statistics, Chong Qing University, Chong Qing, P.R. China

Abstract

The existence and uniqueness of a mild solution to stochastic equations with jumps are established, a stochastic Fubini theorem and a type of Burkholder-Davis-Gundy inequality are proved, and the two formulas are used to study the regularity property of the mild solution of a general stochastic evolution equation perturbed by jump processes. As applications, the regularity of a stochastic heat equation with jump is given.

MSC: 76S05; 60H15

Keywords: stochastic evolution equation; Lévy processes; mild solution

1 Introduction

In recent years, there have been many monographs concerning stochastic partial differ-ential equations with Lévy jump and their applications in physics, economics, statistical mechanics, fluid dynamics and finance etc. For this theory and its applications, one can see [–] and references therein. In this article, the existence, uniqueness, regularity for the mild solution of stochastic partial differential equations with Lévy jump are studied. There are a lot of works dealing with existence and uniqueness for stochastic partial differ-ential equations with jump processes. In [], the existence and uniqueness for solutions of stochastic reaction diffusion equations driven by Poisson random measures are ob-tained. In [], Malliavin calculus is applied to study the absolute continuity of the law of the solutions of stochastic reaction diffusion equations driven by Poisson random mea-sures. In [], a minimal solution is obtained for the stochastic heat equation driven by non-negative Lévy noise with coefficients of polynomial growth. In [], a weak solution is established for the stochastic heat equation driven by stable noise with coefficients of polynomial growth. In [], the existence and uniqueness for solutions of stochastic gener-alized porous media equations with Lévy jump are obtained. In [], the strong solutions to a large class of stochastic equations with Lévy noise are obtained in variational frame-work. And it is shown in [] that the results can be applied to stochastic reaction-diffusion equations, Burgers-type equations, D Navier-Stokes equations,p-Laplace equations and porous media equations with locally monotone perturbations.

The main aim of this paper is to study the existence, uniqueness and regularity of the stochastic equation

dX(t) =AX(t)+BX(t)dt+QX(t)dW(t) +

Z

FX(t–),xN(dt,dx). (.)

In [], the intensity measureλofN(dt, dx) is finite, while the intensity measure in this article isσ-finite, and also the classical Lipschitz condition (.) in [] is relaxed to condi-tion (A.) in this article. The author of this article proves the existence and uniqueness of a mild solution of (.), the continuity of the solution with respect to initial data. And then a stochastic Fubini theorem is established for the compensated Poisson random measure whose intensity measure isσ-finite compared to a finite case in []. Furthermore, a new type of the Burkholder-Davis-Gundy inequality, which is more precise than Lemma . in [], is gotten. The two formulas are basic in stochastic analysis. Using the formulas, the author gets the regularity property of a mild solution of (.) without conditions (.) and (.) in [] which are critical there.

The article is organized as follows. In Section , we present the framework. Existence, uniqueness and regularity are proved in Section . In Section , two examples including stochastic heat equations with Lévy jump are given.

2 Some preliminaries

Let (U, U), (H, ) be separable Hilbert spaces andZbe a Banach space. LetL(U,H) de-note the space of all Hilbert-Schmidt operators fromUtoH, and set L:= L(U,H). Let

W(t),t≥ be a cylindrical Wiener process inUon a probability space (,F,P) with a nor-mal filtrationFt,t≥ and letN(dt,dz) be a Poisson random measure associated with the

compensated (Ft)-martingale measure defined asN(dt,dz) :=N(dt,dz) –λ(dz)dt, where

λ(dz) defined on a measurable spaceZis the intensity measure ofN(dt,dz). In this article we study the following equation:

dX(t) =AX(t) +BX(t)dt+QX(t)dW(t)

+

Z

FX(t–),zN(dt,dz), t∈[,T] (.)

with initial conditionX() =η, and the coefficients in equation (.) satisfy the following conditions:

(A.) A:D(A)→His the infinitesimal generator of aC-semigroupS(t),t≥.

(A.) B:H→HisB(H)/B(H)-measurable, and there exists a positive constantC

satisfying

B(x) –B(y)≤Cx–y for allx,y∈H.

(A.) Q:H→L(U,H)is strongly continuous,i.e.the mapping

x→Q(x)u

is continuous fromHtoHfor eachu∈U. (A.) For allt∈],T]andx∈H, we have

S(t)Q(x)∈L(U,H).

(A.) There is a square integrable mappingM: [, +∞[→[,∞[such that

S(t)Q(x) –Q(y)_L

and

S(t)Q(x)_L

 ≤M(t)

 +x

for allt∈],T]andx,y∈H.

(A.) There is a positive constantClike (A.) such that

Z

F(x,z) –F(y,z)λ(dz)≤Cx–y

and

Z

F(x,z)λ(dz)≤C +x,

whereF:H×Z→HisB(H)×B(Z)/B(H)measurable.

For fixedT> , we define:

H_{(T,H) := Hvalued predictable processXwith}

XH:= sup

t∈[,T]

EX(t)/<∞

.

Obviously, (H(T,H), H) is a Banach space. For technical reasons we define _,λ,T,

λ≥ onH_{(T,H) as}

X,λ,T:= sup t∈[,T]

e–λtEX(t)/.

Then, forX∈H_{(T,H) andλ≥, we have}

X,λ,T≤ XH≤eλTX_,λ,T.

For the reader’s convenience, before giving our main results, we cite some theorems here which will be needed later.

Contraction theorem (i)Let(E, E)and(, )be two Banach spaces.Let the

map-ping G:×E→E satisfy

G(λ,x) –G(λ,y)_E≤αx–yE, α∈[, [

for allλ∈and x,y∈E.Then there exists exactly one mappingϕ:→E such that

ϕ(λ) =Gλ,ϕ(λ)

for allλ∈.

(iii)If the mappingλ→G(λ,x)is not only continuous fromto E for all x∈E,but there even exists an L≥such that

G(λ,x) –G(λ,˜ x)_E≤Lλ–λ˜

for all x∈E,then the mappingϕ:→E is Lipschitz continuous.

Theorem[] If a process is adapted toFt,t∈[,T],and stochastically continuous

with values in a Banach space E,then there exists a predictable version of .

Definition . AnH-valued predictable processX(t),t∈[,T] is called a mild solution of (.) if

X(t) =S(t)η+

t



S(t–s)BX(s)ds+

t



S(t–s)QX(s)dW(s)

+

t



Z

S(t–s)FX(s–),zN(ds,dz)

for eacht∈[,T].

3 Existence, uniqueness and regularity

Theorem . Assume that conditions from(A.)to(A.)hold,then there exists a unique mild solution X(η)∈H(T,H)of(.)with the initial condition

η∈L(,F;H) =:L.

In addition we get that the mapping

X:L_→H(T,H), η→X(η)

is Lipschitz continuous.

Proof Fixt∈[,T],η∈L_andX∈H_{(T,H), and define}

F(η,X)(t) :=S(t)η+

t



S(t–s)BX(s)ds+

t



S(t–s)QX(s)dW(s)

+

t



S(t–s)FX(s–),zN(ds,dz).

In the following two steps, it comes to prove that

F:L_×H(T,H)→H(T,H).

The first step: It is proved that the mappingFis well defined. . Leth∈H, as

[,t](s)S(t–s)B

X(s),h_H=BX(s), [,t](s)S*(t–s)h

By the strongly continuity ofS(t), we know [,t](s)S(t–s)B(X(s)) is predictable. And

to-stochastic integral_tS(t–s)Q(X(s))dW(s) is well defined.

. Similarly to , it is easy to see that [,t](s)S(t–s)F(X(s),z) is predictable, and by (A.)

In the above inequalities, we setMT:=supt∈[,T]S(t)L(H)andC¯:=max{C,B()}, where Cis as in (A.). As we can derive that

t/r



S(r– )sBX(s)ds≤ ¯CM(T)

t/r



 +X(s)ds

≤ ¯CM(T)

t



 +X(s)ds

/ < +∞.

Lettingt→t, and thenr→, by (A.), strong continuity ofS(t) and dominated conver-gence theorem, we know, for all mostw∈, we have

t



S(t–s)BX(s)ds→

t



S(t–s)B

X(s)ds for all ≤t<t≤T.

Similarly we can get the same conclusion for ≤t<t≤Twhent↓t, so we have proved

t

S(t–s)B(X(s))dsis continuous a.s. fort∈[,T]. As we have

E

t



S(t–s)BX(s)ds 

≤E

t



S(t–s)BX(s)dsT

≤(T)CM(T)¯ E

t



 +X(s)ds

≤TCM(T¯ ) +X_H

.

Therefore we have proved_tS(t–s)B(X(s))ds∈H_(T,H).

. There is a version of_tS(t–s)Q(X(s))dW(s),t∈[,T], which is inH_(T,H). First we fixr∈], ], then

t/r



S(t–s)QX(s)dW(s) =

t/r



S(t–rs)S(r– )sQX(s)dW(s).

Let us define

r:= [,T](s)S

(r– )sQX(s).

Then it is clear thatr_∈N

W(,T). By (A.), we have

E

T



r(s)_L

ds=E T



[,T](s)S

(r– )sQX(s)_L

ds

≤E

T



M(r– )s +X(s)ds

≤ T



M(r– )sds

 + sup

t∈[,T]

EX(t)

≤ 

r– 

(r–)T



M(s)ds

 + sup

t∈[,T]

Similarly we have [,t](s)S(t–s)Q(X(s))∈NW(,T) and t

S(t–s)Q(X(s))dW(s) isFt -measurable. Next we are going to show

t→

t



S(t–s)QX(s)dW(s)

is continuous in the mean square and therefore stochastically continuous. Indeed, let ≤ t<t≤T, we get

The last step follows by a dominated convergence theorem since (S(t–rs) –S(t– rs))r_(s)e

For ≤t<t≤T, we can get the same conclusion. Therefore by Theorem [] we know

t



S(t–s)QX(s)dW(s)

has a predictable version for allt∈[,T]. Furthermore, by (A.), we have

+C_M(T)

lettingt→tandr↓, by a dominated convergence theorem, we have the following re-sult:

has a predictable version which is an element inH_(T,H). The third step: We are going to show

+

Obviously,

T



M(s)e–λsds→, asλ→+∞.

By the Burkholder-Davis-Gundy inequality and (A.), we have the following estimate for the third term of the right-hand side of (.):

E

t



Z

S(t–s)FX(s–),z–FX(s–),˜ zN(ds,dz) /

≤

E

t



Z

S(t–s)FX(s),z–FX(s),˜ zλ(dz)ds

/

≤C_M(T)

E

t



X(s) –X(s)˜ ds

/

≤C_M(T)

t



eλs_e–λs_{EX(s) –X(s)}˜  ds

/

≤C_M(T)

t

 eλsds

/ sup

t∈[,T]

e–λtEX(t) –X(t)˜ /

=C_M(T)eλt

t

 e–λsds

/

X–X˜,λ,T.

Thus we have

t



Z

S(t–s)FX(s–),z–FX(s–),˜ zN(ds,dz) ,λ,T

≤C_M(T)

t

 e–λsds

/

X–X(s)˜ _,λ,T.

Obviously,

t



e–λsds→, asλ→+∞.

Therefore we have finally proved that there exists ana(λ) <  with

_F(η,X) –Fη,X(s)˜ _,λ,T≤a(λ)X–X˜,λ,T.

So, there exists a unique

X:L_→H(T,H)

satisfying

X(η) =Fη,X(η),

The fourth step : We will show the Lipschitz continuity ofX:L

→H(T,H). By con-traction theorem (iii), we only have to prove that the mapping

F(·,Y) :L_→H(T,H)

is Lipschitz continuous for allY∈H_{(T,H), where the Lipschitz constant does not depend} onY.

But this is obvious for allξ,ξ∈LandY∈H(T,H), as

_F(ξ,Y) –F(ξ,Y)_H=S(·)(ξ–ξ)_H≤MTξ–ξL.

Before giving the regularity of the mild solution of (.), we first need a stochastic Fubini theorem with respect to compensated Poisson measure. SetT×Z= [,T]××Z, letZ

beσ-algebra generated by open subsets inZ,PTbe the predictableσ-algebra of [,T]×,

ds⊗P⊗λ(dz) be the product of the Lebesgue measure,Pandλ(dz) on [,T]××Z. L₍

T×Z) :=L(([,T]××Z,ds⊗P⊗λ(dz));H).

Proposition . Let(Y,Y,μ)be a finite measure space and letψ:T×Y×Z→H be a

PT⊗Y⊗Z-measurable mapping such that

Y

ψ(y)_L₍

T×Z)dμ<∞.

Then

() the process indexed byt∈[,T]

Y

ψ(y)dμ

is progressively measurable and belongs toL₍

T×Z), () the process indexed byy∈Y

T



Z

ψ(s–,z)N(ds,dz)

has anFT⊗Y-measurable versionm:×Y→Hsuch that

P

m(y) =

T



Z

ψ(y)N(ds,dz)

= ,

() we have

P

Y

m(y)dμ=

T



Z

Y

ψ(y)dμ

N(ds,dz)

= .

Proof () follows from the following inequality: Let f be a nonnegativePT⊗Y⊗Z

-measurable function. Then

T×Z

Y

f dμ



ds×P×λ(dz)≤

Y fL₍

because

Y

f(a,y)du

 =

Y×Y

f(a,y)f(a,y)dμdμ

and we get (.) by the Schwarz inequality. Now suppose thatmin () exists. Then, by takinginstead ofT×Zin (.), we get

E

Y

m(y)dμ



≤

Y

Em(y)dμ≤μ(Y)/ψ(y)_L₍

T×Y×Z) (.)

by the Burkholder inequality. Hence

Y

m(y)dμ

is definedP-almost everywhere. Now takeψnsatisfying the assumption of the proposition

such that the sequence of the integrals

Y

ψn(y) –ψ(y)_L₍

T×Z)dμ

converges to zero. Then there exists a subsequence (nk:k∈N) such that: (a) _T_Zψnk(y)N(ds, dz)→

T



Zψ(y)N(ds, dz)inL

_{(;H)forμ-almost ally∈Y.}

(b) _T_Z(_Yψnk(y)dμ)N(ds, dz)→

T



Z(

Yψ(y)dμ)N(ds,dz)inL(;H)for

μ-almost ally∈Y.

Let us introduce the setDof allPT⊗Y⊗Z-measurable processesψwith

Y

ψ(y)_L₍

T×Z)dμ<∞

such that there exists anmsatisfying () and (). It is easy to see thatDis a linear space and if we can findψn∈Dsuch that

Y

ψn(y) –ψ(y)_L₍

T×Z)dμ→,

then we would finish the proof. Indeed, take the corresponding functionsmn. Then the

sequence (mn:n∈N) is Cauchy inL(×Y;H) due to (.) andψbelongs toDdue to (a)

and (b). Now we will show how to construct the approximating sequenceψn. We assume

thatλ(dz) is finite. By Lemma A.. in [] we can find mappingsFnonH. The simple

functionsFnψtake values in the finite dimensional subspace ofH. Moreover,

Y

Fnψ(y) –ψ(y)_L₍

T×Z)dμ↓,

and ifFnψ∈D,n∈N, thenψ∈D. Now to show thatFnψ∈D, we will take advantage of

the fact that eachFnψis bounded inHand

Y

φn(y) –φ(y)_L₍

if and only if

Y

φn(y) –φ(y)_L₍

T×Y×Z)dμ→

forφnuniformly bounded inH. So asFnis of the form

m

k= ICkBk,

where (Ck:k≤m) is aPT⊗Y⊗Z-decomposition ofT×Y×ZandBk,k≤mare

elements in the finite dimensional subspace ofH. We conclude thatFnψ ∈Dprovided

ICkBk∈Ddue to linearity ofD. Another reduction shows that this is true if

I_C

k×Ck×CkBk∈D

for everyC_k∈PT,Ck∈Y,Ck∈ZasICk can be approximated byIC_kinL(T×Y×Z)

whereC

kis a disjoint union of sets of the typeCk×Ck×Ck. Finally, asIC

kis a predictable process, it can be approximated by simple bounded real processes inL₍

T). So, we will

finish the proof by showing that

I_]s,t]×Cs×C

k×C



kBk∈D

fors<t,Cs∈Fs, but this is obvious. Whenλ(dz) isσ-finite, there exists a sequenceAn

satisfyingλ(An) < +∞andAn↑Z. Obviously,

ψ(y)IAn∈D and

Z

ψIAn(z) –ψ(z)_L₍

T×Z)dμ→,

thereforeψ∈D.

In the following we give a type of the Burkholder-Davis-Gundy inequality which will play an important role in proving the regularity property of the mild solution of (.).

Lemma . If A is the infinitesimal generator of pseudo-contraction C-semigroup S(t), t≤T,and g(t,z)is an H-valued progressively measurable mapping with respect to{Ft}t≥ such that

T



Z

g(t,z)λ(dz)dt<∞, a.s.,

then for any t∈[,T],

E

sup ≤s≤t

s



Z

S(s–u)g(u–,z)N(du,dz) 

≤ert_E t



Z

g(s,z)λ(dz)ds

Proof Let us define

Assume thatg(t,z) takes values inD(A) and

E

SinceAis the infinitesimal generator of aC-semigroup,Ais closed. And by (.) we can check that

By Itô’s formula, we get

SinceAis pseudo-contraction, there existsr≥ such that

So, by Gronwall inequality, we have

By the Fatou lemma, we have

EY(t)˜ ≤ertE

t



Z

g(s,z)λ(dz)ds. (.)

Without assumingg(t,z) takes values inD(A), we definegn(t,z) =nR(n,A)g(t,z) where

R(n,A),n∈N, is the resolvent ofA, then we knowgn(t,z) takes values inD(A) and satisfies

(.) under the following condition:

E

t



Z

g(s,z)λ(dz)ds< +∞. (.)

Define

Yn(t) = t



Z

S(t–s)gn(s–,z)N(ds, dz), Y˜n(t) = sup

≤s≤t Yn(t).

Then instead Y˜(t),Y˜n(t) satisfies (.). By the Burkholder-Davis-Gundy inequality and

dominated convergence theorem, we have

E

t



Z

gn(s,z) –g(s,z)



λ(dz)ds→, asn→+∞.

Then it is easy to check

E

sup ≤s≤t

Yn(s) –Y(s)



→, asn→+∞.

So, under condition (.), (.) also follows without assumption thatg(t,z) takes values inD(A). In order to relax (.), we define stopping times

τn:=inf

t

t



Z

g(s,z)λ(dz)ds≥n

.

Then

E

sup

{≤s≤t∧τn}

_s_ZS(s–u)g(u–,z) 

N(du,dz)

≤ertE

t∧τn



Z

_g(s,z) λ(dz)ds

≤ertE

t



Z

g(s,z)λ(dz)ds

.

Therefore by Fatou lemma, the assertion follows.

In order to study the regularity property of the mild solution of (.), we introduce an approximation system of (.) in the following:

X(t) =R(l)η+

t



AX(s)ds+R(l)BX(s)ds+

t



R(l)QX(s)dW(s)

+

t



Z

wherel∈ρ(A) which is the resolvent set ofA, andR(l) :=lR(l,A),R(l,A) is the resolvent ofA. We say a stochastic process isRCLLif each of it’s sample path is right continuous with left limit. So a stochastic processX(t) is called a strong solution of (.), if it isRCLL,

Ft-adapted,X(t)∈D(A) and satisfies (.).

Theorem . Letη∈L_(,H)andF

-measurable.In addition to assumptions in Theo-rem.we assume A is dissipative with x∈H.Then the mild solution of(.)is RCLL.

Proof Letl∈ρ(A), obviouslyAR(l) =AlR(l,A) =l–l_{R(l,A) are bounded operators. So} (.) has unique strong solution. In fact by Theorem . we know (.) has a unique mild solution denoted byXl_{(t) and the following hold:}

T



t



AS(t–s)R(l)BXl(s)ds dt<∞,

E

T



t



AS(t–s)R(l)QXl(s)_L ds dt

<∞,

E

T



t



Z

AS(t–s)R(l)FXl(s),zλ(dz)ds dt

<∞.

Thus by Fubini theorem, we have

t



s



AS(u–s)R(l)BXl(s)ds du

=

t



t

s

AS(u–s)R(l)BXl(s)du ds

=

t



S(t–s)R(l)BXl(s)ds–

t



R(l)BXl(s)ds.

On the other hand, by the stochastic Fubini theorem forQ-Wiener processes in [], we have

t



u



AS(u–s)R(l)QXl(s)dW(s)du

=

t



t

s

AS(u–s)R(l)QXl(s)du dW(s)

=

t



S(t–s)R(l)QXl(s)dW(s) –

t



R(l)QXl(s)dW(s).

And by Proposition .

t



u



Z

AS(u–s)R(l)FXl(s–),zN(ds,dz)du

=

t



Z t

s

AS(u–s)R(l)FXl(s–),zduN(ds, dz)

=

t



Z

S(t–s)R(l)FXl(s–),zN(ds,dz) –

t



Z

Hence,AXl_{(t) is integrable almost surely and}

t



AXl(s)ds=S(t)R(l)η–R(l)η+

t



S(t–s)R(l)BXl(s)ds–

t



R(l)BXl(s)ds

+

t



S(t–s)R(l)QXl(s)dW(s) –

t



R(l)QXl(s)dW(s)

+

t



Z

S(t–s)R(l)FXl(s–),zN(ds,dz)

–

t



Z

R(l)FXl(s–),zN(ds,dz)

=Xl_{(t) –R(l)η–} t



R(l)BXl_(s)ds– t



R(l)QXl_(s)dW(s)

–

t



Z

R(l)FXl(s–),zN(ds,dz).

So far, we proveXl_{(t)∈D(A),t∈[,T], is the unique strong solution of (.). LetX(t) be}

the mild solution of (.). Then we consider

X(t) –Xl(t) =S(t)η–R(l)η

+

t



S(t–s)BX(s)–R(l)BXl(s)ds

+

t



S(t–s)QX(s)–R(l)QXl(s)dW(s)

+

t



Z

S(t–s)FX(s–),z–R(l)FXl(s–),zN(ds,dz)

for anyt≥. We have that for anyT≥,

E sup ≤t≤T

X(t) –Xl(t)

≤E sup ≤t≤T

_tS(t–s)R(l)BX(s)–BXl(s)ds 

+ E sup ≤t≤T

_tS(t–s)R(l)QX(s)–QXl(s)dW(s) 

+ E sup ≤t≤T

t



Z

S(t–s)R(l)FX(s–),z–FXl(s–),zN(ds,dz) 

+ 

E sup ≤t≤T

S(t)η–R(l)η+

t



S(t–s)I–R(l)BX(s)ds

+

t



S(t–s)I–R(l)QX(s)dW(s)

+

t



Z

S(t–s)I–R(l)FX(s–),zN(ds,dz) 

We consider

Then we consider

I:=E sup

Finally we consider

It is easy to check

E sup ≤t≤T

S(t)η–R(l)η→, asl→ ∞.

By dominated convergence theorem, we have

E sup ≤t≤T

t



S(t–s)I–R(l)BX(s)ds 

≤E sup ≤t≤T

t



I–R(l)S(t–s)BX(s)ds



≤M_TE

T



_I–R(l)BX(s)ds



≤TM_T

T



EI–R(l)BX(s)ds→, asl→ ∞.

By Burkholder-Davis-Gundy inequality, we have

E sup ≤t≤T

_tS(t–s)I–R(l)QX(s)dW(s) 

≤E

T



I–R(l)S(t–s)QX(s)_L

ds

≤ ˜CE

T



S(t–s)QX(s)_L

ds< +∞ (.)

forC˜ > . So, by dominated convergence theorem and the fact that

I–R(l)h→, asl→+∞

forh∈H, we get

(.) =E

T



i

I–R(l)S(t–s)QX(s)eids→, asl→ ∞.

By Lemma ., we get

E sup ≤t≤T

t



Z

S(t–s)I–R(l)FX(s–),zN(ds,dz) 

≤M_TE

T



Z

_I–R(l)

FX(s),zλ(dz)ds→, asl→ ∞.

After discussing aboutIi,i= , , , , we can get the following estimate

E sup ≤t≤T

X(t) –Xl(t)

≤C(T) +C(l) +CMM_T

T

 E sup

≤u≤s

whereg(l)→ asl→ ∞. By Gronwall theorem, we have

E sup ≤t≤T

X(t) –Xl(t)≤g(l)e[C(T)+C(l)+CMMT]T→_{, asl}→ ∞_.

Therefore we knowX(t) isRCLLon [,T] a.s.

4 Example

Example . Consider the following stochastic heat equation

dy(x,t) = ∂ _y(x,t)

∂x dt+b(x)f

y(·,t)dBt+

|z|≤

zxN(dt, dz), t≥,  <x<  (.)

and

y(,t) =y(,t) = , t≥; y(x, ) =y(x), ≤x≤, (.)

whereBt,t≥, is a real standard Brownian motion,N(dt, dz) is a Poisson compensated

martingale measure with character measureλ(dz) onR,b∈L(, ) :=Handf is a real Lipschitz continuous function onHsatisfying|f(u)| ≤cufor somec>  andu∈H. Let A=_∂∂_x withD(A) =H(, )∩H(, ), then we have

u,AuH≤–πu foru∈D(A).

For the sake of simplicity, we assumeb= . It is easy to check that (A.)-(A.) are satis-fied. So Theorem . and Theorem . hold true for (.).

Example . DenoteX(t,x) a process satisfyingX(,x) =x. Then we consider the semi-linear stochastic partial differential equation:

dX(t,x) =a∂ 

∂xX(t,x)dt+

ϕ(t)X(t,x)dt+bX(t,x)dW(t)

+

|z|≤c

zX(t–,x)N(dz,dt), (.)

X(t, ) =X(t,π) =  fort≥,a> ,c> ,x∈H:=L(,π)

for some negative constanta< , and someM> ,ϕ(·) :R+→Ris a bounded function with|ϕ(t)| ≤Mfor allt≥,b(x) :H→R _{is nonlinear and Lipschitz continuous with} b() = ,|b(x)| ≤Mx,W(t),t≥, is an H-valuedQ-wiener process with covariance operatorQ,trQ<∞.

LetC=_|z|≤cz_{λ(dz) < +∞andA=a}∂

∂x with domain

D(A) =

h∈H:∂h ∂x,

∂h

∂x ∈H,h() =h(π) = 

.

So, it is easy to deduce

It is easy to check that (A.)-(A.) are satisfied. So, Theorem . and Theorem . hold true for (.).

Competing interests

The author declares that he has no competing interests.

Author’s contributions

The author read and approved the final manuscript.

Acknowledgements

The author is thankful to the referee for careful reading and insightful comments which led to many improvements of the earlier version. The work was supported by Tianyuan Foundation of NSF (Grant No. 11126079), CPSF (Grant

No. 2013M530559) and Fundamental Research Funds for the Central Universities (Grant No. CDJRC10100011).

Received: 23 April 2013 Accepted: 28 May 2013 Published: 18 June 2013

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doi:10.1186/1687-1847-2013-175