Stability of analytical and numerical solutions of nonlinear stochastic delay differential equations

0.26 MB·English

by  Siqing Gan

Stability of analytical and numerical solutions of nonlinear
stochastic delay differential equations∗
Siqing Gan†
School of Mathematics and Statistics
3 State Key Laboratory of High Performance Complex Manufacturing
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Central South University, Changsha, Hunan 410083, China
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2
n Aiguo Xiao
a School of Mathematics and Computational Science
J
Hunan Key Laboratory for Computation and Simulation in Science and Engineering
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2 Xiangtan University, Xiangtan, Hunan 411105, China

]
A Desheng Wang
N Division of Mathematical Sciences, School of Physical and Mathematical Sciences
. Nanyang Technological University, Singapore, 637371
h
t
a
m
[
Abstract

1
v This paper deals with stability of analytical and numerical solutions of nonlinear stochastic
9 delay differential equations (SDDEs). Sufficient conditions for stability in mean square,
8
contractivity in mean square and asymptotic contractivity in mean square of the solutions
3
of nonlinear SDDEs are given, which provide a unified theoretical treatment for constant
5
. delay and variable delay, bounded delay and unbounded delay. Stability in mean square,
1
contractivity in mean square and asymptotic contractivity in mean square of the backward
0
Euler method are investigated. It is shown that the backwardEuler method can inherit the
3
1 properties of the underlying system. The main theorems in this paper are different from
: Razumikhin-type theorems. It is not requiredto constructor find an appropriate Lyapunov
v
functional when the theorems in this paper are applied.
i
X
AMS subject classification: 34K50,65C20, 65C30.
r
a Key Words: Nonlinearstochasticdelaydifferentialequation,stabilityinmeansquare,con-
tractivityinmeansquare,asymptoticcontractivityinmeansquare,backwardEulermethod
1 Introduction
Many real-life phenomena in physics, engineering, economics, etc can be modeled by stochastic
differential equations (SDEs). The rate of change of such a system depends only on its present
state and some noisy input. However, in many situations therate of change of thestate depends
∗This work was partially supported by NSF of China (No.11171352, 11271311) and State Key Laboratory of
High Performance Complex Manufacturing.
†Corresponding author. E-mail address: [email protected]
1
notonlyonthepresentbutalsoonthepaststatesofthesystem. Stochasticfunctionaldifferential
equations(SFDEs)giveamathematicalformulationforsuchsystem. FormoredetailsonSFDEs,
the reader may refer to [11, 12, 13] and the references therein.
SFDEs also can be regarded as a generalization of deterministic functional differential equa-
tionsifstochasticeffectsaretakenintoaccount. FordeterministicVolterrafunctionaldifferential
equations (VFDEs), Li [8] discussed stability, contractivity and asymptotic stability of the solu-
tions in Banach spaces. The main techniques used in [8] are as follows: The author introduced
a so-called 1-perturbed problem and constructed an auxiliary function Q(t). The former is
n
beneficial to deal with a wide variety of delay arguments and the latter is the key to obtain
main results in that paper. [8] provided a unified framework for stability analysis of solutions to
nonlinear stiff problems in ordinary differential equations (ODEs), delay differential equations
(DDEs), integro-differential equations (IDEs) and VFDEs of other type appeared in practice.
In 2008, Wang [17] extended successfully the theory in [8] to nonlinear Volterra neutral func-
tional differential equations (VNFDEs). Wang and Zhang [18] proved implicit Euler method
can preserve the stability of VFDEs and VNFDEs.
It is natural to ask whether the solutions of SFDEs possess the properties similar to those
presentedin[8]andwhichmethodscanreproducetheproperties. Duetofeaturesofthestochas-
tic calculus the numerical analysis of SFDEs differs in some key areas from the already well-
developed area of the numerical analysis of their deterministic counterparts. In the literature,
much attention on numerical stability has been focused on a special class of SFDEs, that is,
stochastic delay differential equations (SDDEs). The stability includes mean-square stability,
asymptotic stability, exponential mean-square stability and so on. Some results on numerical
stability can befound in [1, 10, 16, 19, 20, 21]. However, most of the existing numerical stability
theory ofSDDEs only dealwithboundedlags. Far less is knownforlong-runbehavior ofSDDEs
with unbounded lags. Very recently, Fan, Song and Liu [3] discussed the mean square stability
of semi-implicit Euler methods for linear stochastic pantograph equation. Moreover, to the best
knowledge of authors, there is no work on contractivity analysis of numerical methods for SD-
DEs. These motivate our work. This is the first paper to investigate stability and contractivity
of general SDDEs with boundedand unboundedlags and study whether numerical methods can
preserve the properties.
The main contributions of this paper could be summarized as follows:
(a)Sufficientconditionsforstabilityinmeansquare,contractivityinmeansquareandasymp-
totic contractivity in mean square of the solutions of nonlinear SDDEs are given, which provide
a unified theoretical treatment for constant and variable delay, bounded and unbounded delay.
Applicability of the theory is illustrated by linear and nonlinear SDDEs with a wide variety
of delay arguments such as constant delays, piecewise constant arguments, proportional delays
and so on. The theorems established in this paper work for some SDDEs to which the existing
results cannot be applied. The main results of analytic solution in this paper can beregarded as
a generalization of those in [8] restricted in finite-dimensional Hilbert spaces and finitely many
delays to the stochastic version.
(b) It is proved that the backward Euler method can reproduce stability in mean square,
contractivity in mean square and asymptotic contractivity in mean square of the underlying
system. In particularly, Theorem 4.2 and Theorem 4.4 show that the backward Euler method
caninheritthecontractivity andtheasymptoticcontractivity withoutanyconstraintonstepsize.
WepointoutthatthemaintheoremsinthepresentpaperaredifferentfromtheRazumikhin-
type theorems in the literature [1, 11]. It is not required to construct or find an appropriate
2
Lyapunov functional when the theorems in this paper are applied to prove the stability, in
contrast to the Razumikhin-type theorems. In this sense, it is more convenient to use the
theorems in this paper than the Razumikhin-type theorems.
The rest of the paper is arranged as follows. In section 2, we introduce some necessary nota-
tions and assumptions. In section 3, some criteria for the stability in mean square, contractivity
in mean square and asymptotic contractivity in mean square of solutions of nonlinear SDDEs
are established. The main results obtained in this section are applied to SDDEs with bounded
and unboundedlags, respectively. In section 4, sufficient conditions on stability in mean square,
contractivity inmeansquareandasymptoticcontractivity inmeansquareofthebackwardEuler
method are derived. Stability of analytic and numerical solutions of SDDEs with several delays
is discussed in section 5.
2 Stochastic delay differential equations
Let (Ω,F,{F } ,P) be a complete probability space with a filtration {F } satisfying the
t t≥a t t≥a
usual conditions (i.e., it is right continuous and F contains all the P-null sets). Let w(t) :=
a
(w (t),...,w (t))T be an m-dimensional Wiener process defined on the probability space. And
1 m
|·|is used to denote both the norm in complex Hilbert space Cd and the trace norm (F-norm) in
Cd×m. Also, C([t ,t ];Cd) is used to represent the family of continuous mappings ψ from [t ,t ]
1 2 1 2
toCd. Finally, Lp ([t ,t ];Cd)isusedtodenoteafamilyofF -measurable,C([t ,t ];Cd)-valued
random variablesFtψ =1 {2ψ(u) : t ≤ u ≤ t } such that kψktp := sup E|1ψ(u2)|p < ∞. E
1 2 E t1≤u≤t2
denote mathematical expectation with respect to P.
Consider the following initial value problems of SDDEs in the sense of Itˆo
dx(t) = f(t,x(t),x(t−τ(t)))dt+g(t,x(t),x(t−τ(t)))dw(t), t ∈ [a,b], (2.1a)
x(t) =ξ(t), t ∈ [a−τ,a], (2.1b)
(cid:26)
where a,b,τ are constants, −∞ < a < b < +∞,τ ≥ 0,τ(t) ≥ 0, inf (t − τ(t)) ≥ a − τ,
a≤t≤b
ξ ∈ Lp ([a−τ,a];Cd),p > 2, f : [a,b]×Cd ×Cd → Cd,g : [a,b]×Cd ×Cd → Cd×m are given
Fa
continuous mappings. We assume that f and g satisfy the following conditions.
For each R > 0 there exists a constant C , depending only on R, such that
R
(2.2)
|f(t,x˜ ,y˜)−f(t,x˜ ,y˜)| ≤ C |x˜ −x˜ |, |x˜ |∨|x˜ |∨|y˜| ≤ R,
1 2 R 1 2 1 2
ℜ x −x ,f(t,x ,y)−f(t,x ,y) ≤ α(t)|x −x |2, (2.3)
1 2 1 2 1 2
|f(t,x,y )−f(t,x,y )| ≤ β(t)|y −y |, (2.4)
(cid:10) 1 2 1(cid:11) 2
|g(t,x ,y )−g(t,x ,y )| ≤ γ (t)|x −x |+γ (t)|y −y |, (2.5)
1 1 2 2 1 1 2 2 1 2
forallt ∈ [a,b],x,x ,x ,y,y ,y ∈ Cd. Hereα(t),β(t),γ (t)andγ (t)arecontinuousreal-valued
1 2 1 2 1 2
functions. We introduce the following notations:
(0) (0)
µ = inf τ(t) ≥ 0, µ (t ,t )= inf t−τ(t) ≥ a−τ,
1 2 1 2
a≤t≤b t1≤t≤t2
∀t ,t :a ≤ t ≤(cid:0)t ≤ b.(cid:1)
1 2 1 2
Forconvenience, wedenotebySD(α,β,γ ,γ )theallproblems(2.1)whichsatisfytheconditions
1 2
(2.2)-(2.5). Examples of linear and nonlinear equations which satisfy the conditions (2.2)-(2.5)
are given in the next section (see Example 3.16 and Example 3.20).
3
In order to deal with a wide variety of delay arguments in this paper, we introduced the
so-called 1-perturbed problem of (2.1) which first introduced by Li[8] for VFDEs. Let
n
f (t,x,ψ) = f(t,x,ψ(n,t)), g (t,x,ψ) = g(t,x,ψ(n,t)), (2.6)
n n
for t ∈ [a,b],x ∈ Cd,ψ ∈ C([a−τ,b];Cd), where ψ(n,t) is defined by
ψ(u), u∈ [a−τ,t− 1],
ψ(n,t)(u) = n (2.7)
ψ(t− 1), u∈ (t− 1,b].
(cid:26) n n
Then the initial problem
dx(t) = f t,x(t),x(t−τ(t)) dt+g t,x(t),x(t−τ(t)) dw(t), t ∈[a,b], (2.8a)
n n
x(t) = ξ(t), t ∈ [a−τ,a] (2.8b)
(cid:26) (cid:0) (cid:1) (cid:0) (cid:1)
is said to be the 1-perturbed problem of the problem (2.1). Here the natural number n > 1
n τ
can be arbitrarily given. Notice that for the case of τ = 0, we can replace it by some positive
number τ˜ and define ξ(u) = ξ(a) for u ∈ [a−τ˜,a]. Therefore, without lose of generality, we
always assume τ > 0 .
It is easy to verify that, if problem (2.1) ∈ SD(α,β,γ ,γ ), then its 1-perturbed problem
1 2 n
(2.8) ∈ SD(α,β,γ ,γ ). Let τ˜(t) =max τ(t), 1 , we have
1 2 n
(cid:8) (cid:9) 1
(0)
µ˜ = inf τ˜(t) ≥ . (2.9)
1 a≤t≤b n
3 Stability analysis of SDDEs
We will discuss three types of stability of SDDEs in this section and here give definitions of
them.
Definition 3.1 The solution of problem (2.1) is said to be stable in mean square if
E|x(t)−y(t)|2 ≤ C sup E|ξ(θ)−η(θ)|2, (3.1)
a−τ≤θ≤a
where y(t) is a solution of any given perturbed problem
dy(t) = f(t,y(t),y(t−τ(t)))dt+g(t,y(t),y(t−τ(t)))dw(t), t ∈ [a,b], (3.2a)
y(t) = η(t), t ∈ [a−τ,a], η ∈ Lp ([a−τ,a];Cd). (3.2b)
( Fa
Definition 3.2 The solution of problem (2.1) is said to be contractive in mean square if (3.1)
with C ≤ 1 holds.
Definition 3.3 The solution of problem (2.1) is said to be asymptotically contractive in mean
square if
lim E|x(t)−y(t)|2 = 0,
t→+∞
where [a,b] replaced by [a,+∞).
4
Remark 3.4 In the strict sense, the solution of (2.1) should be called generalized contractive in
mean square if (3.1) with C ≤ 1 holds. For convenience, we call it contractive in mean square
for brevity.
There exist well-known stability definitions in the literature which are closely related to those
presented in this paper, but there are differences among them. The existing notions of stability
include mean-square stability for stochastic differential equations, that is, lim E|x(t)|2 = 0 (cf.
t→+∞
[14]); exponential mean-square contraction of trajectories for stochastic differential equations
with jumps (cf. [6]). The contractivity in mean square is weaker than that in [6].
The continuity of f and g implies that
|f(t,0,0)| ≤ C, |g(t,0,0)| ≤ C, t ∈ [a,b], (3.3)
where C only depends on f,g and the interval [a,b]. We note that condition (2.5) implies that
the diffusion coefficient g satisfies local Lipschitz condition
|g(t,x ,y )−g(t,x ,y )| ≤ C |x −x |+|y −y | ,
1 1 2 2 R 1 2 1 2
(3.4)
t ∈[a,b],x1,x2,y1,y2 ∈ Cd,(cid:0)|x1|∨|x2|∨|y1|∨|y(cid:1)2| ≤ R.
In fact, we can choose any C with C ≥ max{max γ (t), max γ (t)}. Using (2.3), (2.4) and
R R 1 2
a≤t≤b a≤t≤b
(3.3), we have
ℜhx,f(t,x,y)i = ℜ x−0,f(t,x,y)−f(t,0,y)+f(t,0,y)−f(t,0,0)+f(t,0,0)
D E
≤ α(t)|x|2 +β(t)|x||y|+C|x|≤ C 1+|x|2+|y|2 , (3.5)
1
(cid:16) (cid:17)
where C only depends on C, max α(t) and max β(t). By (2.5), we have
1
a≤t≤b a≤t≤b
|g(t,x,y)|2 ≤ 2|g(t,x,y)−g(t,0,0)|2 +2|g(t,0,0)|2 ≤ C 1+|x|2+|y|2 , (3.6)
1
(cid:16) (cid:17)
where C only depends on C, max γ2(t) and max γ2(t).
1 1 2
a≤t≤b a≤t≤b
3.1 Finite interval
In order to prove the main theorems in this section, we prepare the following lemmas.
Lemma 3.5 Assume that problem (2.1) ∈ SD(α,β,γ ,γ ). Then for each p ≥ 2 there is
1 2
C¯ = C¯(p,a,b,α,β,γ ,γ ) such that
1 2
E sup |x(t)|p ≤ C¯ 1+E( sup |ξ(t)|p) =:A. (3.7)
a≤t≤b a−τ≤t≤a
(cid:0) (cid:1) (cid:0) (cid:1)
Proof. For every integer k ≥ 1, define the stopping time
ρ = inf{t ∈ [a,b] : sup |x(θ)| ≥ k}, (3.8)
k
a−τ≤θ≤t
5
where we use the convention inf∅ = b. Clearly, ρ ↑ b almost surely as k → +∞. Let xk(t) =
k
x(t∧ρ ). Using the Itˆo formula, we have for a ≤ t ≤ b
k
p p
1+|xk(t)|2 2 = 1+|ξ(a)|2 2
(cid:16) t (cid:17) (cid:16) p−2 (cid:17)
+p 1+|xk(s)|2 2 ℜhxk(s),f(s,xk(s),xk(s−τ(s)))iI ds
[[a,ρk]]
Za (cid:16) (cid:17)
p t p−2
+ 1+|xk(s)|2 2 |g(s,xk(s),xk(s−τ(s)))|2I ds
2 [[a,ρk]]
Za (cid:16) (cid:17)
p(p−2) t p−4
+ 1+|xk(s)|2 2 |(xk(s))Tg(s,xk(s),xk(s−τ(s)))|2I ds
2 [[a,ρk]]
Za (cid:16) (cid:17)
t p−2
+p 1+|xk(s)|2 2 ℜhxk(s),g(s,xk(s),xk(s−τ(s)))iI dw(s).
[[a,ρk]]
Za (cid:16) (cid:17)
It follows from (3.5), (3.6) and this expression that
p t p−2
1+|xk(t)|2 2 ≤ 2p−22 1+|ξ(a)|p +2pC1 1+|xk(s)|2 2 1+ sup |xk(u)|2 ds
(cid:16) (cid:17) (cid:16) (cid:17) Za (cid:16) (cid:17) (cid:16) s−τ(s)≤u≤s (cid:17)
t p−2
+pC 1+|xk(s)|2 2 1+ sup |xk(u)|2 ds
1
Za (cid:16) (cid:17) (cid:16) s−τ(s)≤u≤s (cid:17)
t p−4
+p(p−2)C 1+|xk(s)|2 2 |xk(s)|2 1+ sup |xk(u)|2 ds
1
Za (cid:16) (cid:17) (cid:16) s−τ(s)≤u≤s (cid:17)
t p−2
+p 1+|xk(s)|2 2 ℜhxk(s),g(s,xk(s),xk(s−τ(s)))iI dw(s)
[[a,ρk]]
Za (cid:16) (cid:17)
t p
≤ 2p−22 1+|ξ(a)|p +p(p+1)C1 1+ sup |xk(u)|2 2ds
(cid:16) (cid:17) Za (cid:16) s−τ(s)≤u≤s (cid:17)
t p−2
+p 1+|xk(s)|2 2 ℜhxk(s),g(s,xk(s),xk(s−τ(s)))iI dw(s),
[[a,ρk]]
Za (cid:16) (cid:17)
which yields
p
E sup 1+|xk(s)|2 p2 ≤ 2p−22 1+E sup |ξ(s)|p +C2E at 1+ sup |xk(u)|2 2ds
a≤s≤t a−τ≤s≤a a−τ≤u≤s
(cid:16) (cid:17) (cid:16) (cid:17)
+pE(cid:0) sup s (cid:1)1+|xk(u)|2 p−22ℜhxk(u),g(u,xk(u),xk(uR−τ(u)))iI dw(u) .
a [[a,ρk]]
a≤s≤t
(cid:16) (cid:17)
R (cid:0) (cid:1)
6
Applying the Burkholder-Davis-Gundy inequality to the third term at the right hand side of
this expression, we obtain the bound
s
p−2
pE sup 1+|xk(u)|2 2 ℜhxk(u),g(u,xk(u),xk(u−τ(u)))iI dw(u)
[[a,ρk]]
(cid:16)a≤s≤tZa (cid:17)
t (cid:0) (cid:1) 1
≤ C E 1+|xk(u)|2 p−2|xk(u)|2|g(u,xk(u),xk(u−τ(u)))|2du 2
p
(cid:16)Za (cid:17)
(cid:0) (cid:1) p t p−4 1
≤ C E sup 1+|xk(u)|2 2 1+|xk(u)|2 2 |xk(u)|2|g(u,xk(u),xk(u−τ(u)))|2du 2
p
(cid:16)a≤u≤t Za (cid:17)
≤ 1E sup 1(cid:0)+|xk(u)|2 p2(cid:1)+ Cp22(cid:0)C E t 1+(cid:1) sup |xk(s)|2 p2du.
1
2 2
a≤u≤t Za (cid:16) a−τ≤s≤u (cid:17)
(cid:0) (cid:1)
Consequently,
t p
E sup 1+|xk(s)|2 p2 ≤ 2p2 1+E sup |ξ(s)|p +C3E 1+ sup |xk(s)|2 2du
a≤s≤t a−τ≤s≤a Za (cid:16) a−τ≤s≤u (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1) t p
= 2p2 1+E sup |ξ(s)|p +C3E sup 1+|xk(s)|2 2du.
a−τ≤s≤a Za a−τ≤s≤u(cid:16) (cid:17)
(cid:0) (cid:1)
Further, we notice that
p p p
E sup 1+|xk(s)|2 2 ≤ E sup 1+|ξ(s)|2 2 +E sup 1+|xk(s)|2 2
a−τ≤s≤t a−τ≤s≤a a≤s≤t
(cid:0) (cid:1) ≤ 2p−22(1+E(cid:0) sup |ξ((cid:1)s)|p)+E su(cid:0)p 1+|xk(s(cid:1))|2 p2.
a−τ≤s≤a a≤s≤t
(cid:0) (cid:1)
Therefore,
E sup 1+|xk(s)|2 p2 ≤ 32p2(1+E sup |ξ(s)|p)+C3 tE sup 1+|xk(s)|2 p2du.
2
a−τ≤s≤t a−τ≤s≤a Za a−τ≤s≤u(cid:16) (cid:17)
(cid:0) (cid:1)
Now the Gronwall’s inequality yields that
E sup |xk(s)|p ≤ E sup (1+|xk(s)|2)p2 ≤ 32p2 1+E sup |ξ(s)|p eC3(t−a).
2
a−τ≤s≤t a−τ≤s≤t a−τ≤s≤a
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Letting k → +∞ and applying the Fatou’s lemma, we obtain the desired result.
Lemma 3.6 Assume that problem (2.1) ∈ SD(α,β,γ ,γ ). Then there exists a unique solution
1 2
x(t) to equation (2.1).
Proof. Using (2.2), (2.4), (3.4), (3.5) and (3.6), we can complete the proof in a similar manner
to those of Theorem 5.2.7 and Theorem 2.3.5 in [11].
Lemma 3.7 Assume that problem (2.1) ∈ SD(α,β,γ ,γ ). Then we have
1 2
lim sup E|x(t)−x(s)|2 = 0. (3.9)
t−s→0a≤s<t≤b
7
Proof. The proof uses the techniques employed in the proof of Theorem 2.2 in [5]. Integrating
(2.1) gives for a ≤ s < t ≤ b
t t
x(t)−x(s) = f(u,x(u),x(u−τ(u)))du+ g(u,x(u),x(u−τ(u)))dw(u). (3.10)
Zs Zs
Let e(s,t) = x(t)−x(s),
ρ = inf{t ∈ [a,b] : sup |x(θ)| ≥R},
R
a−τ≤θ≤t
where inf∅= b. Using the Young inequality: for r−1+q−1 =1
δ 1
ab ≤ ar + bq, ∀a,b,δ > 0
r qδq/r
and letting r = p,q = p , we thus have for any δ > 0
2 p−2
E |e(s,t)|2 = E |e(s,t)|2I +E |e(s,t)|2I
{ρR≥b} {ρR<b} (3.11)
(cid:0) (cid:1) (cid:0) 2δ (cid:1) (cid:0) 1− 2 (cid:1)
≤ E |e(s,t)|2I + E |e(s,t)|p + p P(ρ < b),
{ρR≥b} p δ2/(p−2) R
(cid:0) (cid:1) (cid:0) (cid:1)
where p > 2. It follows from (3.7) that
|x(ρ )|p 1 A
P(ρ < b)= E I R ≤ E sup |x(t)|p ≤ (3.12)
R {ρR<b} Rp Rp Rp
(cid:18) (cid:19) a≤t≤b
E |e(s,t)|p ≤2p−1E sup |x(s)|p + (cid:0)sup |x(t)|p) (cid:1)≤ 2pA. (3.13)
a≤s≤b a≤t≤b
(cid:0) (cid:1) (cid:0) (cid:1)
We then have
2p+1δA (p−2)A
E |e(s,t)|2 ≤ E |e(s,t)|2I + + . (3.14)
{ρR≥b} p pδ2/(p−2)Rp
(cid:0) (cid:1) (cid:0) (cid:1)
Further, using the Ho¨lder’s inequality and the Itˆo isometry we obtain
E |e(s,t)|2I
{ρR≥b}
t t 2
= E(cid:0) f(u,x(u),x(cid:1)(u−τ(u)))du+ g(u,x(u),x(u−τ(u)))dw(u) I
{ρR≥b}
(cid:16)(cid:12)Zs Zs (cid:12) (cid:17)
(cid:12) t t (cid:12)
≤ 2E(cid:12) f(u,x(u),x(u−τ(u)))du 2+ g(u,x(u),x(u−τ(u)))d(cid:12)w(u) 2 I
{ρR≥b}
(cid:16)(cid:16)(cid:12)Zs t (cid:12) (cid:12)Zs t (cid:12) (cid:17) (cid:17)
≤ 2(t−(cid:12)s)E I |f(u,x(u),x(u(cid:12)−τ(cid:12)(u)))|2du +2E |g(u,x(u),x(u(cid:12) −τ(u)))|2du
{ρR≥b}
(cid:16) Zs (cid:17) Zs
t
≤ 2(t−s)E I |f(u,x(u),x(u−τ(u)))−f(u,0,0)+f(u,0,0)|2du
{ρR≥b}
(cid:16) Zs (cid:17)
t
+2E |g(u,x(u),x(u−τ(u)))|2du.
Zs
8
By (2.2), (2.4), (3.3), (3.6) and Lemma 3.5, we have
E |e(s,t)|2I
{ρR≥b}
t t
≤ 8C(cid:0)2(t−s) E|(cid:1)x(u)|2 +E|x(u−τ(u))|2 du+4(t−s)E |f(u,0,0)|2du
R
Zs (cid:16) (cid:17) Zs
t
+2C 1+E|x(u)|2+E|x(u−τ(u))|2 du ≤ C (t−s),
1 4
Zs (cid:16) (cid:17)
where C is independent of s and t. A combination of this expression and (3.14) leads to
4
2p+1δA (p−2)A
E |e(s,t)|2 = C (t−s)+ + . (3.15)
4
p pδ2/(p−2)Rp
(cid:0) (cid:1)
Therefore, for any given ǫ > 0, we can choose δ and R such that
2p+1δA 1 (p−2)A 1
< ǫ, < ǫ, (3.16)
p 3 pδ2/(p−2)Rp 3
and then choose t−s sufficiently small such that C (t−s)< 1ǫ. Hence, we have
4 3
lim sup E |x(t)−x(s)|2 = lim sup E |e(s,t)|2 =0.
t−s→0a≤s<t≤b t−s→0a≤s<t≤b
(cid:0) (cid:1) (cid:0) (cid:1)
The proof is complete.
Lemma 3.8 Assume that problem (2.1) ∈ SD(α,β,γ ,γ ). Then for any t ,t : a ≤ t ≤ t ≤
1 2 1 2 1 2
b,
E|x(t2)−y(t2)|2 ≤ eRtt12(2α(t)+β(t)+γ1(t)γ2(t)+γ12(t))dtE|x(t1)−y(t1)|2
+ t2 β(s)+γ1(s)γ2(s)+γ22(s) eRst2(2α(u)+β(u)+γ1(u)γ1(u)+γ12(u))duds (3.17)
Zt1
· (cid:0) sup E|x(θ)−y((cid:1)θ)|2,
µ(20)(t1,t2)≤θ≤t2−µ(10)
where y(t) is a solution of any given perturbed problem (3.2).
Proof. Let
V(t,x(t)) =p(t) |x(t)|2 +δq(t) , t ≤ t ≤ t , (3.18)
1 2
where (cid:0) (cid:1)
p(t)= eRat(−2α(u)−β(u)−γ1(u)γ2(u)−γ12(u))du,
(3.19)
t
q(t)= −(p(t))−1 β(u)+γ (u)γ (u)+γ2(u) p(u)du,
1 2 2
Za (cid:16) (cid:17)
δ is a constant to be determined. For convenience, from now on we use σ(t) and ̺(t) to denote
2α(t)+β(t)+γ (t)γ (t)+γ2(t) and β(t)+γ (t)γ (t)+γ2(t), respectively, that is
1 2 1 1 2 2
σ(t) = 2α(t)+β(t)+γ (t)γ (t)+γ2(t),
1 2 1 (3.20)
̺(t) = β(t)+γ (t)γ (t)+γ2(t).
1 2 2
9
Then we have p′(t) = −σ(t)p(t), p(t)q(t) ′ = −p(t)̺(t). By (3.18),(3.19) and the Itˆo formula,
one can derive that, for a ≤ t ≤ t ≤ b,
1 2
(cid:0) (cid:1)
EV t ,x(t )−y(t ) = EV t ,x(t )−y(t )
2 2 2 1 1 1
t2
+ (cid:0) −σ(t)p(t)E(cid:1)|x(t)−(cid:0)y(t)|2 −δp(t)̺(t)(cid:1)
Zt1 (cid:26)
+2p(t)Eℜ x(t)−y(t),f(t,x(t),x(t−τ(t)))−f(t,y(t),y(t−τ(t)))
(cid:10) (cid:11)
+p(t)E g(t,x(t),x(t−τ(t)))−g(t,y(t),y(t−τ(t))) 2 dt
(cid:27)
≤ EV t(cid:12)(cid:12) ,x(t )−y(t ) + t2 −σ(t)p(t)E|x(t)−(cid:12)(cid:12)y(t)|2 −δp(t)̺(t)
1 1 1
Zt1 (cid:26)
+2p(t(cid:0))Eℜ x(t)−y(t),(cid:1)f(t,x(t),x(t−τ(t)))−f(t,y(t),x(t−τ(t)))
+2p(t)Eℜ(cid:10) x(t)−y(t),f(t,y(t),x(t−τ(t)))−f(t,y(t),y(t−τ(t)))(cid:11)
(cid:10) (cid:11)
+p(t)E|g(t,x(t),x(t−τ(t)))−g(t,y(t),y(t−τ(t)))|2 dt.
(cid:27)
Using the conditions (2.3)-(2.5), we obtain
EV t ,x(t )−y(t ) ≤ EV t ,x(t )−y(t )
2 2 2 1 1 1
t2
+ (cid:0) −σ(t)p(t)E(cid:1)|x(t)−(cid:0)y(t)|2−δp(t)̺(t)(cid:1)
Zt1 (cid:26)
+2p(t)α(t)E|x(t)−y(t)|2+2p(t)β(t)E |x(t)−y(t)||x(t−τ(t))−y(t−τ(t))|
2
+p(t)E γ (t)|x(t)−y(t)|+γ (t)|x(t−(cid:0) τ(t))−y(t−τ(t))| dt (cid:1)
1 2
(cid:27)
(cid:16) (cid:17)
t2
≤ EV t ,x(t )−y(t ) + −σ(t)p(t)E|x(t)−y(t)|2 −δp(t)̺(t)
1 1 1
Zt1 (cid:26)
(cid:0) (cid:1)
+2p(t)α(t)E|x(t)−y(t)|2+p(t)β(t) E|x(t)−y(t)|2+E|x(t−τ(t))−y(t−τ(t))|2
(cid:16) (cid:17)
+p(t)γ2(t)E|x(t)−y(t)|2+p(t)γ2(t)E|x(t−τ(t))−y(t−τ(t))|2
1 2
+2p(t)γ (t)γ (t)E |x(t)−y(t)||x(t−τ(t))−y(t−τ(t))| dt
1 2
(cid:27)
(cid:0) (cid:1)
which yields
EV t ,x(t )−y(t )
2 2 2
≤ EV t ,x(t )−y(t ) + t2p(t)β(t) E|x(t−τ(t))−y(t−τ(t))|2 −δ dt
(cid:0) 1 1 (cid:1)1 t1
(3.21)
+ t2p(cid:0)(t)γ2(t) E|x(t−(cid:1)τ(tR))−y(t−τ(cid:16)(t))|2 −δ dt (cid:17)
t1 2
+Rtt12p(t)γ1(t)γ(cid:16)2(t) E|x(t−τ(t))−y(t−τ(t))|2(cid:17)−δ dt.
(cid:16) (cid:17)
R
10