Numerical Analysis for Stochastic Partial Differential Delay Equations with Jumps

Abstract

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,{ℱ_𝑡}_𝑡≥0,ℙ) be a complete probability space with some filtration {ℱ_𝑡}_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 ‖𝜑‖_𝐷=sup_{−𝜏≤𝜃≤0}‖𝜑(𝜃)‖_𝐻. 𝐷^𝑏_ℱ0([−𝜏,0],𝐻) denotes the family of almost surely bounded, ℱ₀-measurable, 𝐷([−𝜏,0],𝐻)-valued random variables. For all 𝑡≥0, 𝑋_𝑡={𝑋(𝑡+𝜃)∶−𝜏≤𝜃≤0} is regarded as 𝐷([−𝜏,0],𝐻)-valued stochastic process.

Let 𝑇 be a positive constant. For given 𝜏≥0, consider the following stochastic partial differential delay equations with jumps: (𝑡)=(𝑡)(𝑋(𝑡)((𝑋(𝑡)((𝑡)+𝑑𝑋𝐴𝑋+𝑓,𝑋𝑡−𝜏))𝑑𝑡+𝑔,𝑋𝑡−𝜏))𝑑𝑊_ℤℎ(𝑋(𝑡),𝑋(𝑡−𝜏),𝑢)𝑁(𝑑𝑡,𝑑𝑢) (1) on 𝑡∈[0,𝑇] with initial datum 𝑋(𝑡)=𝜉(𝑡)∈𝐷^𝑏_ℱ0([−𝜏,0],𝐻), −𝜏≤𝑡≤0. Here (𝐴,𝐷(𝐴)) is a self-adjoint operator on 𝐻. {𝑊(𝑡),𝑡≥0} is 𝐾-valued {ℱ_𝑡}_𝑡≥0-Wiener process defined on the probability space {Ω,ℱ,{ℱ_𝑡}_𝑡≥0,ℙ} with covariance operator 𝑄. We assume that −𝐴 and the covariance operator 𝑄 of the Wiener process have the same eigenbasis {𝑒_𝑚}_𝑚≥1 of 𝐻; that is, −𝐴𝑒_𝑚=𝜆_𝑚𝑒_𝑚,𝑄𝑒_𝑚=𝛼_𝑚𝑒_𝑚,𝑚=1,2,3,…, (2) where {𝜆_𝑚,𝑚∈ℕ} are the discrete spectrum of −𝐴 and 0≤𝜆₁≤𝜆₂⋯≤lim_𝑚→∞𝜆_𝑚=∞, {𝛼_𝑚,𝑚∈ℕ} are the eigenvalues of 𝑄. Then, 𝑊(𝑡) is defined by 𝑊(𝑡)=^∞_𝑛=1√𝛼_𝑚𝛽_𝑚(𝑡)𝑒_𝑚,𝑡≥0, (3) where 𝛽_𝑚(𝑡)(𝑚=1,2,3,…) is a sequence of real-valued standard Brownian motions mutually independent of the probability space (Ω,ℱ,{ℱ_𝑡}_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=𝑄^1/2(𝐾) be the subspace of 𝐾 with the inner product ⟨𝑢,𝑣⟩_𝐾0=⟨𝑄^−1/2𝑢,𝑄^−1/2𝑣⟩_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‖²_ℒ02=tr((Ψ𝑄^1/2)(Ψ𝑄^1/2)^∗).

Let Φ : (0,∞)→ℒ⁰₂ be a predictable, ℱ_𝑡-adapted process such that ^𝑡₀𝔼‖Φ(𝑠)‖_ℒ⁰₂𝑑𝑠≤∞,∀𝑡>0. (4) Then, the 𝐻-valued stochastic integral ∫^𝑡₀Φ(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 𝑁(𝑑𝑡,𝑑𝑢)=𝑁(𝑑𝑡,𝑑𝑢)−𝜌𝑑𝑡𝜋(𝑑𝑢), (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 ^𝑇_0ℤ𝔼‖ℎ(𝑡,𝑢)‖²_𝐻𝑑𝑡𝜋(𝑑𝑢)<∞. (6) Then, the 𝐻-valued stochastic integral ^𝑇_0ℤℎ(𝑡,𝑢)𝑁(𝑑𝑡,𝑑𝑢) (7) 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 ℱ_𝑡, 𝑡≥0, and has càdlàg path on 𝑡≥0 almost surely, (ii)for arbitrary 𝑡∈[0,𝑇], ℙ{𝑤∶∫^𝑡₀‖𝑋(𝑠)‖²_𝐻𝑑𝑠<∞}=1, and almost surely 𝑋(𝑡)=𝑒^𝑡𝐴𝜉(0)+^𝑡₀𝑒^{(𝑡−𝑠)𝐴}𝑓(𝑋(𝑠)(+,𝑋𝑠−𝜏))𝑑𝑠_𝑡0𝑒^{(𝑡−𝑠)𝐴}𝑔(𝑋(𝑠)((𝑠)+,𝑋𝑠−𝜏))𝑑𝑊_𝑡0ℤ𝑒^{(𝑡−𝑠)𝐴}ℎ(𝑋(𝑠),𝑋(𝑠−𝜏),𝑢)𝑁(𝑑𝑠,𝑑𝑢) (8) for any 𝑋(𝑡)=𝜉(𝑡)∈𝐷^𝑏_ℱ0([−𝜏,0],𝐻), −𝜏≤𝑡≤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≤𝜆₁≤𝜆₂≤⋯≤lim_𝑚→∞𝜆_𝑚=∞ with corresponding eigenbasis {𝑒_𝑚}_𝑚≥1 of 𝐻. In this case 𝐴 generates a compact 𝐶₀-semigroup 𝑒^𝑡𝐴,𝑡≥0, such that ‖𝑒^𝑡𝐴‖≤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 𝐿₁>0 and arbitrary 𝑥,𝑦,𝑥₁,𝑦₁,𝑥₂,𝑦₂∈𝐻 and 𝑢∈ℤ: 𝑓𝑥₁,𝑦₁−𝑓𝑥₂,𝑦₂²_𝐻∨𝑔𝑥₁,𝑦₁−𝑔𝑥₂,𝑦₂²_ℒ20≤𝐿₁𝑥₁−𝑥₂²_𝐻+𝑦₁−𝑦₂²_𝐻,ℎ𝑥₁,𝑦₁,𝑢−ℎ𝑥₂,𝑦₂,𝑢²_𝐻≤𝐿₁𝑥₁−𝑥₂²_𝐻+𝑦₁−𝑦₂²_𝐻. (9) This further implies the linear growth condition; that is, 𝑓(𝑥,𝑦)²_𝐻+𝑔(𝑥,𝑦)²_ℒ02≤𝐿₀1+‖𝑥‖²_𝐻+𝑦²_𝐻, (10) where 𝐿₀∶=2𝐿₂∨𝑓(0,0)²_𝐻∨𝑔(0,0)²_ℒ02. (11)(H3)There exists 𝐿₂>0 satisfying ℎ(𝑥,𝑦,𝑢)²_𝐻≤𝐿₂‖𝑥‖²_𝐻+𝑦²_𝐻, (12) for each 𝑥,𝑦∈𝐻 and 𝑢∈ℤ. (H4)For 𝜉∈𝐷^𝑏_ℱ0([−𝜏,0],𝐻), there exists a constant 𝐿₃>0 such that 𝔼𝜉(𝑠)−𝜉(𝑡)²≤𝐿₃|𝑡−𝑠|²,𝑡,𝑠∈[−𝜏,0]. (13)

We now describe our Euler-Maruyama scheme for the approximation of (1). For any 𝑛≥1, let 𝜋_𝑛∶𝐻→𝐻_𝑛=span{𝑒₁,𝑒₂,…,𝑒_𝑛} be the orthogonal projection; that is, 𝜋_𝑛𝑥=∑^𝑛_𝑖=1⟨𝑥,𝑒_𝑖⟩_𝐻𝑒_𝑖, 𝑥∈𝐻, 𝐴_𝑛=𝜋_𝑛𝐴, 𝑓_𝑛=𝜋_𝑛𝑓, 𝑔_𝑛=𝜋_𝑛𝑔, and ℎ_𝑛=𝜋_𝑛ℎ.

Consider the following stochastic differential delay equations with jumps on 𝐻_𝑛: 𝑑𝑋^𝑛(𝑡)=𝐴_𝑛𝑋^𝑛(𝑡)+𝑓_𝑛𝑋^𝑛(𝑡),𝑋^𝑛(𝑡−𝜏)𝑑𝑡+𝑔_𝑛𝑋^𝑛(𝑡),𝑋^𝑛()(𝑡)+𝑡−𝜏𝑑𝑊_ℤℎ_𝑛𝑋^𝑛(𝑡),𝑋^𝑛()𝑁(),𝑋𝑡−𝜏,𝑢𝑑𝑡,𝑑𝑢_𝑛(𝜃)=𝜋_𝑛𝜉(𝜃),𝜃∈[−𝜏,0]. (14) This spatial approximation (14) is called the Galerkin approximation of (1). Due to the fact that 𝜋_𝑛𝐴𝑥=𝜋_𝑛𝐴(∑^𝑛_𝑖=1⟨𝑥,𝑒_𝑖⟩_𝐻𝑒_𝑖)=−∑^𝑛_𝑖=1𝜆_𝑖⟨𝑥,𝑒_𝑖⟩_𝐻𝑒_𝑖,𝑥∈𝐻_𝑛, it follows that for 𝑥∈𝐻_𝑛, 𝐴_𝑛𝑥=𝐴𝑥, 𝑒^{𝑡𝐴𝑛𝑥}=𝑒^𝑡𝐴𝑥.

By (H2) and (H3) and the property of the projection operator, we have that 𝐴_𝑛𝑥−𝐴_𝑛𝑦²_𝐻=𝐴_𝑛𝑥−𝑦²_𝐻≤𝜆²_𝑛𝑥−𝑦_𝐻,𝑓_𝑛𝑥₁,𝑦₁−𝑓_𝑛𝑥₂,𝑦₂²_𝐻∨𝑔𝑥₁,𝑦₁−𝑔𝑥₂,𝑦₂²_ℒ02=𝑓𝑥₁,𝑦₁−𝑓𝑥₂,𝑦₂²_𝐻∨𝑔𝑥₁,𝑦₁−𝑔𝑥₂,𝑦₂²_ℒ02≤𝐿₁𝑥₁−𝑥₂²_𝐻+𝑦₁−𝑦₂²_𝐻,ℎ_𝑛𝑥₁,𝑦₁,𝑢−ℎ_𝑛𝑥₂,𝑦₂,𝑢²_𝐻=ℎ𝑥₁,𝑦₁,𝑢−ℎ𝑥₂,𝑦₂,𝑢²_𝐻≤𝐿₁𝑥₁−𝑥₂²_𝐻+𝑦₁−𝑦₂²_𝐻,ℎ_𝑛(𝑥,𝑦,𝑢)²_𝐻=ℎ(𝑥,𝑦,𝑢)²_𝐻≤𝐿₂‖𝑥‖²_𝐻+𝑦²_𝐻 (15) for arbitrary 𝑥,𝑦,𝑥₁,𝑥₂,𝑦₁,𝑦₂∈𝐻_𝑛 and 𝑢∈ℤ. Hence, (14) admits a unique solution 𝑋^𝑛(𝑡) on 𝐻_𝑛.

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, for some sufficiently large integer 𝑁>𝜏. For any integer 𝑘≥0, the time discretization scheme applied to (14) produces approximations 𝑌^𝑛(𝑘Δ)≈𝑋^𝑛(𝑘Δ) by forming 𝑌^𝑛((𝑘+1)Δ)=𝑒^Δ𝐴𝑛𝑌^𝑛(𝑘Δ)+𝑓_𝑛𝑌^𝑛(𝑘Δ),𝑌^𝑛(𝑘Δ−𝜏)Δ+𝑔_𝑛𝑌^𝑛(𝑘Δ),𝑌^𝑛(𝑘Δ−𝜏)Δ𝑊_𝑘+_ℤℎ_𝑛𝑌^𝑛(𝑘Δ),𝑌^𝑛(𝑘Δ−𝜏),𝑢Δ𝑁_𝑘(𝑢),𝑌^𝑛(𝜃)=𝜋_𝑛𝜉(𝜃),𝜃∈[−𝜏,0], (16) where Δ𝑊_𝑘=𝑊((𝑘+1)Δ)−𝑊(𝑘Δ) and Δ𝑁_𝑘(𝑑𝑢)=𝑁((0,(𝑘+1)Δ],𝑑𝑢)−𝑁((0,𝑘Δ],𝑑𝑢).

The continuous-time version of this scheme associated with (14) is defined by 𝑌^𝑛(𝑡)=𝑒^𝑡𝐴𝑛𝑌^𝑛(0)+^𝑡₀𝑒^{(𝑡−⌊𝑠⌋)𝐴𝑛}𝑓_𝑛𝑌^𝑛(⌊𝑠⌋),𝑌^𝑛𝑠⌋)+(⌊−𝜏𝑑𝑠_𝑡0𝑒^{(𝑡−⌊𝑠⌋)𝐴𝑛}𝑔_𝑛𝑌^𝑛(⌊𝑠⌋),𝑌^𝑛𝑠⌋(𝑠)+(⌊−𝜏)𝑑𝑊_𝑡0ℤ𝑒^{(𝑡−⌊𝑠⌋)𝐴𝑛}ℎ_𝑛𝑌^𝑛(⌊𝑠⌋),𝑌^𝑛()(),𝑌⌊𝑠⌋−𝜏,𝑢×𝑁𝑑𝑠,𝑑𝑢_𝑛(𝜃)=𝜋_𝑛𝜉(𝜃),𝜃∈[−𝜏,0], (17) where ⌊𝑡⌋=[𝑡/Δ]Δ with [𝑡/Δ] denotes the integer of 𝑡/Δ.

From (16) and (17), we have 𝑌^𝑛(𝑘Δ)=𝑌^𝑛(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,𝐿₁,𝐿₂, and 𝐿₃ but is independent of Δ, such that sup_{0≤𝑡≤𝑇}𝔼‖𝑋(𝑡)‖²_𝐻^1/2∨sup_{0≤𝑡≤𝑇}𝔼𝑌^𝑛(𝑡)²_𝐻^1/2≤𝐶. (18)

Proof. Due to the fact that (𝔼‖⋅‖²_𝐻)^1/2 is a norm, we have from (8) that 𝔼‖𝑋(𝑡)‖²_𝐻^1/2≤𝔼𝑒^𝑡𝐴𝑛𝜉(0)²_𝐻^1/2+𝔼^𝑡₀𝑒^{(𝑡−𝑠)𝐴}𝑓(𝑋(𝑠),𝑋(𝑠−𝜏))𝑑𝑠²_𝐻^1/2+𝔼^𝑡₀𝑒^{(𝑡−𝑠)𝐴}𝑔(𝑋(𝑠),𝑋(𝑠−𝜏))𝑑𝑊(𝑠)²_𝐻^1/2+𝔼^𝑡₀𝑒^{(𝑡−𝑠)𝐴}ℎ(𝑋(𝑠),𝑋(𝑠−𝜏),𝑢)𝑁(𝑑𝑠,𝑑𝑢)²_𝐻^1/2=⁴_𝑖=1𝐼_𝑖(𝑡). (19)
Recall the property of the operator 𝐴 (see [18]): (−𝐴)^𝛿1𝑒^𝐴𝑡≤𝐶𝑡^−𝛿1,(−𝐴)^𝛿21−𝑒^𝐴𝑡≤𝐶𝑡^𝛿2,𝛿₁≥0,𝛿₂∈[],()0,1−𝐴^𝛼+𝛽𝑥=(−𝐴)^𝛼(−𝐴)^𝛽𝑥,𝑥∈𝐷(−𝐴)^𝑟, (20) for 𝛼,𝛽∈ℝ, where 𝑟=max{𝛼,𝛽,𝛼+𝛽}.
By (H1) and (H2), together with the Minkowski integral inequality, we derive that 𝐼₂(𝑡)≤^𝑡₀𝔼𝑒^{(𝑡−𝑠)𝐴}𝑓(𝑋(𝑠),𝑋(𝑠−𝜏))²_𝐻^1/2𝑑𝑠≤𝐶^𝑡₀1+𝔼‖𝑋(𝑠)‖²_𝐻^1/2+𝔼‖𝑋(𝑠−𝜏)‖²_𝐻^1/2𝑑𝑠≤𝐶+𝐶^𝑡₀𝔼‖𝑋(𝑠)‖²_𝐻^1/2𝑑𝑠. (21) By (H1), (H2), and (H3) and using the Itô isometry, we have 𝐼₃(𝑡)+𝐼₄(𝑡)≤^𝑡₀𝔼𝑒^{(𝑡−𝑠)𝐴}𝑔(𝑋(𝑠),𝑋(𝑠−𝜏))²_ℒ02𝑑𝑠^1/2+𝔼^𝑡_0𝑍𝑒^{(𝑡−𝑠)𝐴}ℎ(𝑋(𝑠),𝑋(𝑠−𝜏),𝑢)𝑁(𝑑𝑠,𝑑𝑢)+𝜌^𝑡_0ℤ𝑒^{(𝑡−𝑠)𝐴}ℎ×(𝑋(𝑠),𝑋(𝑠−𝜏),𝑢)𝜋(𝑑𝑢)²_𝐻^1/2≤^𝑡₀𝐿₀1+𝔼‖𝑋(𝑠)‖²_𝐻+𝔼‖𝑋(𝑠−𝜏)‖²_𝐻𝑑𝑠^1/2+𝔼^𝑡_0𝑍𝑒^{(𝑡−𝑠)𝐴}ℎ×(𝑋(𝑠),𝑋(𝑠−𝜏),𝑢)𝑁(𝑑𝑠,𝑑𝑢)²_𝐻^1/2+𝜌𝔼^𝑡_0ℤ𝑒^{(𝑡−𝑠)𝐴}ℎ×(𝑋(𝑠),𝑋(𝑠−𝜏),𝑢)𝜋(𝑑𝑢)𝑑𝑠²_𝐻^1/2. (22) Using Hölder inequality and (H3), for the last term of (22), we have 𝜌𝔼^𝑡_0𝑍𝑒^{(𝑡−𝑠)𝐴}ℎ(𝑋(𝑠),𝑋(𝑠−𝜏),𝑢)𝜋(𝑑𝑢)𝑑𝑠²_𝐻^1/2≤𝐶𝔼^𝑡_0𝑍‖ℎ(𝑋(𝑠),𝑋(𝑠−𝜏),𝑢)‖²_𝐻𝜋(𝑑𝑢)𝑑𝑠^1/2≤𝐶𝐿₂^𝑡₀𝔼‖𝑋(𝑠)‖²_𝐻+𝔼‖𝑋(𝑠−𝜏)‖²_𝐻𝑑𝑠^1/2≤𝐶𝐿₂√𝜏𝔼𝜉_𝐻+𝑝𝐶2𝐿₂^𝑡₀𝔼‖𝑋(𝑠)‖²_𝐻𝑑𝑠^1/2. (23) Moreover, by using the Itô isometry and (H3), we obtain that 𝔼^𝑡_0ℤ𝑒^{(𝑡−𝑠)𝐴}ℎ(𝑋(𝑠),𝑋(𝑠−𝜏),𝑢)𝑁(𝑑𝑠,𝑑𝑢)²_𝐻^1/2≤^𝑡_0ℤ𝔼‖ℎ(𝑋(𝑠),𝑋(𝑠−𝜏),𝑢)‖²_𝐻𝜋(𝑑𝑢)𝑑𝑠^1/2≤𝐿₂^𝑡₀𝔼‖𝑋(𝑠)‖²_𝐻+𝔼‖𝑋(𝑠−𝜏)‖²_𝐻𝑑𝑠^1/2≤𝐿₂√𝜏𝔼𝜉_𝐻+2𝐿₂^𝑡₀(𝔼‖𝑋(𝑠)‖²_𝐻𝑑𝑠^1/2. (24) Substituting (23) and (24) into (22), it follows that 𝐼₃(𝑡)+𝐼₄(𝑡)≤𝐶+𝐶𝔼𝜉_𝐻+𝐶^𝑡₀𝔼‖𝑋(𝑠)‖²_𝐻𝑑𝑠^1/2. (25) Hence, 𝔼‖𝑋(𝑡)‖²_𝐻^1/2≤𝐶+𝐶𝔼𝜉_𝐻+𝐶^𝑡₀𝔼‖𝑋(𝑠)‖²_𝐻𝑑𝑠^1/2. (26) Applying the Gronwall inequality, we have sup_{0≤𝑡≤𝑇}𝔼‖𝑋(𝑡)‖²_𝐻^1/2≤𝐶. (27) Using the similar argument, the second assertion of (18) follows.

Lemma 3. Let (H1)–(H4) hold; for sufficiently small Δ, sup_{0≤𝑡≤𝑇}𝔼‖𝑋(𝑡)−𝑋(⌊𝑡⌋)‖²_𝐻^1/2≤𝐶Δ^1/2, (28) where 𝐶>0 is constant dependent on 𝑇,𝜉,𝐿₁,𝐿₂,𝐿₃, and 𝐿₄, while being independent of Δ.

Proof. For any 𝑡∈[0,𝑇], we have from (8) that 𝑋(𝑡)−𝑋(⌊𝑡⌋)=𝑒^{⌊𝑡⌋𝐴}𝑒^{(𝑡−⌊𝑡⌋)𝐴}𝜉(0)+−𝟏_0⌊𝑡⌋𝑒^{(𝑡−⌊𝑡⌋)𝐴}−𝟏𝑒^{(⌊𝑡⌋−𝑠)𝐴}𝑓(𝑋(𝑠)(+,𝑋𝑠−𝜏))𝑑𝑠_{𝑡⌊𝑡⌋}𝑒^{(𝑡−𝑠)𝐴}𝑓(𝑋(𝑠)(+,𝑋𝑠−𝜏))𝑑𝑠_0⌊𝑡⌋𝑒^{(𝑡−⌊𝑡⌋)𝐴}−𝟏𝑒^{(⌊𝑡⌋−𝑠)𝐴}𝑔(𝑋(𝑠)((𝑠)+,𝑋𝑠−𝜏))𝑑𝑊_{0⌊𝑡⌋ℤ}𝑒^{(𝑡−⌊𝑡⌋)𝐴}−𝟏𝑒^{(⌊𝑡⌋−𝑠)𝐴}(𝑋(𝑠)())𝑁()+×ℎ,𝑋𝑠−𝜏,𝑢𝑑𝑠,𝑑𝑢_{𝑡⌊𝑡⌋}𝑒^{(𝑡−𝑠)𝐴}𝑔(𝑋(𝑠),𝑋(𝑠−𝜏))𝑑𝑊(𝑠)+^𝑡_{⌊𝑡⌋ℤ}𝑒^{(𝑡−𝑠)𝐴}ℎ(𝑋(𝑠)())𝑁()=,𝑋𝑠−𝜏,𝑢𝑑𝑠,𝑑𝑢_7𝑖=1𝐽_𝑖(𝑡). (29) Since (𝔼‖⋅‖²_𝐻)^1/2 is a norm, it follows that 𝔼‖𝑋(𝑡)−𝑋(⌊𝑡⌋)‖²_𝐻^1/2≤⁷_𝑖=1𝔼𝐽_𝑖(𝑡)²_𝐻^1/2. (30) Recalling the fundamental inequality 1−𝑒^−𝑦≤𝑦, 𝑦>0, we get from (H1) that 𝑒^{(𝑡−⌊𝑡⌋)𝐴}−𝟏𝑥²_𝐻=^∞_𝑖=1𝑒^{−𝜆𝑖(𝑡−⌊𝑡⌋)}−1𝑥,𝑒_𝑖𝑒_𝑖²_𝐻≤1−𝑒^{−𝜆1(𝑡−⌊𝑡⌋)2}‖𝑥‖²_𝐻≤𝜆²₁Δ²‖𝑥‖²_𝐻. (31) Therefore, 𝔼𝐽₁(𝑡)²_𝐻^1/2=𝔼𝑒^{⌊𝑡⌋𝐴}𝑒^{(𝑡−⌊𝑡⌋)𝐴}−𝟏𝜉(0)²_𝐻^1/2≤𝜆₁𝔼𝜉(0)²_𝐻^1/2Δ. (32) By (H1), (H2), and the Minkowski integral inequality, we obtain that ³_𝑖=2𝔼𝐽_𝑖(𝑡)²_𝐻^1/2≤₀^⌊𝑡⌋𝑒^{(𝑡−⌊𝑡⌋)𝐴}−𝟏𝑒^{(⌊𝑡⌋−𝑠)𝐴}×𝔼𝑓(𝑋(𝑠),𝑋(𝑠−𝜏))²_𝐻^1/2+𝑑𝑠_{𝑡⌊𝑡⌋}𝔼𝑓(𝑋(𝑠),𝑋(𝑠−𝜏))²_𝐻^1/2𝑑𝑠. (33) Together with (31), we arrive at ³_𝑖=2𝔼𝐽_𝑖(𝑡)²_𝐻^1/2≤𝜆₁Δ₀^⌊𝑡⌋𝑑𝑠+Δ𝐶sup_{0≤𝑡≤𝑇}𝔼𝑓(𝑋(𝑡),𝑋(𝑡−𝜏))²_𝐻^1/2≤𝐶1+sup_{0≤𝑡≤𝑇}𝔼‖𝑋(𝑡)‖²_𝐻^1/2Δ. (34) Following the argument of (22), we derive that ⁷_𝑖=4𝔼𝐽_𝑖(𝑡)²_𝐻^1/2≤₀^⌊𝑡⌋𝑒^{(𝑡−⌊𝑡⌋)𝐴}−1²𝑒^{(⌊𝑡⌋−𝑠)𝐴2}×𝔼𝑔(𝑋(𝑠),𝑋(𝑠−𝜏))²_ℒ⁰₂𝑑𝑠^1/2+𝐶₀^⌊𝑡⌋_ℤ𝑒^{(𝑡−⌊𝑡⌋)𝐴}−1²𝑒^{(⌊𝑡⌋−𝑠)𝐴2}×𝔼‖ℎ(𝑋(𝑠),𝑋(𝑠−𝜏),𝑢)‖²_𝐻𝜋(𝑑𝑢)𝑑𝑠^1/2+^𝑡_⌊𝑡⌋𝑒^{(𝑡−𝑠)𝐴2}𝔼𝑔(𝑋(𝑠),𝑋(𝑠−𝜏))²_ℒ⁰₂𝑑𝑠^1/2+𝐶^𝑡_{⌊𝑡⌋ℤ}𝑒^{(𝑡−𝑠)𝐴2}×𝔼‖ℎ(𝑋(𝑠),𝑋(𝑠−𝜏),𝑢)‖²_𝐻𝜋(𝑑𝑢)𝑑𝑠^1/2≤𝐶1+sup_{0≤𝑡≤𝑇}𝔼‖𝑋(𝑡)‖²_𝐻^1/2Δ^1/2. (35) Substituting (32), (34), and (35) into (30), we arrive at 𝔼‖𝑋(𝑡)−𝑋(⌊𝑡⌋)‖²_𝐻^1/2≤𝐶1+sup_{0≤𝑡≤𝑇}𝔼‖𝑋(𝑡)‖²_𝐻^1/2Δ^1/2. (36) Therefore, by Lemma 2, the required assertion (28) follows.

Now, we state our main result in this paper as follows.

Theorem 4. Let (H1)–(H4) hold, and 𝐿₁2𝛼⁻¹+𝜌+3(2𝛼)^−1/2<1. (37) Then, sup_{0≤𝑡≤𝑇}𝔼𝑋(𝑡)−𝑌^𝑛(𝑡)²_𝐻^1/2≤𝐶𝜆_𝑛^−1/2+Δ^1/2, (38) where 𝐶>0 is a constant dependent on 𝑇,𝜉,𝐿₁,𝐿₂,𝐿₃, and 𝐿₄, while being independent of 𝑛 and Δ.

Proof. By (8) and (17), we obtain 𝑋(𝑡)−𝑌^𝑛(𝑡)=𝑒^𝑡𝐴1−𝜋_𝑛𝜉(0)+^𝑡₀𝑒^{(𝑡−𝑠)𝐴}𝑓(𝑋(𝑠),𝑋(𝑠−𝜏))−𝑓_𝑛(𝑋(𝑠)(+,𝑋𝑠−𝜏))𝑑𝑠_𝑡0𝑒^{(𝑡−𝑠)𝐴}𝑓_𝑛(𝑋(𝑠),𝑋(𝑠−𝜏))−𝑓_𝑛(𝑋𝑠𝑠⌋+(⌊⌋),𝑋(⌊−𝜏))𝑑𝑠_𝑡0𝑒^{(𝑡−𝑠)𝐴}𝑔_𝑛(𝑋(𝑠),𝑋(𝑠−𝜏))−𝑔_𝑛(𝑋(⌊𝑠⌋),𝑋(⌊𝑠⌋−𝜏))𝑑𝑊(𝑠)+^𝑡₀𝑒^{(𝑡−𝑠)𝐴}𝑓_𝑛(𝑋(⌊𝑠⌋),𝑋(⌊𝑠⌋−𝜏))−𝑓_𝑛𝑌^𝑛(⌊𝑠⌋),𝑌^𝑛𝑠⌋)+(⌊−𝜏𝑑𝑠_𝑡0𝑒^{(𝑡−𝑠)𝐴}𝑔_𝑛(𝑋(⌊𝑠⌋),𝑋(⌊𝑠⌋−𝜏))−𝑔_𝑛𝑌^𝑛(⌊𝑠⌋),𝑌^𝑛𝑠⌋(𝑠)+(⌊−𝜏)𝑑𝑊_𝑡0𝑒^{(𝑡−𝑠)𝐴}𝑔(𝑋(𝑠),𝑋(𝑠−𝜏))−𝑔_𝑛(𝑋(𝑠),𝑋(𝑠−𝜏))𝑑𝑊(𝑠)+^𝑡₀𝑒^{(𝑡−𝑠)𝐴}1−𝑒^{(𝑠−⌊𝑠⌋)𝐴}𝑓_𝑛𝑌^𝑛(⌊𝑠⌋),𝑌^𝑛𝑠⌋)+(⌊−𝜏𝑑𝑠_𝑡0𝑒^{(𝑡−𝑠)𝐴}1−𝑒^{(𝑠−⌊𝑠⌋)𝐴}×𝑔_𝑛𝑌^𝑛(⌊𝑠⌋),𝑌^𝑛(⌊𝑠⌋−𝜏)𝑑𝑊(𝑠)+^𝑡_0ℤ𝑒^{(𝑡−𝑠)𝐴}ℎ(𝑋(𝑠),𝑋(𝑠−𝜏),𝑢)−ℎ_𝑛(𝑋(𝑠)()𝑁()+,𝑋𝑠−𝜏,𝑢)𝑑𝑠,𝑑𝑢_𝑡0ℤ𝑒^{(𝑡−𝑠)𝐴}ℎ_𝑛(𝑋(𝑠),𝑋(𝑠−𝜏),𝑢)−ℎ_𝑛(𝑋𝑠𝑠⌋)𝑁()+(⌊⌋),𝑋(⌊−𝜏,𝑢)𝑑𝑠,𝑑𝑢_𝑡0ℤ𝑒^{(𝑡−𝑠)𝐴}ℎ_𝑛(𝑋(⌊𝑠⌋),𝑋(⌊𝑠⌋−𝜏),𝑢)−ℎ_𝑛×𝑌^𝑛(⌊𝑠⌋),𝑌^𝑛𝑠⌋)𝑁()+(⌊−𝜏,𝑢𝑑𝑠,𝑑𝑢_𝑡0𝑍𝑒^{(𝑡−𝑠)𝐴}1−𝑒^{(𝑠−⌊𝑠⌋)𝐴}ℎ_𝑛×𝑌^𝑛(⌊𝑠⌋),𝑌^𝑛𝑠⌋)𝑁()=(⌊−𝜏,𝑢𝑑𝑠,𝑑𝑢_13𝑖=1𝐾_𝑖(𝑡). (39) Noting that (𝔼‖⋅‖²_𝐻)^1/2 is a norm, we have 𝔼𝑋(𝑡)−𝑌^𝑛(𝑡)²_𝐻^1/2≤¹³_𝑖=1𝔼𝐾_𝑖(𝑡)²_𝐻^1/2. (40) By (H1) and the nondecreasing spectrum {𝜆_𝑚}_𝑚≥1, it easily follows that 𝔼𝑒^𝑡𝐴1−𝜋_𝑛𝜉(0)_𝐻=𝔼^∞_𝑚=𝑛+1𝑒^{−2𝜆𝑚𝑡}𝜉(0),𝑒_𝑚²_𝐻^1/2=𝔼^∞_𝑚=𝑛+1𝑒^{−2𝜆𝑚𝑡}𝜆²_𝑚𝜆²_𝑚𝜉(0),𝑒_𝑚²_𝐻^1/2≤1𝜆_𝑛𝔼𝐴𝜉(0)_𝐻. (41) By (H2), the Minkowski integral inequality, and Lemma 2, we have 𝔼𝐾₂(𝑡)²_𝐻^1/2≤^𝑡₀𝔼𝑒^{(𝑡−𝑠)𝐴}1−𝜋_𝑛𝑓(𝑋(𝑠),𝑋(𝑠−𝜏))²_𝐻^1/2=𝑑𝑠_𝑡0𝔼^∞_𝑚=𝑛+1𝑒^{−2𝜆𝑚(𝑡−𝑠)}𝑓(𝑋(𝑠),𝑋(𝑠−𝜏)),𝑒_𝑚²_𝐻^1/2≤𝑑𝑠_𝑡0𝑒^{−𝜆𝑛(𝑡−𝑠)}𝔼^∞_𝑚=𝑛+1𝑓(𝑋(𝑠),𝑋(𝑠−𝜏)),𝑒_𝑚²_𝐻^1/2𝑑𝑠≤𝐶^𝑡₀𝑒^{−𝜆𝑛(𝑡−𝑠)}×1+𝔼‖𝑋(𝑠)‖²_𝐻^1/2+𝔼‖𝑋(𝑠−𝜏)‖²_𝐻^1/2𝑑𝑠≤𝐶𝜆_𝑛⁻¹. (42) Applying (H1), (H2), and Lemma 3 and combining the Minkowski integral inequality and the Itô isometry yield⁶_𝑖=3𝔼𝐾_𝑖(𝑡)²_𝐻^1/2≤𝐿₁^𝑡₀𝑒^{(𝑡−𝑠)𝐴}𝔼‖𝑋(𝑠)−𝑋(⌊𝑠⌋)‖²_𝐻+‖𝑋(𝑠−𝜏)−𝑋(⌊𝑠⌋−𝜏)‖²_𝐻^1/2+𝑑𝑠𝐿₁^𝑡₀𝑒^{(𝑡−𝑠)𝐴}𝔼𝑋(⌊𝑠⌋)−𝑌^𝑛(⌊𝑠⌋)²_𝐻+𝑋(⌊𝑠⌋−𝜏)−𝑌^𝑛(⌊𝑠⌋−𝜏)²_𝐻^1/2+𝑑𝑠𝐿₁^𝑡₀𝑒^{(𝑡−𝑠)𝐴2}𝔼‖𝑋(𝑠)−𝑋(⌊𝑠⌋)‖²_𝐻+𝑋(𝑠−𝜏)−𝑌^𝑛(⌊𝑠⌋−𝜏)²_𝐻𝑑𝑠^1/2+𝐿₁^𝑡₀𝑒^{(𝑡−𝑠)𝐴2}𝔼𝑋(⌊𝑠⌋)−𝑌^𝑛(⌊𝑠⌋)²_𝐻+𝑋(⌊𝑠⌋−𝜏)−𝑌^𝑛(⌊𝑠⌋−𝜏)²_𝐻𝑑𝑠^1/2≤𝐶Δ^1/2+𝐿₁sup_{0≤𝑠≤𝑡}𝔼𝑋(𝑠)−𝑌^𝑛(𝑠)²_𝐻^1/2×^𝑡₀𝑒^{−𝛼(𝑡−𝑠)}+𝑑𝑠𝐿₁sup_{0≤𝑠≤𝑡}𝔼𝑋(𝑠)−𝑌^𝑛(𝑠)²_𝐻^1/2𝑡₀𝑒^{−2𝛼(𝑡−𝑠)}𝑑𝑠^1/2+𝐿₁sup_{0≤𝑠≤𝑡}𝔼𝑋(𝑠)−𝑌^𝑛(𝑠)²_𝐻^{1/2𝑡−𝜏}_−𝜏𝑒^{−𝛼(𝑡−𝑠−𝜏)}+𝑑𝑠𝐿₁sup_{0≤𝑠≤𝑡}𝔼𝑋(𝑠)−𝑌^𝑛(𝑠)²_𝐻^{1/2𝑡−𝜏}_−𝜏𝑒^{−2𝛼(𝑡−𝑠−𝜏)}𝑑𝑠^1/2≤𝐶Δ^1/2+𝐿₁sup_{0≤𝑠≤𝑡}𝔼𝑋(𝑠)−𝑌^𝑛(𝑠)²_𝐻^1/2×2𝛼⁻¹+2(2𝛼)^−1/2. (43)By the Itô isometry and a similar argument to that of (42), we deduce that 𝔼𝐾₇(𝑡)²_𝐻^1/2≤^𝑡₀𝔼𝑒^{(𝑡−𝑠)𝐴}1−𝜋_𝑛𝑔(𝑋(𝑠),𝑋(𝑠−𝜏))²_ℒ02𝑑𝑠^1/2≤𝐶^𝑡₀𝑒^{−2𝜆𝑛(𝑡−𝑠)}𝔼𝑔(𝑋(𝑠),𝑋(𝑠−𝜏))²_ℒ⁰₂𝑑𝑠^1/2≤𝐶𝜆_𝑛^−1/2. (44) Moreover, by (31), (H2), and Lemma 2 and combining the Minkowski integral inequality and the Itô isometry, we have ⁹_𝑖=8𝔼𝐾_𝑖(𝑡)²_𝐻^1/2≤^𝑡₀𝔼𝑒^{(𝑡−𝑠)𝐴}1−𝑒^{(𝑠−⌊𝑠⌋)𝐴2}×𝑓_𝑛(𝑌^𝑛(⌊𝑠⌋),𝑌^𝑛(⌊𝑠⌋−𝜏))²_𝐻^1/2+𝑑𝑠_𝑡0𝔼𝑒^{(𝑡−𝑠)𝐴}1−𝑒^{(𝑠−⌊𝑠⌋)𝐴2}×𝑔_𝑛𝑌^𝑛(⌊𝑠⌋),𝑌^𝑛(⌊𝑠⌋−𝜏)²_𝐻𝑑𝑠^1/2≤𝐶Δ^𝑡₀𝔼𝑓_𝑛𝑌^𝑛(⌊𝑠⌋),𝑌^𝑛(⌊𝑠⌋−𝜏)²_𝐻^1/2𝑑𝑠+𝐶Δ^𝑡₀𝔼𝑔_𝑛𝑌^𝑛(⌊𝑠⌋),𝑌^𝑛(⌊𝑠⌋−𝜏)²_ℒ02𝑑𝑠^1/2≤𝐶Δ. (45) By (31) and the Itô isometry, we obtain that 𝔼𝐾₁₀(𝑡)²_𝐻^1/2≤𝔼^𝑡_0ℤ𝑒^{(𝑡−𝑠)𝐴}1−𝜋_𝑛ℎ×(𝑋(𝑠),𝑋(𝑠−𝜏),𝑢)𝑁(𝑑𝑠,𝑑𝑢)²_𝐻^1/2+𝜌𝔼^𝑡_0ℤ𝑒^{(𝑡−𝑠)𝐴}1−𝜋_𝑛ℎ×(𝑋(𝑠),𝑋(𝑠−𝜏),𝑢)𝜋(𝑑𝑢)𝑑𝑠²_𝐻^1/2≤^𝑡_0ℤ𝑒^{(𝑡−𝑠)𝐴}1−𝜋_𝑛²𝔼×‖ℎ(𝑋(𝑠),𝑋(𝑠−𝜏),𝑢)‖²_𝐻𝜋(𝑑𝑢)𝑑𝑠^1/2+𝜌^𝑡_0ℤ𝑒^{(𝑡−𝑠)𝐴}1−𝜋_𝑛²×𝔼‖ℎ(𝑋(𝑠),𝑋(𝑠−𝜏),𝑢)‖²_𝐻𝜋(𝑑𝑢)𝑑𝑠^1/2≤𝐶^𝑡₀𝑒^{−2𝜆𝑛(𝑡−𝑠)}𝔼‖𝑋(𝑠)‖²_𝐻+𝔼‖𝑋(𝑠−𝜏)‖²_𝐻𝑑𝑠^1/2≤𝐶𝜆_𝑛^−1/2. (46) Carrying out the similar arguments to those of (43) and (45), we derive that 𝔼𝐾₁₁(𝑡)²_𝐻^1/2+𝔼𝐾₁₂(𝑡)²_𝐻^1/2≤𝐶Δ^1/2+(2𝛼)^−1/2×𝜌+1𝐿₁sup_{0≤𝑠≤𝑡}𝔼𝑋(𝑠)−𝑌^𝑛(𝑠)²_𝐻^1/2,𝔼𝐾₁₃(𝑡)²_𝐻^1/2≤𝐶Δ. (47) As a result, putting (41)–(47) into (40) gives that sup_{0≤𝑠≤𝑡}𝔼𝑋(𝑠)−𝑌^𝑛(𝑠)²_𝐻^1/2≤𝐶𝜆_𝑛^−1/2+𝐶Δ^1/2+𝐿₁2𝛼⁻¹+𝜌+3(2𝛼)^−1/2×sup_{0≤𝑠≤𝑡}𝔼𝑋(𝑠)−𝑌^𝑛(𝑠)²_𝐻^1/2, (48) 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.