Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 128625, 8 pages
http://dx.doi.org/10.1155/2013/128625
Research Article
Numerical Analysis for Stochastic Partial Differential Delay
Equations with Jumps
Yan Li1,2 and Junhao Hu3
1
Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
College of Science, Huazhong Agriculture University, Wuhan 430074, China
3
College of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China
2
Correspondence should be addressed to Junhao Hu; hujunhao@yahoo.com.cn
Received 3 January 2013; Accepted 21 March 2013
Academic Editor: Xuerong Mao
Copyright © 2013 Y. Li and J. Hu. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We investigate the convergence rate of Euler-Maruyama method for a class of stochastic partial differential delay equations driven by
both Brownian motion and Poisson point processes. We discretize in space by a Galerkin method and in time by using a stochastic
exponential integrator. We generalize some results of Bao et al. (2011) and Jacob et al. (2009) in finite dimensions to a class of
stochastic partial differential delay equations with jumps in infinite dimensions.
1. Introduction
The theory and application of stochastic differential equations have been widely investigated [1–7]. Liu [2] studied
the stability of infinite dimensional stochastic differential
equations. For the numerical analysis of stochastic partial
differential equations, Gyöngy and Krylov [8] discussed the
numerical approximations for linear stochastic partial differential equations in whole space. Jentzen et al. [9] studied the
numerical simulations of nonlinear parabolic stochastic partial differential equations with additive noise. Kloeden et al.
[10] gave the error analysis for the pathwise approximation of
a general semilinear stochastic evolution equations.
By contrast, stochastic partial differential equations with
jumps have begun to gain attention [11–15]. Röckner and
Zhang [15] considered the existence, uniqueness, and large
deviation principles of stochastic evolution equation with
jump. In [12], the successive approximation of neutral SPDEs
was studied. There are few papers on the convergence
rate of numerical solutions for stochastic partial differential
equations with jump, although there are some papers on
the convergence rate of numerical solutions for stochastic
differential equations with jump in finite dimensions [16, 17].
Being motivated by the papers [16, 17], we will discuss
the convergence rate of Euler-Maruyama scheme for a class
of stochastic partial delay equations with jump, where the
numerical scheme is based on spatial discretization by
Galerkin method and time discretization by using a stochastic
exponential integrator. In consequence, we generalize some
results of Bao et al. (2011) and Jacob et al. (2009) in finite
dimensions to a class of stochastic partial delay equations
with jump in infinite dimensions. The rest of this paper is
arranged as follows. We give some preliminary results of
Euler-Maruyama scheme in Section 2. The convergence rate
is discussed in Section 3.
2. Preliminary Results
Throughout this paper, let (Ω, F, {F𝑡 }𝑡≥0 , P) be a complete
probability space with some filtration {F𝑡 }𝑡≥0 satisfying the
usual conditions (i.e., it is right continuous and F0 contains
all P-null sets). Let (𝐻, ⟨⋅, ⋅⟩𝐻, ‖ ⋅ ‖𝐻) and (𝐾, ⟨⋅, ⋅⟩𝐾 , ‖ ⋅ ‖𝐾 ) be
two real separable Hilbert spaces. We denote by (L(𝐾, 𝐻),
‖ ⋅ ‖) the family of bounded linear operators. Let 𝜏 > 0 and
𝐷 ([−𝜏, 0], 𝐻) denote the family of right-continuous function
and left-hand limits 𝜑 from [−𝜏, 0] to 𝐻 with the norm
𝑏
([−𝜏, 0], 𝐻) denotes the fam‖𝜑‖𝐷 = sup−𝜏≤𝜃≤0 ‖𝜑(𝜃)‖𝐻. 𝐷F
0
ily of almost surely bounded, F0 -measurable, 𝐷 ([−𝜏, 0], 𝐻)valued random variables. For all 𝑡 ≥ 0, 𝑋𝑡 = {𝑋(𝑡 + 𝜃) : −𝜏 ≤
𝜃 ≤ 0} is regarded as 𝐷 ([−𝜏, 0], 𝐻)-valued stochastic process.
2
Abstract and Applied Analysis
Let 𝑇 be a positive constant. For given 𝜏 ≥ 0, consider the
following stochastic partial differential delay equations with
jumps:
(1)
+ ∫ ℎ (𝑋 (𝑡) , 𝑋 (𝑡 − 𝜏) , 𝑢) 𝑁 (𝑑𝑡, 𝑑𝑢)
Z
on 𝑡 ∈ [0, 𝑇] with initial datum 𝑋(𝑡) = 𝜉(𝑡) ∈
𝑏
([−𝜏, 0], 𝐻), −𝜏 ≤ 𝑡 ≤ 0. Here (𝐴, 𝐷(𝐴)) is a self-adjoint
𝐷F
0
operator on 𝐻. {𝑊(𝑡), 𝑡 ≥ 0} is 𝐾-valued {F𝑡 }𝑡≥0 -Wiener
process defined on the probability space {Ω, F, {F𝑡 }𝑡≥0 , P}
with covariance operator 𝑄. We assume that −𝐴 and the
covariance operator 𝑄 of the Wiener process have the same
eigenbasis {𝑒𝑚 }𝑚≥1 of 𝐻; that is,
−𝐴𝑒𝑚 = 𝜆 𝑚 𝑒𝑚 ,
𝑄𝑒𝑚 = 𝛼𝑚 𝑒𝑚 ,
𝑚 = 1, 2, 3, . . . ,
∞
𝑊 (𝑡) = ∑ √𝛼𝑚 𝛽𝑚 (𝑡) 𝑒𝑚 ,
𝑡 ≥ 0,
(3)
𝑛=1
where 𝛽𝑚 (𝑡) (𝑚 = 1, 2, 3, . . .) is a sequence of real-valued
standard Brownian motions mutually independent of the
probability space (Ω, F, {F𝑡 }𝑡≥0 , P).
According to Da Prato and Zabczyk [1], we define
stochastic integrals with respect to the 𝑄-Wiener process
𝑊(𝑡). Let 𝐾0 = 𝑄1/2 (𝐾) be the subspace of 𝐾 with the
inner product ⟨𝑢, V⟩𝐾0 = ⟨𝑄−1/2 𝑢, 𝑄−1/2 V⟩𝐾 . Obviously, 𝐾0
is a Hilbert space. Denote by L02 = L(𝐾0 , 𝐻) the family
of Hilbert-Schmidt operators from 𝐾0 into 𝐻 with the norm
∗
‖Ψ‖2L0 = tr((Ψ𝑄1/2 )(Ψ𝑄1/2 ) ).
2
→
𝑡
L02 be a predictable, F𝑡 -adapted
∫ E‖Φ(𝑠)‖L02 𝑑𝑠 ≤ ∞,
0
∀𝑡 > 0.
(4)
is a centred square-integrable martingale.
We recall the definition of the mild solution to (1) as
follows.
Definition 1. A stochastic process {𝑋(𝑡) : 𝑡 ∈ [0, 𝑇]} is called
a mild solution of (1) if
(i) 𝑋(𝑡) is adapted to F𝑡 , 𝑡 ≥ 0, and has càdlàg path on
𝑡 ≥ 0 almost surely,
𝑡
(ii) for arbitrary 𝑡 ∈ [0, 𝑇], P{𝑤 : ∫0 ‖𝑋(𝑠)‖2𝐻 𝑑𝑠 < ∞} =
1, and almost surely
𝑡
𝑋 (𝑡) = 𝑒𝑡𝐴𝜉 (0) + ∫ 𝑒(𝑡−𝑠)𝐴 𝑓 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏)) 𝑑𝑠
𝑡
+ ∫ 𝑒(𝑡−𝑠)𝐴 𝑔 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏)) 𝑑𝑊 (𝑠)
̃ (𝑑𝑡, 𝑑𝑢) = 𝑁 (𝑑𝑡, 𝑑𝑢) − 𝜌𝑑𝑡𝜋 (𝑑𝑢) ,
𝑁
(5)
where 0 < 𝜌 < ∞ is known as the jump rate and 𝜋(⋅) is the
jump distribution (a probability measure). Let Z ∈ B(𝐾−{0})
be the measurable set. Denote by P2 ([0, 𝑇] × Z, 𝐻) the space
of all predictable mappings ℎ : [0, 𝑇] × Z → 𝐻 for which
∫ ∫
0
Z
E‖ℎ (𝑡, 𝑢)‖2𝐻𝑑𝑡𝜋 (𝑑𝑢)
< ∞.
(6)
(8)
0
𝑡
+ ∫ ∫ 𝑒(𝑡−𝑠)𝐴 ℎ (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏) , 𝑢) 𝑁 (𝑑𝑠, 𝑑𝑢)
0
Z
𝑏
for any 𝑋(𝑡) = 𝜉(𝑡) ∈ 𝐷F
([−𝜏, 0], 𝐻), −𝜏 ≤ 𝑡 ≤ 0.
0
For the existence and uniqueness of the mild solution to
(1) (see [11]), we always make the following assumptions.
(H1) (𝐴, 𝐷(𝐴)) is a self-adjoint operator on 𝐻 such that
−𝐴 has discrete spectrum 0 ≤ 𝜆 1 ≤ 𝜆 2 ≤ ⋅ ⋅ ⋅ ≤
lim𝑚 → ∞ 𝜆 𝑚 = ∞ with corresponding eigenbasis
{𝑒𝑚 }𝑚≥1 of 𝐻. In this case 𝐴 generates a compact 𝐶0 semigroup 𝑒𝑡𝐴 , 𝑡 ≥ 0, such that ‖𝑒𝑡𝐴‖ ≤ 𝑒−𝛼𝑡 .
(H2) The mappings 𝑓 : 𝐻 × 𝐻 → 𝐻, 𝑔 : 𝐻 ×
𝐻 → L(𝐾, 𝐻), and ℎ : 𝐻 × 𝐻 × Z → 𝐻 are
Borel measurable and satisfy the following Lipschitz
continuity condition for some constant 𝐿 1 > 0 and
arbitrary 𝑥, 𝑦, 𝑥1 , 𝑦1 , 𝑥2 , 𝑦2 ∈ 𝐻 and 𝑢 ∈ Z:
󵄩󵄩
󵄩2
󵄩󵄩𝑓 (𝑥1 , 𝑦1 ) − 𝑓 (𝑥2 , 𝑦2 )󵄩󵄩󵄩𝐻
󵄩
󵄩2
∨ 󵄩󵄩󵄩𝑔 (𝑥1 , 𝑦1 − 𝑔 (𝑥2 , 𝑦2 ))󵄩󵄩󵄩L2
0
󵄩2 󵄩
󵄩2
󵄩
≤ 𝐿 1 (󵄩󵄩󵄩𝑥1 − 𝑥2 󵄩󵄩󵄩𝐻 + 󵄩󵄩󵄩𝑦1 − 𝑦2 󵄩󵄩󵄩𝐻) ,
𝑡
Then, the 𝐻-valued stochastic integral ∫0 Φ(𝑠)𝑑𝑊(𝑠) is a
continuous square martingale. Let 𝑁(𝑑𝑡, 𝑑𝑢) be the Poisson
measure which is independent of the 𝑄-Wiener process 𝑊(𝑡).
Denote the compensated or centered Poisson measure as
𝑇
(7)
Z
0
(2)
where {𝜆 𝑚 , 𝑚 ∈ N} are the discrete spectrum of −𝐴 and
0 ≤ 𝜆 1 ≤ 𝜆 2 ⋅ ⋅ ⋅ ≤ lim𝑚 → ∞ 𝜆 𝑚 = ∞, {𝛼𝑚 , 𝑚 ∈ N} are the
eigenvalues of 𝑄. Then, 𝑊(𝑡) is defined by
Let Φ : (0, ∞)
process such that
𝑇
̃ (𝑑𝑡, 𝑑𝑢)
∫ ∫ ℎ (𝑡, 𝑢) 𝑁
0
𝑑𝑋 (𝑡) = [𝐴𝑋 (𝑡) + 𝑓 (𝑋 (𝑡) , 𝑋 (𝑡 − 𝜏))] 𝑑𝑡
+ 𝑔 (𝑋 (𝑡) , 𝑋 (𝑡 − 𝜏)) 𝑑𝑊 (𝑡)
Then, the 𝐻-valued stochastic integral
(9)
󵄩2
󵄩󵄩
󵄩󵄩ℎ (𝑥1 , 𝑦1 , 𝑢) − ℎ (𝑥2 , 𝑦2 , 𝑢)󵄩󵄩󵄩𝐻
󵄩2 󵄩
󵄩2
󵄩
≤ 𝐿 1 (󵄩󵄩󵄩𝑥1 − 𝑥2 󵄩󵄩󵄩𝐻 + 󵄩󵄩󵄩𝑦1 − 𝑦2 󵄩󵄩󵄩𝐻) .
This further implies the linear growth condition; that
is,
󵄩2 󵄩
󵄩2
󵄩󵄩
󵄩 󵄩2
2
󵄩󵄩𝑓(𝑥, 𝑦)󵄩󵄩󵄩𝐻 + 󵄩󵄩󵄩𝑔(𝑥, 𝑦)󵄩󵄩󵄩L0 ≤ 𝐿 0 (1 + ‖𝑥‖𝐻 + 󵄩󵄩󵄩𝑦󵄩󵄩󵄩𝐻) ,
2
(10)
where
󵄩2 󵄩
󵄩2
󵄩
𝐿 0 := 2 (𝐿 2 ∨ 󵄩󵄩󵄩𝑓(0, 0)󵄩󵄩󵄩𝐻 ∨ 󵄩󵄩󵄩𝑔(0, 0)󵄩󵄩󵄩L0 ) .
2
(11)
Abstract and Applied Analysis
3
(H3) There exists 𝐿 2 > 0 satisfying
󵄩2
󵄩 󵄩2
󵄩󵄩
2
󵄩󵄩ℎ(𝑥, 𝑦, 𝑢)󵄩󵄩󵄩𝐻 ≤ 𝐿 2 (‖𝑥‖𝐻 + 󵄩󵄩󵄩𝑦󵄩󵄩󵄩𝐻) ,
(12)
for each 𝑥, 𝑦 ∈ 𝐻 and 𝑢 ∈ Z.
𝑏
([−𝜏, 0], 𝐻), there exists a constant 𝐿 3 > 0
(H4) For 𝜉 ∈ 𝐷F
0
such that
󵄨2
󵄨
E (󵄨󵄨󵄨𝜉 (𝑠) − 𝜉 (𝑡)󵄨󵄨󵄨 ) ≤ 𝐿 3 |𝑡 − 𝑠|2 ,
𝑡, 𝑠 ∈ [−𝜏, 0] .
(13)
for some sufficiently large integer 𝑁 > 𝜏. For any integer
𝑘 ≥ 0, the time discretization scheme applied to (14) produces
𝑛
approximations 𝑌 (𝑘Δ) ≈ 𝑋𝑛 (𝑘Δ) by forming
𝑛
𝑌 ((𝑘 + 1) Δ)
𝑛
+ 𝑔𝑛 (𝑋𝑛 (𝑡) , 𝑋𝑛 (𝑡 − 𝜏)) 𝑑𝑊 (𝑡)
+ ∫ ℎ𝑛 (𝑋𝑛 (𝑡) , 𝑋𝑛 (𝑡 − 𝜏) , 𝑢) 𝑁 (𝑑𝑡, 𝑑𝑢) ,
(14)
Z
𝑛
𝑋 (𝜃) = 𝜋𝑛 𝜉 (𝜃) ,
𝑛
𝑛
𝑛
+ ∫ ℎ𝑛 (𝑌 (𝑘Δ) , 𝑌 (𝑘Δ − 𝜏) , 𝑢) Δ𝑁𝑘 (𝑢)} ,
Z
𝑛
𝑌 (𝜃) = 𝜋𝑛 𝜉 (𝜃) ,
𝜃 ∈ [−𝜏, 0] ,
(16)
where Δ𝑊𝑘 = 𝑊((𝑘 + 1)Δ) − 𝑊(𝑘Δ) and Δ𝑁𝑘 (𝑑𝑢) =
𝑁((0, (𝑘 + 1)Δ], 𝑑𝑢) − 𝑁((0, 𝑘Δ], 𝑑𝑢).
The continuous-time version of this scheme associated
with (14) is defined by
𝑡
This spatial approximation (14) is called the Galerkin
approximation of (1). Due to the fact that 𝜋𝑛 𝐴𝑥 =
𝜋𝑛 𝐴(∑𝑛𝑖=1 ⟨𝑥, 𝑒𝑖 ⟩𝐻𝑒𝑖 ) = − ∑𝑛𝑖=1 𝜆 𝑖 ⟨𝑥, 𝑒𝑖 ⟩𝐻𝑒𝑖 , 𝑥 ∈ 𝐻𝑛 , it follows
that for 𝑥 ∈ 𝐻𝑛 , 𝐴 𝑛 𝑥 = 𝐴𝑥, 𝑒𝑡𝐴 𝑛 𝑥 = 𝑒𝑡𝐴𝑥 .
By (H2) and (H3) and the property of the projection
operator, we have that
= 𝑒𝑡𝐴 𝑛 𝑌𝑛 (0) + ∫ 𝑒(𝑡−⌊𝑠⌋)𝐴 𝑛 𝑓𝑛 (𝑌𝑛 (⌊𝑠⌋) , 𝑌𝑛 (⌊𝑠⌋ − 𝜏)) 𝑑𝑠
0
𝑡
+ ∫ 𝑒(𝑡−⌊𝑠⌋)𝐴 𝑛 𝑔𝑛 (𝑌𝑛 (⌊𝑠⌋) , 𝑌𝑛 (⌊𝑠⌋ − 𝜏)) 𝑑𝑊 (𝑠)
0
𝑡
+ ∫ ∫ 𝑒(𝑡−⌊𝑠⌋)𝐴 𝑛 ℎ𝑛 (𝑌𝑛 (⌊𝑠⌋) , 𝑌𝑛 (⌊𝑠⌋ − 𝜏) , 𝑢)
0
󵄩
󵄩2 󵄩
󵄩2
󵄩󵄩
2󵄩
󵄩󵄩𝐴 𝑛 𝑥 − 𝐴 𝑛 𝑦󵄩󵄩󵄩𝐻 = 󵄩󵄩󵄩𝐴 𝑛 (𝑥 − 𝑦)󵄩󵄩󵄩𝐻 ≤ 𝜆 𝑛 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩𝐻,
󵄩󵄩
󵄩2
󵄩󵄩𝑓𝑛 (𝑥1 , 𝑦1 ) − 𝑓𝑛 (𝑥2 , 𝑦2 )󵄩󵄩󵄩𝐻
󵄩
󵄩2
∨ 󵄩󵄩󵄩𝑔 (𝑥1 , 𝑦1 ) − 𝑔 (𝑥2 , 𝑦2 )󵄩󵄩󵄩L0
2
Z
× 𝑁 (𝑑𝑠, 𝑑𝑢) ,
𝑌𝑛 (𝜃) = 𝜋𝑛 𝜉 (𝜃) ,
𝜃 ∈ [−𝜏, 0] ,
(17)
󵄩
󵄩2
= 󵄩󵄩󵄩𝑓 (𝑥1 , 𝑦1 ) − 𝑓 (𝑥2 , 𝑦2 )󵄩󵄩󵄩𝐻
󵄩2 󵄩
󵄩2
󵄩
≤ 𝐿 1 (󵄩󵄩󵄩𝑥1 − 𝑥2 󵄩󵄩󵄩𝐻 + 󵄩󵄩󵄩𝑦1 − 𝑦2 󵄩󵄩󵄩𝐻) ,
𝑛
𝑌𝑛 (𝑡)
𝜃 ∈ [−𝜏, 0] .
󵄩
󵄩2
∨ 󵄩󵄩󵄩𝑔 (𝑥1 , 𝑦1 ) − 𝑔 (𝑥2 , 𝑦2 )󵄩󵄩󵄩L0
2
𝑛
+ 𝑔𝑛 (𝑌 (𝑘Δ) , 𝑌 (𝑘Δ − 𝜏)) Δ𝑊𝑘
We now describe our Euler-Maruyama scheme for the
approximation of (1). For any 𝑛 ≥ 1, let 𝜋𝑛 : 𝐻 →
𝐻𝑛 = span{𝑒1 , 𝑒2 , . . . , 𝑒𝑛 } be the orthogonal projection; that is,
𝜋𝑛 𝑥 = ∑𝑛𝑖=1 ⟨𝑥, 𝑒𝑖 ⟩𝐻𝑒𝑖 , 𝑥 ∈ 𝐻, 𝐴 𝑛 = 𝜋𝑛 𝐴, 𝑓𝑛 = 𝜋𝑛 𝑓, 𝑔𝑛 = 𝜋𝑛 𝑔,
and ℎ𝑛 = 𝜋𝑛 ℎ.
Consider the following stochastic differential delay equations with jumps on 𝐻𝑛 :
𝑑𝑋𝑛 (𝑡) = [𝐴 𝑛 𝑋𝑛 (𝑡) + 𝑓𝑛 (𝑋𝑛 (𝑡) , 𝑋𝑛 (𝑡 − 𝜏))] 𝑑𝑡
𝑛
= 𝑒Δ𝐴 𝑛 {𝑌 (𝑘Δ) + 𝑓𝑛 (𝑌 (𝑘Δ) , 𝑌 (𝑘Δ − 𝜏)) Δ
(15)
󵄩2
󵄩󵄩
󵄩󵄩ℎ𝑛 (𝑥1 , 𝑦1 , 𝑢) − ℎ𝑛 (𝑥2 , 𝑦2 , 𝑢)󵄩󵄩󵄩𝐻
󵄩
󵄩2
= 󵄩󵄩󵄩ℎ (𝑥1 , 𝑦1 , 𝑢) − ℎ (𝑥2 , 𝑦2 , 𝑢)󵄩󵄩󵄩𝐻
󵄩2 󵄩
󵄩2
󵄩
≤ 𝐿 1 (󵄩󵄩󵄩𝑥1 − 𝑥2 󵄩󵄩󵄩𝐻 + 󵄩󵄩󵄩𝑦1 − 𝑦2 󵄩󵄩󵄩𝐻) ,
󵄩2
󵄩2 󵄩
󵄩 󵄩2
󵄩󵄩
2
󵄩󵄩ℎ𝑛 (𝑥, 𝑦, 𝑢)󵄩󵄩󵄩𝐻 = 󵄩󵄩󵄩ℎ(𝑥, 𝑦, 𝑢)󵄩󵄩󵄩𝐻 ≤ 𝐿 2 (‖𝑥‖𝐻 + 󵄩󵄩󵄩𝑦󵄩󵄩󵄩𝐻)
for arbitrary 𝑥, 𝑦, 𝑥1 , 𝑥2 , 𝑦1 , 𝑦2 ∈ 𝐻𝑛 and 𝑢 ∈ Z. Hence, (14)
admits a unique solution 𝑋𝑛 (𝑡) on 𝐻𝑛 .
We introduce a time discretization scheme for (14) by
using a stochastic exponential integrator. For given 𝑇 ≥ 0 and
𝜏 > 0, the time-step size Δ ∈ (0, 1) is defined by Δ := 𝜏/𝑁,
where ⌊𝑡⌋ = [𝑡/Δ]Δ with [𝑡/Δ] denotes the integer of 𝑡/Δ.
𝑛
From (16) and (17), we have 𝑌𝑛 (𝑘Δ) = 𝑌 (𝑘Δ) for
every 𝑘 ≥ 0. That is, the discrete-time and continuous-time
schemes coincide at the grid points.
3. Convergence Rate
In this section, we shall investigate the convergence rate of the
Euler-Maruyama method. In what follows, 𝐶 > 0 is a generic
constant whose values may change from line to line.
Lemma 2. Let (H1)–(H4) hold; then there is a positive constant
𝐶 > 0 which depends on 𝑇, 𝜉, 𝐿 1 , 𝐿 2 , and 𝐿 3 but is independent
of Δ, such that
1/2
sup (E‖𝑋 (𝑡)‖2𝐻)
0≤𝑡≤𝑇
󵄩2
󵄩
∨ sup (E󵄩󵄩󵄩𝑌𝑛 (𝑡)󵄩󵄩󵄩𝐻)
1/2
0≤𝑡≤𝑇
≤ 𝐶.
(18)
4
Abstract and Applied Analysis
1/2
Proof. Due to the fact that (E‖ ⋅ ‖2𝐻)
from (8) that
(E‖𝑋 (𝑡)‖2𝐻)
is a norm, we have
𝑡
≤ (∫ 𝐿 0 (1 + E‖𝑋 (𝑠)‖2𝐻 + E‖𝑋 (𝑠 − 𝜏)‖2𝐻) 𝑑𝑠)
1/2
0
󵄩󵄩 𝑡
󵄩
+ (E 󵄩󵄩󵄩∫ ∫ 𝑒(𝑡−𝑠)𝐴 ℎ
󵄩󵄩 0 𝑍
1/2
󵄩2 1/2
󵄩
≤ (E󵄩󵄩󵄩󵄩𝑒𝑡𝐴 𝑛 𝜉 (0)󵄩󵄩󵄩󵄩𝐻)
󵄩󵄩 𝑡
󵄩󵄩2 1/2
󵄩
󵄩
+ (E󵄩󵄩󵄩∫ 𝑒(𝑡−𝑠)𝐴 𝑓 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏)) 𝑑𝑠󵄩󵄩󵄩 )
󵄩󵄩 0
󵄩󵄩𝐻
󵄩󵄩 𝑡
󵄩󵄩2 1/2
󵄩
󵄩
+ (E󵄩󵄩󵄩∫ 𝑒(𝑡−𝑠)𝐴 𝑔 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏)) 𝑑𝑊 (𝑠)󵄩󵄩󵄩 )
󵄩󵄩 0
󵄩󵄩𝐻
󵄩󵄩 𝑡
󵄩󵄩2 1/2
󵄩
󵄩
+ (E󵄩󵄩󵄩∫ 𝑒(𝑡−𝑠)𝐴 ℎ (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏) , 𝑢) 𝑁 (𝑑𝑠, 𝑑𝑢)󵄩󵄩󵄩 )
󵄩󵄩 0
󵄩󵄩𝐻
4
= ∑𝐼𝑖 (𝑡) .
𝑖=1
(19)
󵄩2 1/2
̃ (𝑑𝑠, 𝑑𝑢) 󵄩󵄩󵄩󵄩 )
× (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏) , 𝑢) 𝑁
󵄩󵄩𝐻
󵄩󵄩 𝑡
󵄩
+ 𝜌 (E 󵄩󵄩󵄩∫ ∫ 𝑒(𝑡−𝑠)𝐴 ℎ
󵄩󵄩 0 Z
1/2
󵄩󵄩2
× (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏) , 𝑢) 𝜋 (𝑑𝑢) 𝑑𝑠󵄩󵄩󵄩󵄩 ) .
󵄩𝐻
(22)
Using Hölder inequality and (H3), for the last term of (22),
we have
󵄩󵄩2 1/2
󵄩󵄩 𝑡
󵄩
󵄩󵄩
(𝑡−𝑠)𝐴
ℎ (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏) , 𝑢) 𝜋 (𝑑𝑢) 𝑑𝑠󵄩󵄩󵄩 )
𝜌(E󵄩󵄩∫ ∫ 𝑒
󵄩󵄩𝐻
󵄩󵄩 0 𝑍
1/2
𝑡
Recall the property of the operator 𝐴 (see [18]):
≤ 𝐶(E ∫ ∫ ‖ℎ (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏) , 𝑢)‖2𝐻𝜋 (𝑑𝑢) 𝑑𝑠)
󵄩󵄩
󵄩
󵄩󵄩(−𝐴)𝛿1 𝑒𝐴𝑡 󵄩󵄩󵄩 ≤ 𝐶𝑡−𝛿1 ,
󵄩
󵄩
󵄩󵄩
󵄩
󵄩󵄩(−𝐴)𝛿2 (1 − 𝑒𝐴𝑡 )󵄩󵄩󵄩 ≤ 𝐶𝑡𝛿2 , 𝛿1 ≥ 0, 𝛿2 ∈ [0, 1] ,
󵄩
󵄩
≤ 𝐶√𝐿 2 (∫ (E‖𝑋 (𝑠)‖2𝐻 + E‖𝑋 (𝑠 − 𝜏)‖2𝐻) 𝑑𝑠)
(−𝐴)𝛼+𝛽 𝑥 = (−𝐴)𝛼 (−𝐴)𝛽 𝑥,
0
𝑍
𝑡
1/2
0
(20)
1/2
𝑡
󵄩 󵄩
≤ 𝐶√𝐿 2 √𝜏E󵄩󵄩󵄩𝜉󵄩󵄩󵄩𝐻 + 𝑝𝐶√2𝐿 2 (∫ E‖𝑋 (𝑠)‖2𝐻𝑑𝑠)
𝑥 ∈ 𝐷 ((−𝐴)𝑟 ) ,
0
for 𝛼, 𝛽 ∈ R, where 𝑟 = max{𝛼, 𝛽, 𝛼 + 𝛽}.
By (H1) and (H2), together with the Minkowski integral
inequality, we derive that
𝑡
󵄩
󵄩2
𝐼2 (𝑡) ≤ ∫ (E󵄩󵄩󵄩󵄩𝑒(𝑡−𝑠)𝐴 𝑓 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏))󵄩󵄩󵄩󵄩𝐻)
0
𝑡
1/2
Moreover, by using the Itô isometry and (H3), we obtain that
𝑡
1/2
0
1/2
+(E‖𝑋 (𝑠 − 𝜏)‖2𝐻)
𝑡
≤ 𝐶 + 𝐶 ∫ (E‖𝑋 (𝑠)‖2𝐻)
0
1/2
(23)
1/2
󵄩󵄩 𝑡
󵄩󵄩2
󵄩
̃ (𝑑𝑠, 𝑑𝑢)󵄩󵄩󵄩 )
(E󵄩󵄩󵄩∫ ∫ 𝑒(𝑡−𝑠)𝐴 ℎ (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏) , 𝑢) 𝑁
󵄩󵄩
󵄩󵄩 0 Z
󵄩𝐻
𝑑𝑠
≤ 𝐶 ∫ {1 + (E‖𝑋 (𝑠)‖2𝐻)
.
≤ (∫ ∫ E‖ℎ (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏) , 𝑢)‖2𝐻𝜋 (𝑑𝑢) 𝑑𝑠)
0
(21)
} 𝑑𝑠
1/2
Z
𝑡
≤ √𝐿 2 (∫ (E‖𝑋 (𝑠)‖2𝐻 + E‖𝑋 (𝑠 − 𝜏)‖2𝐻) 𝑑𝑠)
1/2
0
𝑑𝑠.
1/2
𝑡
󵄩 󵄩
≤ √𝐿 2 √𝜏E󵄩󵄩󵄩𝜉󵄩󵄩󵄩𝐻 + √2𝐿 2 (∫ (E‖𝑋 (𝑠)‖2𝐻𝑑𝑠)
0
By (H1), (H2), and (H3) and using the Itô isometry, we have
.
(24)
Substituting (23) and (24) into (22), it follows that
𝐼3 (𝑡) + 𝐼4 (𝑡)
1/2
𝑡
󵄩2
󵄩
≤ (∫ E󵄩󵄩󵄩󵄩𝑒(𝑡−𝑠)𝐴 𝑔 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏))󵄩󵄩󵄩󵄩L0 𝑑𝑠)
2
0
󵄩󵄩 𝑡
󵄩
̃ (𝑑𝑠, 𝑑𝑢)
+ (E 󵄩󵄩󵄩∫ ∫ 𝑒(𝑡−𝑠)𝐴 ℎ (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏) , 𝑢) 𝑁
󵄩󵄩 0 𝑍
𝑡
(𝑡−𝑠)𝐴
+ 𝜌∫ ∫ 𝑒
0
Z
ℎ
󵄩󵄩2 1/2
× (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏) , 𝑢) 𝜋 (𝑑𝑢) 󵄩󵄩󵄩󵄩 )
󵄩𝐻
𝑡
󵄩 󵄩
𝐼3 (𝑡) + 𝐼4 (𝑡) ≤ 𝐶 + 𝐶 E󵄩󵄩󵄩𝜉󵄩󵄩󵄩𝐻 + 𝐶(∫ E‖𝑋 (𝑠)‖2𝐻𝑑𝑠)
1/2
0
.
(25)
Hence,
1/2
(E‖𝑋 (𝑡)‖2𝐻)
𝑡
󵄩 󵄩
≤ 𝐶 + 𝐶 E󵄩󵄩󵄩𝜉󵄩󵄩󵄩𝐻 + 𝐶(∫ E‖𝑋 (𝑠)‖2𝐻𝑑𝑠)
0
1/2
.
(26)
Abstract and Applied Analysis
5
Applying the Gronwall inequality, we have
sup (E‖𝑋 (𝑡)‖2𝐻)
1/2
≤ 𝐶.
0≤𝑡≤𝑇
Therefore,
󵄩2 1/2
󵄩
(E󵄩󵄩󵄩𝐽1 (𝑡)󵄩󵄩󵄩𝐻)
(27)
󵄩2
󵄩
= (E󵄩󵄩󵄩󵄩𝑒⌊𝑡⌋𝐴 {𝑒(𝑡−⌊𝑡⌋)𝐴 − 1} 𝜉 (0)󵄩󵄩󵄩󵄩𝐻)
1/2
Using the similar argument, the second assertion of (18)
follows.
󵄩2
󵄩
≤ 𝜆 1 (E󵄩󵄩󵄩𝜉 (0)󵄩󵄩󵄩𝐻)
1/2
Lemma 3. Let (H1)–(H4) hold; for sufficiently small Δ,
sup (E‖𝑋 (𝑡) − 𝑋 (⌊𝑡⌋)‖2𝐻)
1/2
0≤𝑡≤𝑇
≤ 𝐶Δ1/2 ,
(28)
3
󵄩2 1/2
󵄩
∑(E󵄩󵄩󵄩𝐽𝑖 (𝑡)󵄩󵄩󵄩𝐻)
𝑖=2
Proof. For any 𝑡 ∈ [0, 𝑇], we have from (8) that
≤∫
𝑡
⌊𝑡⌋
(𝑒(𝑡−⌊𝑡⌋)𝐴 − 1) 𝑒(⌊𝑡⌋−𝑠)𝐴 𝑓 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏)) 𝑑𝑠
Together with (31), we arrive at
3
+ ∫ 𝑒(𝑡−𝑠)𝐴 𝑓 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏)) 𝑑𝑠
󵄩2 1/2
󵄩
∑(E󵄩󵄩󵄩𝐽𝑖 (𝑡)󵄩󵄩󵄩𝐻)
⌊𝑡⌋
+∫
0
⌊𝑡⌋
+∫
0
(33)
󵄩2 1/2
󵄩
+ ∫ (E󵄩󵄩󵄩𝑓 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏))󵄩󵄩󵄩𝐻) 𝑑𝑠.
𝑡
⌊𝑡⌋
󵄩󵄩 (𝑡−⌊𝑡⌋)𝐴
󵄩
󵄩󵄩
󵄩󵄩𝑒
− 1󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝑒(⌊𝑡⌋−𝑠)𝐴 󵄩󵄩󵄩󵄩
󵄩
󵄩2 1/2
󵄩
× (E󵄩󵄩󵄩𝑓 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏))󵄩󵄩󵄩𝐻) 𝑑𝑠
= 𝑒⌊𝑡⌋𝐴 (𝑒(𝑡−⌊𝑡⌋)𝐴 − 1) 𝜉 (0)
0
⌊𝑡⌋
0
𝑋 (𝑡) − 𝑋 (⌊𝑡⌋)
⌊𝑡⌋
Δ.
By (H1), (H2), and the Minkowski integral inequality, we
obtain that
where 𝐶 > 0 is constant dependent on 𝑇, 𝜉, 𝐿 1 , 𝐿 2 , 𝐿 3 , and 𝐿 4 ,
while being independent of Δ.
+∫
(32)
𝑖=2
(𝑒(𝑡−⌊𝑡⌋)𝐴 − 1) 𝑒(⌊𝑡⌋−𝑠)𝐴 𝑔 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏)) 𝑑𝑊 (𝑠)
≤ (𝜆 1 Δ ∫
⌊𝑡⌋
0
∫ (𝑒(𝑡−⌊𝑡⌋)𝐴 − 1) 𝑒(⌊𝑡⌋−𝑠)𝐴
󵄩2 1/2
󵄩
𝑑𝑠 + Δ) 𝐶 sup (E󵄩󵄩󵄩𝑓 (𝑋 (𝑡) , 𝑋 (𝑡 − 𝜏))󵄩󵄩󵄩𝐻)
0≤𝑡≤𝑇
≤ 𝐶 (1 + sup (E‖𝑋 (𝑡)‖2𝐻)
Z
1/2
0≤𝑡≤𝑇
) Δ.
(34)
× ℎ (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏) , 𝑢) 𝑁 (𝑑𝑠, 𝑑𝑢)
Following the argument of (22), we derive that
𝑡
+ ∫ 𝑒(𝑡−𝑠)𝐴 𝑔 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏)) 𝑑𝑊 (𝑠)
7
󵄩2
󵄩
∑(E󵄩󵄩󵄩𝐽𝑖 (𝑡)󵄩󵄩󵄩𝐻)
⌊𝑡⌋
𝑡
𝑖=4
+ ∫ ∫ 𝑒(𝑡−𝑠)𝐴 ℎ (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏) , 𝑢) 𝑁 (𝑑𝑠, 𝑑𝑢)
⌊𝑡⌋ Z
≤ (∫
⌊𝑡⌋
0
7
= ∑𝐽𝑖 (𝑡) .
𝑖=1
(29)
1/2
Since (E‖ ⋅ ‖2𝐻)
(E‖𝑋 (𝑡) −
⌊𝑡⌋
󵄩
󵄩2
󵄩2 󵄩
∫ 󵄩󵄩󵄩󵄩𝑒(𝑡−⌊𝑡⌋)𝐴 − 1󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝑒(⌊𝑡⌋−𝑠)𝐴 󵄩󵄩󵄩󵄩
Z
× E‖ℎ (𝑋 (𝑠) , 𝑋 (𝑠−𝜏) , 𝑢)‖2𝐻𝜋 (𝑑𝑢) 𝑑𝑠)
𝑖=1
2
≤ (1 − 𝑒−𝜆 1 (𝑡−⌊𝑡⌋) ) ‖𝑥‖2𝐻
1/2
2
(30)
Recalling the fundamental inequality 1 − 𝑒−𝑦 ≤ 𝑦, 𝑦 > 0, we
get from (H1) that
󵄩󵄩 (𝑡−⌊𝑡⌋)𝐴
󵄩2
󵄩󵄩(𝑒
− 1) 𝑥󵄩󵄩󵄩󵄩𝐻
󵄩
󵄩󵄩 ∞
󵄩󵄩2
󵄩󵄩
󵄩󵄩
= 󵄩󵄩󵄩∑ (𝑒−𝜆 𝑖 (𝑡−⌊𝑡⌋) − 1) ⟨𝑥, 𝑒𝑖 ⟩ 𝑒𝑖 󵄩󵄩󵄩
󵄩󵄩𝑖=1
󵄩󵄩
(31)
󵄩
󵄩𝐻
≤ 𝜆21 Δ2 ‖𝑥‖2𝐻.
󵄩2
󵄩
× E󵄩󵄩󵄩𝑔 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏))󵄩󵄩󵄩L0 𝑑𝑠)
0
7
󵄩2 1/2
󵄩
≤ ∑(E󵄩󵄩󵄩𝐽𝑖 (𝑡)󵄩󵄩󵄩𝐻) .
󵄩󵄩 (𝑡−⌊𝑡⌋)𝐴
󵄩2
󵄩2 󵄩
󵄩󵄩𝑒
− 1󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩𝑒(⌊𝑡⌋−𝑠)𝐴 󵄩󵄩󵄩󵄩
󵄩
+ 𝐶 (∫
is a norm, it follows that
1/2
𝑋 (⌊𝑡⌋)‖2𝐻)
1/2
1/2
1/2
𝑡
󵄩
󵄩2 󵄩
󵄩2
+ (∫ 󵄩󵄩󵄩󵄩𝑒(𝑡−𝑠)𝐴 󵄩󵄩󵄩󵄩 E󵄩󵄩󵄩𝑔 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏))󵄩󵄩󵄩L0 𝑑𝑠)
2
⌊𝑡⌋
𝑡
󵄩
󵄩2
+ 𝐶 (∫ ∫ 󵄩󵄩󵄩󵄩𝑒(𝑡−𝑠)𝐴 󵄩󵄩󵄩󵄩
⌊𝑡⌋ Z
× E‖ℎ (𝑋 (𝑠) , 𝑋 (𝑠−𝜏) , 𝑢)‖2𝐻𝜋 (𝑑𝑢) 𝑑𝑠)
≤ 𝐶 (1 + sup (E‖𝑋 (𝑡)‖2𝐻)
0≤𝑡≤𝑇
1/2
1/2
) Δ1/2 .
(35)
6
Abstract and Applied Analysis
𝑡
Substituting (32), (34), and (35) into (30), we arrive at
+ ∫ 𝑒(𝑡−𝑠)𝐴 (𝑔 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏))
0
1/2
(E‖𝑋 (𝑡) − 𝑋 (⌊𝑡⌋)‖2𝐻)
≤ 𝐶 (1 +
−𝑔𝑛 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏))) 𝑑𝑊 (𝑠)
1/2
sup (E‖𝑋 (𝑡)‖2𝐻) ) Δ1/2 .
0≤𝑡≤𝑇
(36)
𝑡
+ ∫ 𝑒(𝑡−𝑠)𝐴 (1 − 𝑒(𝑠−⌊𝑠⌋)𝐴 ) 𝑓𝑛 (𝑌𝑛 (⌊𝑠⌋) , 𝑌𝑛 (⌊𝑠⌋ − 𝜏)) 𝑑𝑠
0
𝑡
Therefore, by Lemma 2, the required assertion (28) follows.
+ ∫ 𝑒(𝑡−𝑠)𝐴 (1 − 𝑒(𝑠−⌊𝑠⌋)𝐴 )
0
× 𝑔𝑛 (𝑌𝑛 (⌊𝑠⌋) , 𝑌𝑛 (⌊𝑠⌋ − 𝜏)) 𝑑𝑊 (𝑠)
Now, we state our main result in this paper as follows.
Theorem 4. Let (H1)–(H4) hold, and
√𝐿 1 (2𝛼−1 + (𝜌 + 3) (2𝛼)−1/2 ) < 1.
𝑡
+ ∫ ∫ 𝑒(𝑡−𝑠)𝐴 {ℎ (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏) , 𝑢)
0
−ℎ𝑛 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏) , 𝑢)} 𝑁 (𝑑𝑠, 𝑑𝑢)
(37)
𝑡
+ ∫ ∫ 𝑒(𝑡−𝑠)𝐴 {ℎ𝑛 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏) , 𝑢)
0
Then,
󵄩2 1/2
󵄩
sup (E󵄩󵄩󵄩𝑋 (𝑡) − 𝑌𝑛 (𝑡)󵄩󵄩󵄩𝐻) ≤ 𝐶 {𝜆−1/2
+ Δ1/2 } ,
𝑛
0≤t≤𝑇
Z
−ℎ𝑛 (𝑋 (⌊𝑠⌋) , 𝑋 (⌊𝑠⌋−𝜏) , 𝑢)} 𝑁 (𝑑𝑠, 𝑑𝑢)
(38)
𝑡
+ ∫ ∫ 𝑒(𝑡−𝑠)𝐴 {ℎ𝑛 (𝑋 (⌊𝑠⌋) , 𝑋 (⌊𝑠⌋ − 𝜏) , 𝑢) − ℎ𝑛
0
where 𝐶 > 0 is a constant dependent on 𝑇, 𝜉, 𝐿 1 , 𝐿 2 , 𝐿 3 , and
𝐿 4 , while being independent of 𝑛 and Δ.
Proof. By (8) and (17), we obtain
Z
Z
× (𝑌𝑛 (⌊𝑠⌋) , 𝑌𝑛 (⌊𝑠⌋ − 𝜏) , 𝑢)} 𝑁 (𝑑𝑠, 𝑑𝑢)
𝑡
+ ∫ ∫ 𝑒(𝑡−𝑠)𝐴 (1 − 𝑒(𝑠−⌊𝑠⌋)𝐴 ) ℎ𝑛
0
𝑍
× (𝑌𝑛 (⌊𝑠⌋) , 𝑌𝑛 (⌊𝑠⌋ − 𝜏) , 𝑢) 𝑁 (𝑑𝑠, 𝑑𝑢)
𝑛
𝑋 (𝑡) − 𝑌 (𝑡)
13
= ∑𝐾𝑖 (𝑡) .
𝑡𝐴
= 𝑒 (1 − 𝜋𝑛 ) 𝜉 (0)
𝑖=1
(39)
𝑡
+ ∫ 𝑒(𝑡−𝑠)𝐴 (𝑓 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏))
0
−𝑓𝑛 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏))) 𝑑𝑠
𝑡
+ ∫ 𝑒(𝑡−𝑠)𝐴 (𝑓𝑛 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏))
0
Noting that (E‖ ⋅ ‖2𝐻)1/2 is a norm, we have
13
󵄩
󵄩2 1/2
󵄩2 1/2
󵄩
(E󵄩󵄩󵄩𝑋 (𝑡) − 𝑌𝑛 (𝑡)󵄩󵄩󵄩𝐻) ≤ ∑(E󵄩󵄩󵄩𝐾𝑖 (𝑡)󵄩󵄩󵄩𝐻) .
(40)
𝑖=1
−𝑓𝑛 (𝑋 (⌊𝑠⌋) , 𝑋 (⌊𝑠⌋ − 𝜏))) 𝑑𝑠
𝑡
+ ∫ 𝑒(𝑡−𝑠)𝐴 (𝑔𝑛 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏))
0
−𝑔𝑛 (𝑋 (⌊𝑠⌋) , 𝑋 (⌊𝑠⌋ − 𝜏))) 𝑑𝑊 (𝑠)
By (H1) and the nondecreasing spectrum {𝜆 𝑚 }𝑚 ≥ 1 , it easily
follows that
󵄩
󵄩
E󵄩󵄩󵄩󵄩𝑒𝑡𝐴 (1 − 𝜋𝑛 ) 𝜉 (0)󵄩󵄩󵄩󵄩𝐻
𝑡
+ ∫ 𝑒(𝑡−𝑠)𝐴 (𝑓𝑛 (𝑋 (⌊𝑠⌋) , 𝑋 (⌊𝑠⌋ − 𝜏))
0
−𝑓𝑛 (𝑌𝑛 (⌊𝑠⌋) , 𝑌𝑛 (⌊𝑠⌋ − 𝜏))) 𝑑𝑠
𝑡
+ ∫ 𝑒(𝑡−𝑠)𝐴 (𝑔𝑛 (𝑋 (⌊𝑠⌋) , 𝑋 (⌊𝑠⌋ − 𝜏))
0
𝑛
𝑛
−𝑔𝑛 (𝑌 (⌊𝑠⌋) , 𝑌 (⌊𝑠⌋ − 𝜏))) 𝑑𝑊 (𝑠)
∞
2
= E( ∑ 𝑒−2𝜆 𝑚 𝑡 ⟨𝜉 (0) , 𝑒𝑚 ⟩𝐻)
1/2
𝑚 = 𝑛+1
∞
1/2
𝑒−2𝜆 𝑚 𝑡 2
2
= E( ∑
𝜆 𝑚 ⟨𝜉 (0) , 𝑒𝑚 ⟩𝐻)
2
𝑚 = 𝑛+1 𝜆 𝑚
≤
1 󵄩󵄩
󵄩
E󵄩𝐴𝜉 (0)󵄩󵄩󵄩𝐻.
𝜆𝑛 󵄩
(41)
Abstract and Applied Analysis
7
By (H2), the Minkowski integral inequality, and Lemma 2, we
have
󵄩2 1/2
󵄩
(E󵄩󵄩󵄩𝐾2 (𝑡)󵄩󵄩󵄩𝐻)
𝑡
󵄩2 1/2
󵄩
≤ ∫ (E󵄩󵄩󵄩󵄩𝑒(𝑡−𝑠)𝐴 (1 − 𝜋𝑛 ) 𝑓 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏))󵄩󵄩󵄩󵄩𝐻) 𝑑𝑠
0
𝑡
∞
0
𝑚 = 𝑛+1
−𝜆 𝑛 (𝑡−𝑠)
≤∫ 𝑒
0
∞
(E ∑ ⟨𝑓 (𝑋 (𝑠) , 𝑋 (𝑠 −
𝑚=𝑛+1
󵄩2 1/2
󵄩
(E󵄩󵄩󵄩𝐾7 (𝑡)󵄩󵄩󵄩𝐻)
𝑡
󵄩2
󵄩
≤ (∫ E󵄩󵄩󵄩󵄩𝑒(𝑡−𝑠)𝐴 (1 − 𝜋𝑛 ) 𝑔 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏))󵄩󵄩󵄩󵄩L0 𝑑𝑠)
2
0
1/2
2
= ∫ (E ∑ 𝑒−2𝜆 𝑚 (𝑡−𝑠) ⟨𝑓 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏)) , 𝑒𝑚 ⟩𝐻)
𝑡
By the Itô isometry and a similar argument to that of (42), we
deduce that
𝑑𝑠
𝑡
󵄩2
󵄩
≤ 𝐶(∫ 𝑒−2𝜆 𝑛 (𝑡−𝑠) E󵄩󵄩󵄩𝑔 (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏))󵄩󵄩󵄩L0 𝑑𝑠)
1/2
2
𝜏)) , 𝑒𝑚 ⟩𝐻) 𝑑𝑠
1/2
1/2
2
0
≤ 𝐶𝜆−1/2
.
𝑛
(44)
𝑡
≤ 𝐶 ∫ 𝑒−𝜆 𝑛 (𝑡−𝑠)
0
× {1 + (E‖𝑋 (𝑠)‖2𝐻)
1/2
1/2
+ (E‖𝑋 (𝑠 − 𝜏)‖2𝐻)
Moreover, by (31), (H2), and Lemma 2 and combining the
Minkowski integral inequality and the Itô isometry, we have
} 𝑑𝑠
≤ 𝐶𝜆−1
𝑛 .
(42)
Applying (H1), (H2), and Lemma 3 and combining the
Minkowski integral inequality and the Itô isometry yield
6
󵄩2 1/2
󵄩
∑(E󵄩󵄩󵄩𝐾𝑖 (𝑡)󵄩󵄩󵄩𝐻 )
9
󵄩2
󵄩
∑(E󵄩󵄩󵄩𝐾𝑖 (𝑡)󵄩󵄩󵄩𝐻)
1/2
𝑖=8
𝑡
󵄩
󵄩2
≤ ∫ (E󵄩󵄩󵄩󵄩𝑒(𝑡−𝑠)𝐴 (1 − 𝑒(𝑠−⌊𝑠⌋)𝐴 )󵄩󵄩󵄩󵄩
0
󵄩
󵄩2 1/2
× 󵄩󵄩󵄩𝑓𝑛 (𝑌𝑛 (⌊𝑠⌋), 𝑌𝑛 (⌊𝑠⌋ − 𝜏))󵄩󵄩󵄩𝐻) 𝑑𝑠
𝑖=3
𝑡
󵄩
󵄩
≤ √𝐿 1 ∫ 󵄩󵄩󵄩󵄩𝑒(𝑡−𝑠)𝐴 󵄩󵄩󵄩󵄩 (E (‖𝑋 (𝑠) − 𝑋 (⌊𝑠⌋)‖2𝐻
0
𝑡
󵄩
󵄩2
+ (∫ E󵄩󵄩󵄩󵄩𝑒(𝑡−𝑠)𝐴 (1 − 𝑒(𝑠−⌊𝑠⌋)𝐴 )󵄩󵄩󵄩󵄩
0
1/2
+‖𝑋(𝑠 − 𝜏) − 𝑋 (⌊𝑠⌋ − 𝜏)‖2𝐻 )) 𝑑𝑠
󵄩2
󵄩
× 󵄩󵄩󵄩𝑔𝑛 (𝑌𝑛 (⌊𝑠⌋) , 𝑌𝑛 (⌊𝑠⌋ − 𝜏))󵄩󵄩󵄩𝐻𝑑𝑠)
𝑡
󵄩
󵄩
󵄩
󵄩2
+ √𝐿 1 ∫ 󵄩󵄩󵄩󵄩𝑒(𝑡−𝑠)𝐴 󵄩󵄩󵄩󵄩 (E (󵄩󵄩󵄩𝑋 (⌊𝑠⌋) − 𝑌𝑛 (⌊𝑠⌋)󵄩󵄩󵄩𝐻
0
󵄩2 1/2
󵄩
+󵄩󵄩󵄩𝑋 (⌊𝑠⌋ − 𝜏) − 𝑌𝑛 (⌊𝑠⌋ − 𝜏)󵄩󵄩󵄩𝐻 )) 𝑑𝑠
𝑡
󵄩
󵄩2
+ √𝐿 1 (∫ 󵄩󵄩󵄩󵄩𝑒(𝑡−𝑠)𝐴 󵄩󵄩󵄩󵄩 (E (‖𝑋 (𝑠) − 𝑋 (⌊𝑠⌋)‖2𝐻
0
(45)
1/2
𝑡
󵄩2 1/2
󵄩
≤ 𝐶Δ ∫ (E󵄩󵄩󵄩𝑓𝑛 (𝑌𝑛 (⌊𝑠⌋) , 𝑌𝑛 (⌊𝑠⌋ − 𝜏))󵄩󵄩󵄩𝐻) 𝑑𝑠
0
1/2
𝑡
󵄩2
󵄩
+󵄩󵄩󵄩𝑋 (𝑠−𝜏) − 𝑌𝑛 (⌊𝑠⌋−𝜏)󵄩󵄩󵄩𝐻 )) 𝑑𝑠)
1/2
󵄩2
󵄩
+ 𝐶Δ(∫ E󵄩󵄩󵄩𝑔𝑛 (𝑌𝑛 (⌊𝑠⌋) , 𝑌𝑛 (⌊𝑠⌋ − 𝜏))󵄩󵄩󵄩L0 𝑑𝑠)
2
0
≤ 𝐶Δ.
𝑡
󵄩
󵄩2 󵄩
󵄩2
+ √𝐿 1 (∫ 󵄩󵄩󵄩󵄩𝑒(𝑡−𝑠)𝐴 󵄩󵄩󵄩󵄩 (E󵄩󵄩󵄩𝑋 (⌊𝑠⌋) − 𝑌𝑛 (⌊𝑠⌋)󵄩󵄩󵄩𝐻
0
By (31) and the Itô isometry, we obtain that
󵄩
󵄩2
+󵄩󵄩󵄩𝑋(⌊𝑠⌋ − 𝜏) − 𝑌𝑛 (⌊𝑠⌋ − 𝜏)󵄩󵄩󵄩𝐻 ) 𝑑𝑠)
≤ 𝐶Δ
1/2
1/2
󵄩2 1/2
󵄩
+ √𝐿 1 sup (E󵄩󵄩󵄩𝑋 (𝑠) − 𝑌𝑛 (𝑠)󵄩󵄩󵄩𝐻 )
0≤𝑠≤𝑡
𝑡
× ∫ 𝑒−𝛼(𝑡−𝑠) 𝑑𝑠
󵄩󵄩2 1/2
󵄩
̃
× (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏) , 𝑢) 𝑁 (𝑑𝑠, 𝑑𝑢) 󵄩󵄩󵄩 )
󵄩󵄩𝐻
0
1/2
𝑡
󵄩2 1/2
󵄩
+ √𝐿 1 sup (E󵄩󵄩󵄩𝑋 (𝑠) − 𝑌𝑛 (𝑠)󵄩󵄩󵄩𝐻 ) (∫ 𝑒−2𝛼(𝑡−𝑠) 𝑑𝑠)
0
0≤𝑠≤𝑡
𝑡−𝜏
󵄩2 1/2
󵄩
+ √𝐿 1 sup (E󵄩󵄩󵄩𝑋 (𝑠) − 𝑌𝑛 (𝑠)󵄩󵄩󵄩𝐻 ) ∫
−𝜏
0≤𝑠≤𝑡
𝑡−𝜏
−𝜏
0≤𝑠≤𝑡
󵄩2
󵄩
≤ 𝐶Δ1/2 + √𝐿 1 sup (E󵄩󵄩󵄩𝑋 (𝑠) − 𝑌𝑛 (𝑠)󵄩󵄩󵄩𝐻 )
󵄩󵄩 𝑡
󵄩
+ 𝜌 (E 󵄩󵄩󵄩∫ ∫ 𝑒(𝑡−𝑠)𝐴 (1 − 𝜋𝑛 ) ℎ
󵄩󵄩 0 Z
𝑒−𝛼(𝑡−𝑠−𝜏) 𝑑𝑠
󵄩2 1/2
󵄩
+ √𝐿 1 sup (E󵄩󵄩󵄩𝑋 (𝑠) − 𝑌𝑛 (𝑠)󵄩󵄩󵄩𝐻 ) (∫
−2𝛼(𝑡−𝑠−𝜏)
𝑒
󵄩2 1/2
󵄩
(E󵄩󵄩󵄩𝐾10 (𝑡)󵄩󵄩󵄩𝐻)
󵄩󵄩 𝑡
󵄩
≤ (E 󵄩󵄩󵄩∫ ∫ 𝑒(𝑡−𝑠)𝐴 (1 − 𝜋𝑛 ) ℎ
󵄩󵄩 0 Z
1/2
󵄩󵄩2
󵄩
× (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏) , 𝑢) 𝜋 (𝑑𝑢) 𝑑𝑠󵄩󵄩󵄩 )
󵄩󵄩𝐻
𝑡
󵄩
󵄩2
≤ (∫ ∫ 󵄩󵄩󵄩󵄩𝑒(𝑡−𝑠)𝐴 (1 − 𝜋𝑛 )󵄩󵄩󵄩󵄩 E
0 Z
1/2
𝑑𝑠)
1/2
0≤𝑠≤𝑡
× (2𝛼−1 + 2(2𝛼)−1/2 ) .
(43)
× ‖ℎ (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏) , 𝑢)‖2𝐻𝜋(𝑑𝑢)𝑑𝑠)
1/2
8
Abstract and Applied Analysis
𝑡
󵄩
󵄩2
+ 𝜌 (∫ ∫ 󵄩󵄩󵄩󵄩𝑒(𝑡−𝑠)𝐴 (1 − 𝜋𝑛 )󵄩󵄩󵄩󵄩
0 Z
× E‖ℎ (𝑋 (𝑠) , 𝑋 (𝑠 − 𝜏) , 𝑢)‖2𝐻𝜋(𝑑𝑢)𝑑𝑠)
𝑡
≤ 𝐶(∫ 𝑒−2𝜆 𝑛 (𝑡−𝑠) (E‖𝑋 (𝑠)‖2𝐻 + E‖𝑋 (𝑠 − 𝜏)‖2𝐻) 𝑑𝑠)
1/2
1/2
0
≤
𝐶𝜆−1/2
.
𝑛
(46)
Carrying out the similar arguments to those of (43) and (45),
we derive that
󵄩2 1/2
󵄩2 1/2
󵄩
󵄩
(E󵄩󵄩󵄩𝐾11 (𝑡)󵄩󵄩󵄩𝐻) + (E󵄩󵄩󵄩𝐾12 (𝑡)󵄩󵄩󵄩𝐻)
≤ 𝐶Δ1/2 + (2𝛼)−1/2 (𝜌 + 1)
󵄩2 1/2
󵄩
× √𝐿 1 sup (E󵄩󵄩󵄩𝑋 (𝑠) − 𝑌𝑛 (𝑠)󵄩󵄩󵄩𝐻) ,
(47)
0≤𝑠≤𝑡
󵄩2 1/2
󵄩
(E󵄩󵄩󵄩𝐾13 (𝑡)󵄩󵄩󵄩𝐻) ≤ 𝐶Δ.
As a result, putting (41)–(47) into (40) gives that
󵄩2 1/2
󵄩
sup (E󵄩󵄩󵄩𝑋 (𝑠) − 𝑌𝑛 (𝑠)󵄩󵄩󵄩𝐻)
0≤𝑠≤𝑡
≤ 𝐶𝜆−1/2
+ 𝐶Δ1/2 + √𝐿 1 (2𝛼−1 + (𝜌 + 3) (2𝛼)−1/2 ) (48)
𝑛
󵄩2 1/2
󵄩
× sup (E󵄩󵄩󵄩𝑋 (𝑠) − 𝑌𝑛 (𝑠)󵄩󵄩󵄩𝐻) ,
0≤𝑠≤𝑡
and therefore the desired assertion follows.
Remark 5. For finite-dimensional Euler-Maruyama method,
the condition (37) can be deleted by the Gronwall inequality
[16, 17].
Acknowledgment
This work is partially supported by National Natural Science
Foundation of China Under Grants 60904005 and 11271146.
References
[1] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite
Dimensions, vol. 44 of Encyclopedia of Mathematics and Its
Applications, Cambridge University Press, Cambridge, UK,
1992.
[2] K. Liu, Stability of Infinite Dimensional Stochastic Differential
Equations with Applications, Chapman & Hall/CRC, Boca
Raton, Fla, USA, 2004.
[3] Q. Luo, F. Deng, J. Bao, B. Zhao, and Y. Fu, “Stabilization of
stochastic Hopfield neural network with distributed parameters,” Science in China F, vol. 47, no. 6, pp. 752–762, 2004.
[4] Q. Luo, F. Deng, X. Mao, J. Bao, and Y. Zhang, “Theory and
application of stability for stochastic reaction diffusion systems,”
Science in China F, vol. 51, no. 2, pp. 158–170, 2008.
[5] X. Mao, Stochastic Differential Equations and Applications,
Horwood Publishing, Chichester, UK, 2007.
[6] Y. Shen and J. Wang, “An improved algebraic criterion for global
exponential stability of recurrent neural networks with timevarying delays,” IEEE Transactions on Neural Networks, vol. 19,
no. 3, pp. 528–531, 2008.
[7] Y. Shen and J. Wang, “Almost sure exponential stability of
recurrent neural networks with markovian switching,” IEEE
Transactions on Neural Networks, vol. 20, no. 5, pp. 840–855,
2009.
[8] I. Gyöngy and N. Krylov, “Accelerated finite difference schemes
for linear stochastic partial differential equations in the whole
space,” SIAM Journal on Mathematical Analysis, vol. 42, no. 5,
pp. 2275–2296, 2010.
[9] A. Jentzen, P. E. Kloeden, and G. Winkel, “Efficient simulation
of nonlinear parabolic SPDEs with additive noise,” The Annals
of Applied Probability, vol. 21, no. 3, pp. 908–950, 2011.
[10] P. E. Kloeden, G. J. Lord, A. Neuenkirch, and T. Shardlow, “The
exponential integrator scheme for stochastic partial differential
equations: pathwise error bounds,” Journal of Computational
and Applied Mathematics, vol. 235, no. 5, pp. 1245–1260, 2011.
[11] J. Bao, A. Truman, and C. Yuan, “Stability in distribution of mild
solutions to stochastic partial differential delay equations with
jumps,” Proceedings of The Royal Society of London A, vol. 465,
no. 2107, pp. 2111–2134, 2009.
[12] B. Boufoussi and S. Hajji, “Successive approximation of neutral
functional stochastic differential equations with jumps,” Statistics and Probability Letters, vol. 80, no. 5-6, pp. 324–332, 2010.
[13] E. Hausenblas, “Finite element approximation of stochastic partial differential equations driven by Poisson random measures
of jump type,” SIAM Journal on Numerical Analysis, vol. 46, no.
1, pp. 437–471, 2007/08.
[14] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations
with Lévy Noise: An Evolution Equation Approach, vol. 113 of
Encyclopedia of Mathematics and Its Applications, Cambridge
University Press, Cambridge, UK, 2007.
[15] M. Röckner and T. Zhang, “Stochastic evolution equations of
jump type: existence, uniqueness and large deviation principles,” Potential Analysis, vol. 26, no. 3, pp. 255–279, 2007.
[16] J. Bao, B. Böttcher, X. Mao, and C. Yuan, “Convergence
rate of numerical solutions to SFDEs with jumps,” Journal of
Computational and Applied Mathematics, vol. 236, no. 2, pp.
119–131, 2011.
[17] N. Jacob, Y. Wang, and C. Yuan, “Numerical solutions of
stochastic differential delay equations with jumps,” Stochastic
Analysis and Applications, vol. 27, no. 4, pp. 825–853, 2009.
[18] A. Pazy, Semigroups of Linear Operators and Applications to
Partial Differential Equations, vol. 44 of Applied Mathematical
Sciences, Springer, New York, NY, USA, 1983.
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