R E S E A R C H
Open Access
Local stability and parameter dependence of
mild solutions for stochastic differential
equations
Lijun Pan
**Correspondence:
School of Mathematics, Jia Ying University, Meizhou, Guangdong 514015, P.R. China
Abstract
For nonlinear stochastic equationsdx(t) = [Ax(t) +f(t,x(t),
λ
)]dt+g(t,x(t),λ
)dω
(t) with parameterλ
in a Hilbert space, we show the existence and uniqueness of mild solutions. Provided thatfsatisfies a locally Lipschitz condition andgis a uniformly Lipschitz function, some sufficient conditions forp(p≥2) moment locallyexponential stability as well as almost surely exponential stability of mild solutions are obtained under a sufficiently small initial value
ξ
. Meanwhile, we also consider parameter dependence of stable mild solutions for the stochastic system iff,gare sufficiently small Lipschitz perturbations in the parameterλ
.Keywords: mild solutions; exponential stability; parameter dependence
1 Introduction
Stochastic differential equation has attracted a great attention from both theoretical and applied disciplines since it has been successfully applied to problems in mechanics, eco-nomics, physics and several fields in engineering. For details, see [–] and the refer-ences therein. In recent years, existence, uniqueness, stability, invariant measures and other quantitative and qualitative properties of solutions to stochastic partial differen-tial equations have been extensively considered by many authors. For example, in , Haussmann [] studied asymptotic stability of Itô equations in infinite dimensions. Cara-ballo [] extended the results from Haussmann to stochastic partial differential equations with delay. In [], Mao considered exponential stability in the mean square sense for the strong solutions of linear stochastic differential equations. Caraballo and Real [] con-sidered the stability for the strong solutions of semilinear stochastic evolution equations based on the ideas in []. Govindan [, ] studied the existence and stability of mild so-lutions for stochastic partial differential equations by the comparison theorem. Caraballo and Liu [] discussed the exponential stability for mild solutions of stochastic partial differential equations with delays by employing the well-known Gronwall inequality and stochastic analysis technique under the Lipschitz condition. Liu and Mao [] established Razuminkhin-type exponential stability for mild solution of stochastic partial functional differential equations. Da Pratoet al.[] and Dawson [] developed this topic by the semigroup approach []. In [], Taniguchiet al.discussed the existence, uniqueness and asymptotic behavior of solutions to stochastic partial functional differential equations in a Hilbert space.
Recently, Burton [] has successfully extended the fixed point theorem to study the stability for deterministic systems. By employing the contraction mapping principle and stochastic analysis, Luo [] has obtained some sufficient conditions which ensure expo-nential stability inp(p≥) moment and almost sure exponential stability for mild so-lutions of stochastic partial differential equations with delays. Following the ideas of Luo, Sakthivel and Luo investigated the existence and asymptotic stability in thepth moment of mild solutions of nonlinear impulsive stochastic differential equations with infinite delay in [].
Our main objective is to obtain local stability of mild solutions for stochastic parameter systems, with two novelties when compared with the former work in this area:
. We considerp(p≥)th moment locally exponential stability as well as almost surely exponential stability of mild solutions under sufficiently small nonlinear perturbations and sufficiently small initial value.
. We first discuss parameter dependence of stable mild solutions under sufficiently small Lipschitz perturbation in the parameter spaceY.
The rest of this paper is organized as follows. In Section , stochastic differential equa-tions with parameter are presented together with some definiequa-tions of mild soluequa-tions. In Section , existence, uniqueness, stability and parameter dependence of mild solutions are derived. An example is given to illustrate our results in Section . At last, we present some conclusions of our paper in Section .
2 Preliminaries
LetHbe a real separable Hilbert space with the inner product (·,·) and the norm · , and letKbe another real separable Hilbert space with the inner product (·,·)Kand the norm
· K.L(K,H) denotes the space of bounded operators fromKtoH. Let (,F,P) be a
complete probability space equipped with a complete family of right-continuous increas-ing subσ-algebras{Ft,t≥}satisfyingFt⊂F.
Letβn(t),n= , , . . . , be a sequence of real-valued one-dimensional standard Brownian
motions mutually independent over (,F,P). Set
w(t) = ∞
n=
λnβn(t)ξn, t≥,
whereλn≥ (n= , , . . .) are nonnegative real numbers and{ξn}(n= , , . . .) is a complete
orthonormal basis inK. LetQ∈L(K,K) be an operator defined byQξn=λnξnwith a finite
traceTr(Q) =∞n=λn<∞. Then the aboveK-valued stochastic processω(t) is called aQ
-Wiener process.
Letϕ∈L(K,H) and define
ϕ L =Tr
ϕQϕ∗=
∞
n=
λnϕξn
.
IfϕL
<∞, thenϕis called aQ-Hilbert-Schmidt operator, whereL
(K,H) denotes the space of allQ-Hilbert-Schmidt operatorsϕ:K→H.
We denote byLp(,H) (p≥) the collection of all strongly-measurable, p-integrable H-valued variables with the normx(·)Lp= (Ex(·;ω)p
H)
p, whereEis defined byE(h) =
We consider the following stochastic differential equations with parameter in a Hilbert space:
dx(t) =Ax(t) +ft,x(t),λdt+gt,x(t),λdω(t), t≥,
x() =ξ,
()
whereξis anH-valued random variable,Ais the infinitesimal generator of a semigroup of bounded linear operatorsS(t),t≥,f :R+×H×Y→H,g:R+×H×Y→L(K,H) are all Borel measurable,Y is a Banach space (which is the parameter space).
Now, we recall the mild solutions for Eq. () as follows.
Definition . A stochastic process{x(t,ξ,λ) :t∈[,T],λ∈Y}, ≤T<∞, is called a mild solution of Eq. () if
(a) x(t,ξ,λ)is adapted toFt,t≥;
(b) x(t,ξ,λ)∈Hhas a càdlàg path ont∈[,T]almost surely; (c) for arbitraryt∈[,T],
x(t,ξ,λ) =S(t)ξ+ t
S(t–s)fs,x(s),λds+ t
S(t–s)gs,x(s),λdω(s). ()
We assume that the following conditions hold:
(i) Ais the infinitesimal generator of a semigroup of bounded linear operatorsS(t),
t≥, and there existM≥,η> such thatS(t)H≤Me–ηtfort≥;
(ii) there exist constantsc> ,q> such that fort≥,x,y∈H,λ,μ∈Y,
f(t,x,λ) –f(t,y,λ)H≤cx–yH
xqH+yqH,
and
f(t,x,λ) –f(t,x,λ)H≤cλ–μHxqH+;
(iii) there existsd> such that fort≥,x,y∈H,λ,μ∈Y,
g(t,x,λ) –g(t,y,λ)L
≤dx–yH,
and
g(t,x,λ) –g(t,x,μ)L
≤dλ–μYxH;
(iv) q+ ≤p; (v) pMpdpC
p(η)
p
< , whereCp= (p(p– ))p.
For example, if
f(t,x,λ) =
m
j=
n
i=
and
g(t,x,λ) =
m
j=
n
i=
λjgi(t,x)
for some functionsfi,gi:R+×H→H, and there exist constantsci> ,di> ,i= , , . . . ,n,
q> such that fort≥,x,y∈H,
fi(t,x) –fi(t,y)H≤cix–yH
xqH+yqH,
and
gi(t,x) –gi(t,y)L
≤dix–yH
for eacht≥,x,y∈H, then conditions (ii) and (iii) hold forλ,μin a small neighborhood of zero.
SetB(δ) ={x∈H:ExpH≤δ},δ> . Then we have the following definitions of locally exponential behavior for mild solutions.
Definition . Letp≥ be an integer, mild solutionx(t,ξ,λ) of Eq. () is said to bepth moment locally exponentially stable if there existγ> ,L≥ such that
Ex(t,ξ,λ)H≤Le–γtEξp H
for any initial valueξ∈B(δ),λ∈Yandt≥.
Definition . The mild solutionx(t,ξ,λ) of Eq. () is said to be almost surely exponen-tially stable if for any initial valueξ∈B(δ) and for anyλ∈Y, there existsν> such that
lim sup
t→+∞
tlogx(t,ξ,λ)H≤–ν, a.s.
Lemma .[] For any r≥and for arbitrary L
(K,H)-valued predictable processΦ(·),
sup
s∈[,t]
E
s
Φ(u)dω(u) r
≤Cr
t
EΦ(s)Lr
rds r
, t≥,
where Cr= (r(r– ))r.
3 Stability and parameter dependence
In this section, we present and prove our main results. We consider the setZ(δ) ={(t,ξ) : ≤t≤T,ξ∈B(δ)}.
Theorem . Assume that(i)-(v)hold.Then,for anyδ> sufficiently small and for any
λ∈Y,given (t,ξ)∈Z(δ),there exists a unique stochastic process x(t,ξ,λ)satisfying().
. There exists <α<pηsuch that
pMpdpCp
p
(pη–α) p
< ; ()
. x(t,ξ,λ)ispth moment locally exponentially stable,that is,there existsL≥such that
Ex(t,ξ,λ)Hp ≤Le–αtEξpH ()
for anyλ∈Yand(t,ξ)∈Z(δ); . There existsK> such that
Ex(t,ξ,λ) –x(t,ξ,μ)pH≤Kλ–μpYe–αt ()
for anyλ,μ∈Yand(t,ξ)∈Z(δ).
Proof By (iv), it follows that there exists <α<ηsuch that () holds. We consider the spaceχof allF-adapted processesx=x(t,ξ,λ) : [, +∞)×B(δ)×Y→Hsuch that
. x(,ξ,λ) =ξfor anyλ∈Y;
. the normxχ=M supt≥[eαtEx(t,ξ,λ)pH]satisfiesxχ≤δ.
We can easily see thatχis a complete metric space. Define an operatorJλ:χ→χby
Jλ(x)(t,ξ) =S(t)ξ() + t
S(t–s)fs,x(s,ξ,λ),λds
+ t
S(t–s)gs,x(s,ξ,λ),λdω(s)
=
i=
Ii(t) ()
for anyx∈χ. We first verify the continuity ofJλinχ. Letx∈χ,t≥, and||be suffi-ciently small, then
EJλ(x)(t+) –Jλ(x)(t)
p H≤
p–
i=
EIi(t+) –Ii(t)
p
H. ()
Obviously,
EIi(t+) –Ii(t)
p
H→, i= , , ()
as→. By Lemma ., we have
EI(t+) –I(t)
p H
≤p–E
t
S(t+–s) –S(t–s)
gs,x(s,ξ,λ),λdω(s)
p
H
+ p–E
t+
t
S(t+–s)g
s,x(s,ξ,λ),λdω(s)
p
≤p–Cp by the Hölder inequality,
Ex∗yH≤ExrHrEys
where∗represents the transpose, and the Lyapunov inequality
By (iii), we obtain
≤p–
E
t
Me–η(t–s)fs,x(s,ξ,λ),λ–fs,y(s,ξ,λ),λ Hds
p
+Cp
t
Mpe–pη(t–s)Egs,x(s,ξ,λ),λ–gs,x(s,ξ,λ),λp L
pds p
≤
p+qMp+q+cpδqe–ηpt
t
eηs–(q+)αp sds p
+ pMp+dpCpe–pηt
t
epηsp–αsds p
x–yχ
≤
p+qMp+q+cpδq p pη–α(q+ )
p+ pMp+dpCp
p
(pη–α) p
×e–αtx–yχ. ()
Therefore
Jλ(x) –Jλ(y)χ≤θx–yχ,
whereθ= p+q–Mp+qcpδq| p pη–α(q+)|
p+ p–MpdpC
p[(pηp–α)] p
. Takingδsufficiently small
so thatθ<, the operatorJλbecomes a contraction. In addition, we can obtain
Jλ(x)χ≤S(t)ξχ+θxχ≤
δ+
δ=δ. ()
This shows thatJλ(χ)⊂χ. Thus, there exists a unique functionx∈χsuch thatJλ(x) =x. By the above inequality, we havexχ≤
ξpH
(–θ), which means that () holds. Writingyλ=
x(·,ξ,λ), we have
yλ–yμ=Jλyλ–Jμyμ=Jλyλ–Jλyμ+Jλyμ–Jμyμ ()
for anyλ,μ∈Y. Thus
yλ–yμχ≤θyλ–yμχ+Jλyμ–Jμyμχ. ()
It follows that
yλ–yμχ≤( –θ)–Jλyμ–Jμyμχ. ()
Using the Lyapunov inequality
ExmHm ≤Exn H
n, x∈H, <m≤n< +∞, ()
from (ii), (iii) and (), we get
a(s) =Efs,x(s,ξ,μ),λ–fs,x(s,ξ,μ),μ H
≤cλ–μYEx(s,ξ,μ) q+
≤cλ–μY
which yields (). This completes the proof of the theorem.
Theorem . Assume that(i)-(v)hold.Then,for anyδ> sufficiently small,mild solution
It follows from (ii) and () that
Ef(s,x(s,ξ,λ),λH≤cEx(s,ξ,λ)q+
From (iii) and (), we have the following estimation:
Eg(s,x(s,ξ,λ),λp L ≤d
pEx(s,ξ,λ)p H≤d
and
Consequently, from the Borel-Cantelli lemma, there existsT(ω) > such that for anyt≥ T(ω),
x(t,ξ,λ)Hp ≤e–αpt, a.s. ()
4 Example
In this section, an example is provided to illustrate our results.
Example . Consider the following stochastic partial equation:
dZ(t,x) =
∂
∂xZ(t,x) +
λZ(t,x) +|Z(t,x)|
dt+ λZ(t,x)
( +|Z(t,x)|)dω(t), t≥,
Z(t, ) =Z(t,π) = , t≥,
Z(,x) =ξ(x),
()
whereω(t) is a standard one-dimensional Brownian motion,λ∈Y= [,] is a parameter. LetA= ∂
∂x with the domain
D(A) =
u∈H: ∂
∂u, ∂
∂u ∈H,u() =u(π) =
, ()
thenS(t)H≤e–t,∀t≥, which implies thatM= . It is easy to see that (ii)-(iii) hold and c= ,q= ,d=. Takingp= , we may show that pdpCp(
π)
p
< . Then the mild
solu-tion of Eq. () has locally stable behavior and parameter dependence forδ> sufficiently small.
5 Conclusion
This paper is devoted to locally stable behavior and parameter dependence for stochastic differential equations. Our results are derived under sufficiently small Lipschitz pertur-bation in the parameter system. Meanwhile, a lot of stochastic inequality techniques are used to estimate the bound of mild solution. The future research topics would be extend-ing locally stable behavior and parameter dependence to the stochastic delayed parameter system.
Competing interests
The author declares that they have no competing interests.
Author’s contributions
The author contributed to the manuscript. The author read and approved the final manuscript.
Acknowledgements
The author would like to thank the referee and the associate editor for their very helpful suggestions.
Received: 19 May 2014 Accepted: 8 October 2014 Published:27 Oct 2014
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