R E S E A R C H
Open Access
On the asymptotic stability of a class of
jump-diffusions of neutral type with impulses
Feng Jiang
1*, Hua Yang
2and Xinquan Zhao
1*Correspondence:
1School of Statistics & Mathematics,
Zhongnan University of Economics and Law, Wuhan, 430073, China Full list of author information is available at the end of the article
Abstract
This paper is concerned with the asymptotic stability in thepth moment for a class of jump-diffusions of neutral type with impulses. Sufficient conditions ensuring the stability of jump-diffusions of neutral type with impulses are established by means of the Banach fixed point theorem. The results obtained here generalize and improve some well-known results.
Keywords: impulses; Poisson jump; mild solution; asymptotic stability
Introduction
Recently, the existence, uniqueness and stability of solutions of stochastic differential equations, especially stochastic partial differential equations [–], have been considered by many authors []. Besides stochastic effects, impulsive effects also occur in real systems. The study of impulsive systems in a separable Hilbert space is motivated by modeling some evolution phenomena arising in physics, communications, engineering,etc.[–].
In addition, many dynamical systems not only depend on present and past states, but also involve derivative with delays, and neutral systems are often used to describe such systems. It should pointed out that there are a few works about the existence and stability of mild solutions of neutral systems [–]. Meanwhile, there are also a few works on jump diffusions, and some results on the existence, uniqueness, stability and qualitative properties of solutions have been obtained. For example, Baoet al.[] studied almost sure asymptotic stability of stochastic partial differential equations with jumps. Baoet al.
[] discussed stability in distribution of mild solutions to stochastic partial differential equations with jumps. Cuiet al.[] discussed exponential stability for neutral stochastic partial differential equations with delays and Poisson jumps. Peszat and Zabczyk [] dis-cussed the theory of stochastic partial differential equations with Lévy noise. Motivated by the above papers, in the paper we aim to study the existence and asymptotic stability of a class of jump-diffusions of neutral type with impulses by means of the Banach fixed point theorem, the results obtained here generalize the main results from Mahmudov [], Jiang and Shen [], Sakthivel [].
The organization of the paper is as follows. In the next section, we introduce some nota-tions and defininota-tions of mild solution and asymptotic stability. Then we give sufficient con-ditions ensuring the stability of jump-diffusions of neutral type with impulses by means of the Banach fixed point theorem.
Preliminaries
Throughout this paper, let (,F,{Ft}t≥,P) be a complete probability space with a filtra-tion{Ft}t≥satisfying the usual conditions (i.e., it is increasing and right-continuous while Fcontains allP-null sets) []. Moreover, letX,Ybe two real separable Hilbert spaces, and letL(Y,X) denote the space of all bounded linear operators fromYintoX.
For simplicity, we use the notation| · |to denote the norm inX,Y and · to denote the operator norm inL(X,X) andL(Y,X). Let·X,·Ydenote the inner products ofX,Y, respectively. Let{w(t) :t≥}denote anY-valued Wiener process defined on the probabil-ity space (,F,{Ft}t≥,P) with the covariance operatorQ, that is,Ew(t),xYw(s),yY= (t∧s)Qx,yY for allx,y∈Y, whereQis a positive, self-adjoint, trace class operator onY. In particular, we denote byw(t) aY-valuedQ-Wiener process with respect to{Ft}t≥. We assume that there exists a complete orthonormal system{ei}inY, a bounded sequence of nonnegative real numbers λisuch that Qei=λiei,i= , , . . . , and a sequence{βi}i≥ of independent Brownian motions such that w(t),e=∞i=√λiei,eβi(t), e∈Y, and Ft=Fw
t , whereFtwis theσ-algebra generated by{w(s) : ≤s≤t}. LetL=L(Q/Y;X) be the space of all Hilbert-Schmidt operators from Q/Y toX with the inner product
u,ξL
=tr[uQξ]; see, for example, [].
Letp= (p(t)),t∈Dpbe a stationaryFt-Poisson point process with characteristic mea-sure λ. Denote by N(dt,dv) the Poisson counting measure associated withp, that is,
N(t,Z) =s∈Dp,s≤tIZ(p(s)) with a measurable setZ∈B(Y–{}), which denotes the Borel
σ-field ofY–{}. LetN(dt,dv) =N(dt,dv) –dtλ(dv) be the compensated Poisson mea-sure, which is independent ofw(t). Denote byP([,T]×Z;X) the space of all predictable mappingsL: [,T]×Z×→Xfor which
T
ZE
L(t,v)dtλ(dv) <∞.
We may then define theX-valued stochastic integralTZL(t,v)N(dt,dv), which is a cen-tered square-integrable martingale []. We always assume thatw(t) andNare independent ofF.
Now consider a class of jump-diffusions of neutral type with impulses of the form
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
d[x(t) –u(t,x(t–ρ(t)))] = [Ax(t) +f(t,x(t–τ(t)))]dt
+g(t,x(t–δ(t)))dw(t)
+Zh(t,x(t–κ(t)),v)N(dt,dv), t≥,t=tk,
x(tk) =x(t+
k) –x(tk–) =Ik(x(tk–)), t=tk,k= , , . . . ,m,
()
with the initial datax(t) =ϕ∈CFb([–τ, ],X), whereu:R+×X→X,f :R+×X→X,g:
R+×X→L(Y,X),h:R+×X×Z→Xare all Borel measurable;ρ:R+→[,τ],τ:R+→ [,τ],δ:R+→[,τ],κ:R+→[,τ] are continuous;Ais the infinitesimal generator of a semigroup of bounded linear operatorsS(t),t≥, inX;Ik:X→X. Furthermore, the fixed moments of timestksatisfy <t<· · ·<tm<limk→∞tk=∞,x(tk+) andx(tk–) represent the right and left limits ofx(t) att=tk, respectively. Alsox(tk) =x(t+k) –x(tk–) represents the jump in the statexat timetkwithIkdetermining the size of the jump.τ > andC=
a family of all almost surely bounded,F-measurable, continuous random variables from [–τ, ] toX.
Suppose that{S(t),t≥}is an analytic semigroup with its infinitesimal generatorA. For a basic reference, the reader is referred to Pazy []. We always assume ∈(A), the resolvent set of –A. For anyα∈[, ], it is possible to define the fractional power (–A)α
which is a closed linear operator with its domainD((–A)α).
Definition A process{x(t),t∈[,T]}, ≤T<∞, is called a mild solution of Eq. () if (i) x(t)is adapted toFt,t≥withT|x(t)|dt<∞a.s.;
(ii) x(t)∈Xhas càdlàg paths ont∈[,T]a.s. and for eacht∈[,T],x(t)satisfies the integral equation
x(t) =S(t)ϕ() –u,x–ρ()+ut,xt–ρ(t) +
t
AS(t–s)us,xs–ρ(s)ds
+
t
S(t–s)fs,xs–τ(s)ds
+
t
S(t–s)gs,xs–δ(s)dw(s)
+
t
ZS(t–s)h
s,xs–κ(s),vN(ds,dv)
+
<tk<t
S(t–tk)Ik
xtk–, ()
andϕ∈Cb
F([–τ, ],X).
Moreover, for the purposes of stability, we always assume thatu(t, ) = ,f(t, ) = ,
g(t, ) = ,h(t, ,v) = ,Ik() = (k= , , . . .). Hence Eq. () has a trivial solution when
ϕ= .
Definition Letp≥ be an integer. The trivial solution of Eq. () or Eq. () itself is said to be stable in thepth moment if for arbitrarily givenε> , there existsδ> such that
ϕC<δguarantees that
Esup t≥
x(t)p<ε.
Definition Letp≥ be an integer. The trivial solution of Eq. () or Eq. () itself is said to be asymptotically stable (or globally asymptotically stable) in thepth moment if it is stable in thepth moment and for anyϕ∈Cb
F([–τ, ],X),
lim T→∞E
sup t≥T
x(t)p
= .
To establish the stability of Eq. (), we employ the following assumptions.
(H) A is the infinitesimal generator of a semigroup of bounded linear operatorsS(t),
t≥, inXsatisfying|S(t)| ≤Me–at,t≥, for some constantsM≥and <a∈R+.
(H) The functionsf,gandhsatisfy the following conditions: there exists a constantK
such that for anyx,y∈Xandt≥,
f(t,x) –f(t,y)≤K|x–y|,
g(t,x) –g(t,y)≤K|x–y|,
Z
h(t,x,v) –h(t,y,v)λ(dv)≤K|x–y|.
(H) There exist a numberα∈[, ]and a positive constantKsuch that for anyx,y∈X
andt≥,u(t,x)∈D((–A)α)and
(–A)αu(t,x) – (–A)αu(t,y)≤K|x–y|.
(H) There exists a constantqksuch that|Ik(x) –Ik(y)|≤qk|x–y|for eachx,y∈X (k= , . . . ,m).
Asymptotic stability
In this section, we consider the asymptotic stability of Eq. () by means of the fixed point theory. LetHbe the space of allF-adapted processesψ(t,w) : [,∞)×→Rwhich is almost certainly continuous intfor fixedw∈. Moreover,ψ(s,w) =ϕ(s) fors∈[–τ, ] andE|ψ(t,w)|→ ast→ ∞.
Now let us state the following well-known lemma [], which will be used in the sequel in the proof of the main result.
Lemma If (H)holds and∈(A),then for anyβ∈(, ], (i) for eachx∈D((–A)β),S(t)(–A)βx= (–A)βS(t)x;
(ii) there exist constantsMβ> anda∈R+such that(–A)βS(t) ≤Mβt–βe–at,t> .
We can now state our main result of this paper.
Theorem If (H)-(H)hold for someα∈(/, ],then Eq. ()is mean square globally asymptotically stable provided
K(–A)–α+K M–αa– α
(α– ) +MKa–+KMa–+ML< /, ()
where L=e–aTE(m k=|qk|).
Proof Define an operatorπ:H→Hbyπ(x)(t) =(t) fort∈[–τ, ] and fort≥, (πx)(t) =S(t)ϕ() –u,x–ρ()+ut,xt–ρ(t)
+
t
+
t
S(t–s)fs,xs–τ(s)ds+
t
S(t–s)gs,xs–δ(s)dw(s)
× t
ZS(t–s)h(s,x
s–κ(s),vN(ds,dv) + <tk<t
S(t–tk)Ik
xt–k
:=
i=
Fi(t). ()
We divide the proof into three steps.
Step . We claim thatπis mean square continuous on [,∞). Letx∈H,t≥, and|h| be sufficiently small, then
E(πx)(t+h) – (πx)(t)
≤
i=
EFi(t+h) –Fi(t)
:=
i=
EFi(t).
We can easily see thatE|Fi(t)|→,i= , , , , , ash→. Moreover, by the proper-ties of the martingales [, ], we have
EF(t)
≤
t
ES(t+h–s) –S(t–s)
gs,xs–δ(s)ds
+
t+h
t
ES(t+h–s)g
s,xs–δ(s)ds
→ ash→.
EF(t)
≤
t
ZE
S(t+h–s) –S(t–s)
hs,xs–κ(s),vλ(dv)ds
+
t+h
t
ZE
S(t+h–s) –S(t–s)
hs,xs–κ(s),vλ(dv)ds
→ ash→.
Consequently,πis mean square continuous on [,∞). Step . We claim thatπ(H)⊂H. From (), we have
E(πx)(t)≤ES(t)ϕ() –u,x–ρ()+ Eut,xt–ρ(t) + E
t
AS(t–s)us,xs–ρ(s)ds
+ E
t
S(t–s)fs,xs–τ(s)ds
+ E
t
+ E
t
ZS(t–s)h(s,x
s–κ(s),vN(ds,dv)
+ E
<tk<t
S(t–tk)Ik
xtk–
:=
i=
Gi(t). ()
Note thatx(t)∈H. By (H), (H), (H) and Lemma , we have
G(t)≤Me–at +
K(–A)–αϕC→ ast→ ∞, ()
G(t)≤K(–A)–α
Ext–ρ(t)→ ast→ ∞, ()
G(t)≤Me–atqkEx
tk–→ ast→ ∞. ()
By Lemma , (H) and the Hölder inequality, we obtain
G(t)≤E
t
(–A)–αS(t–s)(–A)αus,xs–ρ(s)ds
≤M–αK t
e–a(t–s)(t–s)α–ds
t
e–a(t–s)Exs–ρ(s)ds
≤M–αK a–α(α– ) t
e–a(t–s)Exs–ρ(s)ds. ()
For anyx(t)∈H and> , there existst> such thatE|x(t–ρ(t))|< fort≥t. We thus obtain
G(t)≤M–αK a–α(α– ) t
e–a(t–s)Exs–ρ(s)ds
+ M–αK a– α
(α– ). ()
We can seee–at→ ast→ ∞. By (), there existst
≥tsuch that for anyt≥twe obtain
M–αK a– α
(α– )
t
e–a(t–s)Exs–ρ(s)ds
≤– M–αK a– α
(α– ). ()
This, together with (), yields for anyt≥t,G(t)≤. That is,
G(t)→ ast→ ∞. ()
By (H), (H), the Hölder inequality, Lemma and the properties of the martingales [, ], we easily obtain
G(t)≤MKa–
t
e–a(t–s)Exs–τ(s)ds, ()
G(t)≤MK
t
G(t)≤
t
ZE
S(t–s)hs,xs–κ(s),vλ(dv)ds
≤MK
t
e–a(t–s)Exs–κ(s)ds. ()
Further, similar to the proof of (), from (), () and (), we then haveG(t),G(t),
G(t)→ ast→ ∞. Therefore, we haveE|(πx)(t)|→ ast→ ∞. That is,π(H)⊂H. Step . We claim thatπis a contraction mapping. Letx,y∈H, we have
sup t∈[,T]
E(πx)(t) – (πy)(t)
≤
sup t∈[,T]
Eut,xt–ρ(t)–ut,yt–ρ(t)
+ sup
t∈[,T]
E
t
AS(t–s)us,xs–ρ(s)–us,ys–ρ(s)ds
+ sup
t∈[,T]
E
t
S(t–s)fs,xs–τ(s)–fs,ys–τ(s)ds
+ sup
t∈[,T]
E
t
S(t–s)gs,xs–δ(s)–gs,ys–δ(t)dw(s)
+ sup
t∈[,T]
E
t
ZS(t–s)
hs,xs–κ(s),v–hs,ys–κ(s),vN(ds,dv)
+ sup
t∈[,T]
E
<tk<t
S(t–tk)
Ik
xt–k–Ik
yt–k
≤K(–A)–α+K M–αa– α
(α– ) +MKa–+KMa–
+ML sup t∈[,T]
Ex(t) –y(t), ()
whereL=e–aTE(mk=|qk|). Thusπis a contraction mapping. Hence there exists a unique fixed pointx(t) inHwhich is the solution of Eq. () andE|x(t)|→ ast→ ∞. The proof
is complete.
Similarly, we can easily generalize the above result to global asymptotic stability in the
pth moment.
Theorem If (H)-(H)hold for someα∈(/p, ],p≥,and the inequality
p–Kp/(–A)–αp+Kp/M–pαa–p α
+p(α– )/(p– )p–+MpKp/a–p
+cpMpKp/
a(p– )/(p– )–p//a+MpL<
also holds,then Eq. () is globally asymptotically stable in the pth moment, where L=
e–apTE(m
k=|qk|p)and cp= (p(p– )/)p/.
reduces to impulsive stochastic neutral partial differential equations, which is studied in []. If without the neutral term and Poisson jumps, then Eq. () reduces to an impulsive stochastic partial differential equation, which is studied in []. In the sense, the results of this paper are generalized.
Conclusion
This paper discusses the globally asymptotic stability of the mild solutions to jump-diffusions of neutral type with impulses by the fixed point theory. Globally asymptotic stability of the mild solutions to jump-diffusions of neutral type with impulses are derived. Some earlier results are generalized and improved.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
FJ carried out the study of asymptotic stability of this paper and drafted the manuscript. HY participated in the proof of the main result of the paper. XZ provided some constructive suggestions for the improvement of the paper. All authors read and approved the final manuscript.
Author details
1School of Statistics & Mathematics, Zhongnan University of Economics and Law, Wuhan, 430073, China.2School of
Mathematics & Computer Science, Wuhan Polytechnic University, Wuhan, 430023, China.
Acknowledgements
We wish to thank the referees for their valuable suggestions which have considerably improved the presentation of this article. The work is supported by the Fundamental Research Funds for the Central Universities, the National Natural Science Foundation of China under Grant 61304067 and the Natural Science Foundation of Hubei Province of China under Grant 2013CFB443.
Received: 24 December 2012 Accepted: 29 October 2013 Published:25 Nov 2013
References
1. Caraballo, T: Asymptotic exponential stability of stochastic partial differential equations with delay. Stochastics33, 27-47 (1990)
2. Caraballo, T, Liu, K: Exponential stability of mild solutions of stochastic partial differential equations with delays. Stoch. Anal. Appl.17, 743-763 (1999)
3. Da Prato, G, Zabczyk, J: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992) 4. Liu, K: Stability of Infinite Dimensional Stochastic Differential Equations with Applications. Chapman and Hall, London
(2006)
5. Luo, J, Liu, K: Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps. Stoch. Process. Appl.118, 864-895 (2008)
6. Mao, X: Stochastic Differential Equations and Applications. Horwood, Chichester (1997)
7. Benchohra, M, Henderson, J, Ntouyas, S: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York (2006)
8. Hu, J, Liu, X: Existence results of impulsive partial neutral integro-differential inclusions with infinity delay. Nonlinear Anal.71, e1132-e1138 (2009)
9. Svishchuk, AV, Kazmerchuk, YI: Stability of stochastic delay equations of Ito form with jumps and Markovian switchings, and their applications in finance. Theory Probab. Math. Stat.64, 167-178 (2002)
10. Mahmudov, NI: Existence and uniqueness results for neutral SDEs in Hilbert spaces. Stoch. Anal. Appl.24, 79-95 (2006)
11. Bao, J, Hou, Z: Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients. Comput. Math. Appl.59, 207-214 (2010)
12. Govindan, TE: Almost sure exponential stability for stochastic neutral partial functional differential equations. Stochastics77, 139-154 (2005)
13. Luo, J: Fixed points and stability of neutral stochastic delay differential equations. J. Math. Anal. Appl.334, 431-440 (2007)
14. Jiang, F, Shen, Y: Stability of impulsive stochastic neutral partial differential equations with infinite delays. Asian J. Control14, 1706-1709 (2012)
15. Chen, H: Integral inequality and exponential stability for neutral stochastic partial differential equations with delays. J. Inequal. Appl.2009, 1-15 (2009)
16. Sakthivel, R, Luo, J: Asymptotic stability of impulsive stochastic partial differential equations with infinite delays. J. Math. Anal. Appl.356, 1-6 (2009)
17. Bao, J, Truman, A, Yuan, C: Almost sure asymptotic of stochastic partial differential equations with jumps. SIAM J. Control Optim.49, 771-787 (2011)
19. Cui, J, Yan, L, Sun, X: Exponential stability for neutral stochastic partial differential equations with delays and Poisson jumps. Stat. Probab. Lett.81, 1970-1977 (2011)
20. Peszat, S, Zabczyk, J: Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach. Cambridge University Press, Cambridge (2007)
21. Sakthivel, R, Luo, J: Asymptotic stability of nonlinear impulsive stochastic differential equations. Stat. Probab. Lett.79, 1219-1223 (2009)
22. Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1992)
10.1186/1029-242X-2013-561
