Convergence of the Euler Method of Stochastic Differential Equations with Piecewise Continuous Arguments

Abstract

The main purpose of this paper is to investigate the strong convergence of the Euler method to stochastic differential equations with piecewise continuous arguments (SEPCAs). Firstly, it is proved that the Euler approximation solution converges to the analytic solution under local Lipschitz condition and the bounded 𝑝th moment condition. Secondly, the Euler approximation solution converge to the analytic solution is given under local Lipschitz condition and the linear growth condition. Then an example is provided to show which is satisfied with the monotone condition without the linear growth condition. Finally, the convergence of numerical solutions to SEPCAs under local Lipschitz condition and the monotone condition is established.

1. Introduction

Recently, differential equations with piecewise continuous arguments (EPCAs) have attracted much attention, and many useful conclusions have been obtained. These systems have applications in certain biomedical models, control systems with feedback delay in the work of Cooke and Wiener [1]. The general theory and basic results for EPCAs have by now been thoroughly investigated in the book of Wiener [2]. Song et al. [3] deal with the stability analysis of numerical methods for the solution of advanced differential equations with piecewise continuous arguments. A typical EPCA contains arguments that are constant on certain intervals. The solutions are determined by a finite set of initial data, rather than by an initial function, as in the case of general functional differential equation. A solution is defined as a continuous, sectionally smooth function that satisfies the equation within these intervals. Continuity of a solution at a point joining any two consecutive intervals leads to recursion relations for the solution at such points. Hence, EPCAs represent a hybrid of continuous and discrete dynamical systems and combine the properties of both differential and difference equations.

However, up to now there are few people who have considered the influence of noise to EPCAs. Actually, the environment and accidental events may greatly influence the systems. Thus analyzing SEPCAs is an interesting topic both in theory and applications. There is in general no explicit solution to an SEPCA, hence numerical solutions are required in practice. Numerical solutions to stochastic differential equations (SDEs) have been discussed under the local Lipschitz condition and the linear growth condition by many authors (see [4, 5]). Mao [6] discusses numerical solutions to stochastic differential delay equations (SDDEs) under the local Lipschitz condition and the linear growth condition. Mao and Sabanis [7] discuss numerical solutions to SDDEs with variable delay under the local Lipschitz condition and the linear growth condition. Mao discusses numerical solutions to SDEs and SDDEs under the local Lipschitz condition and the monotone condition (see [8]). Dai and Liu [9] give the mean-square stability of the numerical solutions of linear SEPCAs. However, SEPCAs do not have the convergence results. The main aim of this paper is to establish convergence of numerical solution for SEPCAs under the differential conditions.

The paper is organized as follows. In Section 2, we introduce necessary notations and the Euler method. In Section 3, the strong convergence of the Euler-Maruyama method to SEPCAs under local Lipschitz condition and the bounded 𝑝th moment condition will be given. In Section 4, the strong convergence of the Euler-Maruyama method to SEPCAs under local Lipschitz condition and the linear growth condition will be presented. In Section 5, an example is provided to show which is satisfied with the monotone condition without the linear growth condition. In Section 6, we obtain the convergence of numerical solutions to SEPCAs under local Lipschitz condition and the monotone condition is established.

2. Preliminary Notation and Euler Method

In this paper, unless otherwise specified, let |𝑥| be the Euclidean norm in 𝑥∈𝑅^𝑛. If 𝐴 is a vector or matrix, its transpose is defined by 𝐴^𝑇. If 𝐴 is a matrix, its trace norm is defined by |𝐴|=trace(𝐴^𝑇𝐴). For simplicity, we also have to denote by 𝑎∧𝑏=min{𝑎,𝑏},𝑎∨𝑏=max{𝑎,𝑏}.

Let (Ω,ℱ,𝑃) be a complete probability space with a filtration {ℱ_𝑡}_𝑡≥0, satisfying the usual conditions. ℒ¹([0,∞),𝑅^𝑛) and ℒ²([0,∞),𝑅^𝑛) denote the family of all real valued ℱ_𝑡-adapted process 𝑓(𝑡)_𝑡≥0, such that for every 𝑇>0, ∫^𝑇₀|𝑓(𝑡)|𝑑𝑡<∞𝑎.𝑠. and ∫^𝑇₀|𝑓(𝑡)|²𝑑𝑡<∞𝑎.𝑠., respectively. For any 𝑎,𝑏∈𝑅 with 𝑎<𝑏, denote 𝐶([𝑎,𝑏];𝑅^𝑛) the family of continuous functions 𝜙 from [𝑎,𝑏] to 𝑅^𝑛 with the norm ‖𝜙‖=sup_{𝑎≤𝜃≤𝑏}|𝜙(𝜃)|. Denote 𝐶^𝑏_ℱ𝑡([𝑎,𝑏];𝑅^𝑛) the family of all bounded ℱ_𝑡-measurable 𝐶([𝑎,𝑏];𝑅^𝑛)-valued random variables. Let 𝐵(𝑡)=(𝐵₁(𝑡),…,𝐵_𝑑(𝑡))^𝑇 be a 𝑑-dimensional Brownian motion defined on the probability space.

Throughout this paper, we consider stochastic differential equations with piecewise continuous arguments: 𝑑𝑥(𝑡)=𝑓(𝑥(𝑡),𝑥([𝑡]))𝑑𝑡+𝑔(𝑥(𝑡),𝑥([𝑡]))𝑑𝐵(𝑡)∀𝑡≥0, (2.1) with initial data 𝑥(0)=𝑥₀, where 𝑓:𝑅^𝑛×𝑅^𝑛→𝑅^𝑛,𝑔:𝑅^𝑛×𝑅^𝑛→𝑅^𝑛×𝑑,𝑥₀ is a vector, and [⋅] denotes the greatest-integer function. By the definition of stochastic differential, this equation is equivalent to the following stochastic integral equation: 𝑥(𝑡)=𝑥(0)+^𝑡₀𝑓(𝑥(𝑠),𝑥([𝑠]))𝑑𝑠+^𝑡₀𝑔(𝑥(𝑠),𝑥([𝑠]))𝑑𝐵(𝑠)∀𝑡≥0. (2.2) Moreover, we also require the coefficients 𝑓 and 𝑔 to be sufficiently smooth.

To be precise, let us state the following conditions. (H1) The local Lipschitz condition: for every integer 𝑖≥1, there exists a positive constant 𝐿_𝑖 such that 𝑓𝑥,𝑦−𝑓𝑥,𝑦²∨𝑔𝑥,𝑦−𝑔𝑥,𝑦²≤𝐿_𝑖|𝑥−𝑥|²+𝑦−𝑦², (2.3) for those 𝑥,𝑥,𝑦,𝑦∈𝑅^𝑛 with |𝑥|∨|𝑥|∨|𝑦|∨|𝑦|≤𝑖. (H2) Linear growth condition: there exists a positive constant 𝐾 such that 𝑓𝑥,𝑦²∨𝑔𝑥,𝑦²≤𝐾1+|𝑥|²+𝑦², (2.4) for all (𝑥,𝑦)∈𝑅^𝑛×𝑅^𝑛. (H3) Monotone condition: there exists a positive constant 𝐾₁ such that 𝑥^𝑇𝑓𝑥,𝑦+𝑝−12𝑔𝑥,𝑦²≤𝐾₁1+|𝑥|²+𝑦², (2.5) for all (𝑥,𝑦)∈𝑅^𝑛×𝑅^𝑛. (H4) The bounded 𝑝th moment condition: there exists a pair of constants 𝑝>2 and 𝐾₂>0 such that 𝐸sup_{0≤𝑡≤𝑇}|𝑥(𝑡)|^𝑝∨𝐸sup_{0≤𝑡≤𝑇}𝑦(𝑡)^𝑝≤𝐾₂. (2.6)

Let us first give the definition of the solution.

Definition 2.1 (see [10]). An 𝑅^𝑛-valued stochastic process {𝑥(𝑡)} is called a solution of (2.1) on [0,∞), if it has the following properties: (1){𝑥(𝑡)} is continuous on [0,∞) and ℱ_𝑡 adapted; (2){𝑓(𝑥(𝑡),𝑥([𝑡]))}∈ℒ¹([0,∞),𝑅^𝑛) and {𝑔(𝑥(𝑡),𝑥([𝑡]))}∈ℒ²([0,∞),𝑅^𝑛×𝑑);(3)Equation (2.2) is satisfied on each interval [𝑛,𝑛+1)⊂[0,∞) with integral end points almost surely. A solution {𝑥(𝑡)} is said to be unique if any other solution {𝑥(𝑡)} is indistinguishable from {𝑥(𝑡)}, that is, 𝑃{𝑥(𝑡)=𝑥(𝑡)∀𝑡∈[0,∞)}=1. (2.7)Let ℎ=1/𝑚 be a given stepsize with integer 𝑚≥1 and the grid points 𝑡_𝑛 defined by 𝑡_𝑛=𝑛ℎ(𝑛=0,1,2,…). For simplicity, we assume 𝑇=𝑁ℎ. We consider the Euler-Maruyama method to (2.1), 𝑦_𝑛+1=𝑦_𝑛+𝑓𝑦_𝑛,𝑦^ℎ([𝑛ℎ])ℎ+𝑔𝑦_𝑛,𝑦^ℎ([𝑛ℎ])Δ𝐵_𝑛, (2.8) for 𝑛=0,1,2,…, where Δ𝐵_𝑛=𝐵(𝑡_𝑛)−𝐵(𝑡_𝑛−1),𝑦^ℎ([𝑛ℎ]) is approximation to the exact solution 𝑥([𝑛ℎ]). Let 𝑛=𝑘𝑚+𝑙(𝑘=0,1,2,…,𝑙=0,1,2,…,𝑚−1). The adaptation of the Euler method to (2.1) leads to a numerical process of the following type: 𝑦_{𝑘𝑚+𝑙+1}=𝑦_{𝑘𝑚+𝑙}+𝑓𝑦_{𝑘𝑚+𝑙},𝑦_𝑘𝑚ℎ+𝑔𝑦_{𝑘𝑚+𝑙},𝑦_𝑘𝑚Δ𝐵_{𝑘𝑚+𝑙}, (2.9) where Δ𝐵_{𝑘𝑚+𝑙}=𝐵(𝑡_{𝑘𝑚+𝑙})−𝐵(𝑡_{𝑘𝑚+𝑙−1}),𝑦_{𝑘𝑚+𝑙} and 𝑦_𝑘𝑚 are approximations to the exact solution 𝑥(𝑡_{𝑘𝑚+𝑙}) and 𝑥([𝑡_{𝑘𝑚+𝑙}]), respectively. The continuous Euler-Maruyama approximate solution is defined by 𝑦(𝑡)=𝑦(0)+^𝑡₀𝑓(𝑧(𝑠),𝑧([𝑠]))𝑑𝑠+^𝑡₀𝑔(𝑧(𝑠),𝑧([𝑠]))𝑑𝐵(𝑠), (2.10) where 𝑧(𝑡)=𝑦_{𝑘𝑚+𝑙} and 𝑧([𝑡])=𝑦_𝑘𝑚 for 𝑡∈[𝑡_{𝑘𝑚+𝑙},𝑡_{𝑘𝑚+𝑙+1}). It is not difficult to see that 𝑦(𝑡_{𝑘𝑚+𝑙})=𝑧(𝑡_{𝑘𝑚+𝑙})=𝑦_{𝑘𝑚+𝑙} for 𝑘=0,1,2,…,𝑙=0,1,2,…,𝑚−1. For sufficiently large integer 𝑖, define the stopping times 𝜂_𝑖=inf{𝑡≥0:|𝑥(𝑡)|≥𝑖},𝜃_𝑖=inf{𝑡≥0:|𝑦(𝑡)|≥𝑖},𝜏_𝑖=𝜂_𝑖∧𝜃_𝑖.

3. Convergence of the Euler-Maruyama Method under the Bounded pth Moment

We will show the strong convergence of the EM method on (2.1) under local Lipschitz condition and the bounded 𝑝th moment condition. The following lemma shows that both 𝑦(𝑡) and 𝑧(𝑡) are close to each other.

Lemma 3.1. Under the condition (H1), let 𝑇>0 be arbitrary. Then 𝐸sup_{0≤𝑡≤𝑇}𝑦(𝑡)−𝑧(𝑡)²≤𝐶₁𝑥₀,𝑖ℎ, (3.1) where 𝐶₁(𝑥₀,𝑖)=4𝐿_𝑖(𝑇+4)𝐾₂.

Proof. For 𝑡∈[0,𝑇), there are two integers 𝑘 and 𝑙 such that 𝑡∈[𝑡_{𝑘𝑚+𝑙},𝑡_{𝑘𝑚+𝑙+1}). By the Hölder inequality, we compute 𝑦(𝑡)−𝑧(𝑡)²=^𝑡_{𝑡𝑘𝑚+𝑙}𝑓(𝑧(𝑠),𝑧([𝑠]))𝑑𝑠+^𝑡_{𝑡𝑘𝑚+𝑙}𝑔(𝑧(𝑠),𝑧([𝑠]))𝑑𝐵(𝑠)²≤2^𝑡_{𝑡𝑘𝑚+𝑙}𝑓(𝑧(𝑠),𝑧([𝑠]))𝑑𝑠²+2^𝑡_{𝑡𝑘𝑚+𝑙}𝑔(𝑧(𝑠),𝑧([𝑠]))𝑑𝐵(𝑠)²≤2𝑇^𝑡_{𝑡𝑘𝑚+𝑙}𝑓(𝑧(𝑠),𝑧([𝑠]))²𝑑𝑠+2^𝑡_{𝑡𝑘𝑚+𝑙}𝑔(𝑧(𝑠),𝑧([𝑠]))𝑑𝐵(𝑠)². (3.2) This implies that, for any 0≤𝑡₁≤𝑇, 𝐸sup_{0≤𝑡≤𝑡1}𝑦(𝑡)−𝑧(𝑡)²≤2𝑇𝐸sup_{0≤𝑡≤𝑡1}^𝑡_{𝑡𝑘𝑚+𝑙}𝑓(𝑧(𝑠),𝑧([𝑠]))²𝑑𝑠+2𝐸sup_{0≤𝑡≤𝑡1}^𝑡_{𝑡𝑘𝑚+𝑙}𝑔(𝑧(𝑠),𝑧([𝑠]))𝑑𝐵(𝑠)². (3.3) By the Doob martingale inequality, we have 𝐸sup_{0≤𝑡≤𝑡1}𝑦(𝑡)−𝑧(𝑡)²≤2𝑇𝐸^𝑡1_{𝑡𝑘𝑚+𝑙}𝑓(𝑧(𝑠),𝑧([𝑠]))²𝑑𝑠+8𝐸^𝑡1_{𝑡𝑘𝑚+𝑙}𝑔(𝑧(𝑠),𝑧([𝑠]))²𝑑𝑠. (3.4) Using the local Lipschitz conditions 𝐸sup_{0≤𝑡≤𝑡1}𝑦(𝑡)−𝑧(𝑡)²≤2𝐿_𝑖(𝑇+4)𝐸^𝑡1_{𝑡𝑘𝑚+𝑙}|𝑧(𝑠)|²+|𝑧([𝑠])|²𝑑𝑠≤4𝐿_𝑖(𝑇+4)^𝑡1_{𝑡𝑘𝑚+𝑙}𝐸sup_{0≤𝑢≤𝑠}𝑦(𝑢)²𝑑𝑠≤𝐶₁𝑥₀,𝑖ℎ, (3.5) where 𝐶₁(𝑥₀,𝑖)=4𝐿_𝑖(𝑇+4)𝐾₂. The proof is completed.

Theorem 3.2. Under the conditions (H1) and (H4), the EM approximate solution converges to the exact solution of (2.1) in the sense that lim_ℎ→0𝐸sup_{0≤𝑡≤𝑇}𝑦(𝑡)−𝑥(𝑡)²=0. (3.6)

Proof. Fix a 𝑝>2; let 𝑒(𝑡)=𝑥(𝑡)−𝑦(𝑡); it is easy to see that 𝐸sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²=𝐸sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²1_{{𝜃𝑖>𝑇and𝜂𝑖>𝑇}}+𝐸sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²1_{{𝜃𝑖≤𝑇or𝜂𝑖≤𝑇}}=𝐸sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²1_{{𝜏𝑖>𝑇}}+𝐸sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²1_{{𝜃𝑖≤𝑇or𝜂𝑖≤𝑇}}≤𝐸sup_{0≤𝑡≤𝑇}𝑒𝑡∧𝜏_𝑖²+𝐸sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²1_{{𝜃𝑖≤𝑇or𝜂𝑖≤𝑇}}. (3.7) By the Young inequality 𝑥𝑦≤(𝑥^𝑝/𝑝)+(𝑦^𝑞/𝑞), for any 𝑎,𝑏,𝑝,𝑞,𝛿>0,(1/𝑝)+(1/𝑞)=1, we have 𝑎𝑏≤𝑎𝛿^1/𝑝𝑏𝛿^1/𝑝≤𝑎𝛿^1/𝑝𝑝𝑝+𝑏^𝑞𝑞𝛿^𝑞/𝑝=𝑎^𝑝𝛿𝑝+𝑏^𝑞𝑞𝛿^𝑞/𝑝. (3.8) Thus for any 𝛿>0, we have 𝐸sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²1_{{𝜃𝑖≤𝑇or𝜂𝑖≤𝑇}}≤2𝛿𝑝𝐸sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|^𝑝+1−2/𝑝𝛿^2/(𝑝−2)𝑃𝜃_𝑖≤𝑇or𝜂_𝑖≤𝑇. (3.9) By condition (H4), we have 𝑃𝜃_𝑖≤𝑇=𝐸1_{{𝜃𝑖≤𝑇}}𝑦𝜃_𝑖^𝑝𝑖^𝑝≤1𝑖^𝑝𝐸sup_{0≤𝑡≤𝑇}𝑦(𝑡)^𝑝≤𝐾₂𝑖^𝑝. (3.10) Similarly, the result is 𝑃𝜂_𝑖≤𝐾≤𝑇²𝑖^𝑝. (3.11) So that 𝑃𝜃_𝑖≤𝑇or𝜂_𝑖≤𝑇≤𝑃𝜃_𝑖≤𝑇+𝑃𝜂_𝑖≤𝑇≤2𝐾₂𝑖^𝑝. (3.12) Using these bounds, then 𝐸sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|^𝑝≤2^𝑝−1𝐸sup_{0≤𝑡≤𝑇}|𝑥(𝑡)|^𝑝+𝐸sup_{0≤𝑡≤𝑇}𝑦(𝑡)^𝑝≤2^𝑝𝐾₂.𝐸sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²1_{{𝜃𝑖≤𝑇or𝜂𝑖≤𝑇}}≤2^𝑝+1𝛿𝐾₂𝑝+2𝑝−2𝐾₂𝑝𝛿^2/(𝑝−2)𝑖^𝑝. (3.13) By the definitions of 𝑥(𝑡) and 𝑦(𝑡), we have 𝑥𝑡∧𝜏_𝑖−𝑦𝑡∧𝜏_𝑖²≤2^{𝑡∧𝜏𝑖}₀𝑓(𝑥(𝑠),𝑥([𝑠])−𝑓(𝑧(𝑠),𝑧([𝑠]))𝑑𝑠²+2^{𝑡∧𝜏𝑖}₀𝑔(𝑥(𝑠),𝑥([𝑠])−𝑔(𝑧(𝑠),𝑧([𝑠]))𝑑𝐵(𝑠)². (3.14) Thus, for any 𝑡₁∈[0,𝑇]𝐸sup_{0≤𝑡≤𝑡1}𝑥𝑡∧𝜏_𝑖−𝑦𝑡∧𝜏_𝑖²≤2𝐸sup_{0≤𝑡≤𝑡1}^{𝑡∧𝜏𝑖}₀𝑓(𝑥(𝑠),𝑥([𝑠])−𝑓(𝑧(𝑠),𝑧([𝑠]))𝑑𝑠²+2𝐸sup_{0≤𝑡≤𝑡1}^{𝑡∧𝜏𝑖}₀𝑔(𝑥(𝑠),𝑥([𝑠])−𝑔(𝑧(𝑠),𝑧([𝑠]))𝑑𝐵(𝑠)². (3.15) By the Hölder inequality, condition (H1), and Lemma 3.1, one gets 𝐸sup_{0≤𝑡≤𝑡1}^{𝑡∧𝜏𝑖}₀𝑓(𝑥(𝑠),𝑥([𝑠])−𝑓(𝑧(𝑠),𝑧([𝑠]))𝑑𝑠²≤𝑇𝐸^{𝑡1∧𝜏𝑖}₀|𝑓𝑥(𝑠),𝑥([𝑠])−𝑓(𝑧(𝑠),𝑧([𝑠]))²𝑑𝑠=𝑇𝐸^𝑡1₀|𝑓𝑥𝑠∧𝜏_𝑖,𝑥𝑠∧𝜏_𝑖−𝑓𝑧𝑠∧𝜏_𝑖,𝑧𝑠∧𝜏_𝑖²𝑑𝑠≤𝑇𝐿_𝑖𝐸^𝑡1₀𝑥𝑠∧𝜏_𝑖−𝑧𝑠∧𝜏_𝑖²+𝑥𝑠∧𝜏_𝑖−𝑧𝑠∧𝜏_𝑖²𝑑𝑠≤2𝑇𝐿_𝑖^𝑡1₀𝐸𝑥𝑠∧𝜏_𝑖−𝑦𝑠∧𝜏_𝑖²+𝐸𝑥𝑠∧𝜏_𝑖−𝑦𝑠∧𝜏_𝑖²𝑑𝑠+2𝑇𝐿_𝑖^𝑡1₀𝐸𝑦𝑠∧𝜏_𝑖−𝑧𝑠∧𝜏_𝑖|²+𝐸𝑦𝑠∧𝜏_𝑖−𝑧𝑠∧𝜏_𝑖|²𝑑𝑠≤2𝑇𝐿_𝑖^𝑡1₀𝐸𝑥𝑠∧𝜏_𝑖−𝑦𝑠∧𝜏_𝑖²+𝐸𝑥𝑠∧𝜏_𝑖−𝑦𝑠∧𝜏_𝑖²𝑑𝑠+4𝑇²𝐿_𝑖𝐶₁𝑥₀,𝑖ℎ≤4𝑇𝐿_𝑖^𝑡1₀𝐸sup_{0≤𝜈≤𝑠∧𝜏𝑖}𝑥(𝜈)−𝑦(𝜈)²𝑑𝑠+4𝑇²𝐿_𝑖𝐶₁𝑥₀,𝑖ℎ. (3.16) Similarly, by the Burkhôlder-Davis-Gundy inequality, then 𝐸sup_{0≤𝑡≤𝑡1}^{𝑡∧𝜏𝑖}₀𝑔(𝑥(𝑠),𝑥([𝑠]))−𝑔(𝑧(𝑠),𝑧([𝑠]))𝑑𝐵(𝑠)²≤4𝐸^{𝑡∧𝜏𝑖}₀𝑔(𝑥(𝑠),𝑥([𝑠]))−𝑔(𝑧(𝑠),𝑧([𝑠]))²𝑑𝑠≤16𝐿_𝑖^𝑡1₀𝐸sup_{0≤𝜈≤𝑠∧𝜏𝑖}𝑥(𝜈)−𝑦(𝜈)²𝑑𝑠+16𝑇𝐿_𝑖𝐶₁𝑥₀,𝑖ℎ. (3.17) Substituting (3.16) and (3.17) into (3.15) gives 𝐸sup_{0≤𝑡≤𝑡1}𝑥𝑡∧𝜏_𝑖−𝑦𝑡∧𝜏_𝑖²≤8(𝑇+4)𝐿_𝑖^𝑡1₀𝐸sup_{0≤𝜈≤𝑠∧𝜏𝑖}𝑥(𝜈)−𝑦(𝜈)²𝑑𝑠+8𝑇(𝑇+4)𝐿_𝑖𝐶₁𝑥₀,𝑖ℎ. (3.18) By the Gronwall inequality, we must get 𝐸sup_{0≤𝑡≤𝑇}𝑥𝑡∧𝜏_𝑖−𝑦𝑡∧𝜏_𝑖²≤𝐶₂𝑥₀,𝑖ℎ, (3.19) where 𝐶₂(𝑥₀,𝑖)=8𝑇(𝑇+4)𝐿_𝑖𝐶₁(𝑥₀,𝑖)𝑒^{8𝑇(𝑇+4)𝐿𝑖}. So we have 𝐸sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²≤𝐸sup_{0≤𝑡≤𝑇}𝑒𝑡∧𝜏_𝑖²+𝐸sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²1_{{𝜃𝑖<𝑇or𝜂𝑖<𝑇}}≤𝐶₂𝑥₀2,𝑖ℎ+^𝑝+1𝛿𝐾₂𝑝+2𝑝−2𝐾₂𝑝𝛿^2/(𝑝−2)𝑖^𝑝. (3.20) Given any 𝜖>0, we can choose 𝛿 sufficiently small for 2^𝑝+1𝛿𝐾₂𝑝<𝜖3, (3.21) then choose 𝑖 sufficiently large for 2𝑝−2𝐾₂𝑝𝛿^2/(𝑝−2)𝑖^𝑝<𝜖3, (3.22) and finally choose ℎ sufficiently small, so that 𝐶₂𝑥₀𝜖,𝑖ℎ<3. (3.23) Thus 𝐸[sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²]<𝜖. The proof is completed.

4. Convergence of the Euler-Maruyama Method under Linear Growth Condition

We will show the strong convergence of the EM method on (2.1) under local Lipschitz condition and the linear growth condition. In the following we will show that the linear growth condition (H2) implies the bounded 𝑝th moment condition (H4).

Lemma 4.1. Under the linear growth conditions (H2), there exists a positive constant 𝐶₃ such that the solution of (2.1) satisfies 𝐸sup_{0≤𝑡≤𝑇}|𝑥(𝑡)|^𝑝≤𝐶₃1+𝑥₀^𝑝, (4.1) where 𝐶₃=𝐶₃(𝑝,𝑇,𝐾) is a constant independent of ℎ.

Proof. It follows from (2.2) that |𝑥(𝑡)|^𝑝=𝑥(0)+^𝑡₀𝑓(𝑥(𝑠),𝑥([𝑠]))𝑑𝑠+^𝑡₀𝑔(𝑥(𝑠),𝑥([𝑠]))𝑑𝐵(𝑠)^𝑝≤3^𝑝−1|𝑥(0)|^𝑝+^𝑡₀𝑓(𝑥(𝑠),𝑥([𝑠]))𝑑𝑠^𝑝+^𝑡₀𝑔(𝑥(𝑠),𝑥([𝑠]))𝑑𝐵(𝑠)^𝑝. (4.2) By the Hölder inequality, we obtain |𝑥(𝑡)|^𝑝≤3^𝑝−1|𝑥(0)|^𝑝+𝑇^𝑝−1𝑡₀𝑓(𝑥(𝑠),𝑥([𝑠]))^𝑝𝑑𝑠+^𝑡₀𝑔(𝑥(𝑠),𝑥([𝑠]))𝑑𝐵(𝑠)^𝑝. (4.3) This implies that, for any 0≤𝑡₁≤𝑇, 𝐸sup_{0≤𝑡≤𝑡1}|𝑥(𝑡)|^𝑝≤3^𝑝−1|𝑥(0)|^𝑝+𝑇^𝑝−1𝐸sup_{0≤𝑡≤𝑡1}^𝑡₀𝑓(𝑥(𝑠),𝑥([𝑠]))^𝑝𝑑𝑠+𝐸sup_{0≤𝑡≤𝑡1}^𝑡₀𝑔(𝑥(𝑠),𝑥([𝑠]))𝑑𝐵(𝑠)^𝑝. (4.4) By the Burkholder-Davis-Gundy inequality and the Hölder inequality, it is not difficult to show that 𝐸sup_{0≤𝑡≤𝑡1}|𝑥(𝑡)|^𝑝≤3^𝑝−1|𝑥(0)|^𝑝+𝑇^𝑝−1𝐸^𝑡1₀𝑓(𝑥(𝑠),𝑥([𝑠]))^𝑝𝑑𝑠+𝐶_𝑝𝑇^𝑝/2−1𝐸^𝑡1₀𝑔(𝑥(𝑠),𝑥([𝑠]))^𝑝𝑑𝑠, (4.5) where 𝐶_𝑝 is a constant. Note from the linear growth conditions that 𝐸sup_{0≤𝑡≤𝑡1}|𝑥(𝑡)|^𝑝≤3^𝑝−1|𝑥(0)|^𝑝+𝑇^𝑝−1𝐸^𝑡1₀𝐾^𝑝/21+|𝑥(𝑠)|²+|𝑥([𝑠])|^2𝑝/2𝑑𝑠+𝐶_𝑝𝑇^𝑝/2−1𝐸^𝑡1₀𝐾^𝑝/21+|𝑥(𝑠)|²+|𝑥([𝑠])|^2𝑝/2𝑑𝑠≤3^𝑝−1|𝑥(0)|^𝑝+3^{(3𝑝/2)−2}𝐾^𝑝/2𝑇^𝑝−1+𝐶_𝑝𝑇^𝑝/2−1×𝐸^𝑡1₀1+|𝑥(𝑠)|^𝑝+|𝑥([𝑠])|^𝑝𝑑𝑠≤3^𝑝−1|𝑥(0)|^𝑝+3^{(3𝑝/2)−2}𝐾^𝑝/2𝑇^𝑝+𝐶_𝑝𝑇^𝑝/2+2×3^{(3𝑝/2)−2}𝐾^𝑝/2𝑇^𝑝−1+𝐶_𝑝𝑇^𝑝/2−1𝐸sup_{0≤𝑢≤𝑠}|𝑥(𝑢)|^𝑝𝑑𝑠. (4.6) By the Gronwall inequality, we must get 𝐸sup_{0≤𝑡≤𝑇}|𝑥(𝑡)|^𝑝≤𝐶₃1+𝑥₀^𝑝, (4.7) where 𝐶₃=𝐶₃(𝑝,𝑇,𝐾) is a constant independent of ℎ.

The following lemma shows that the continuous Euler-Maruyama approximate solution has bounded 𝑝th moments.

Lemma 4.2. Under the linear growth conditions (H2), there exists a positive constant 𝐶₄ such that the continuous approximate solution of the Euler-Maruyama (2.10) satisfies 𝐸sup_{0≤𝑡≤𝑇}𝑦(𝑡)^𝑝≤𝐶₄1+𝑥₀^𝑝, (4.8) where 𝐶₄=𝐶₄(𝑝,𝑇,𝐾) is a constant independent of ℎ.

Proof. By the inequality |𝑎+𝑏+𝑐|²≤3|𝑎|²+3|𝑏|²+3|𝑐|² and (2.10), in the same way as Lemma 4.1, for any 0≤𝑡₁≤𝑇, we can obtain 𝐸sup_{0≤𝑡≤𝑡1}𝑦(𝑡)^𝑝≤3^𝑝−1𝑦(0)^𝑝+𝑇^𝑝−1𝐸^𝑡1₀𝑓(𝑧(𝑠),𝑧([𝑠]))^𝑝𝑑𝑠+𝐶_𝑝𝑇^(𝑝/2)−1𝐸^𝑡1₀𝑔(𝑧(𝑠),𝑧([𝑠]))^𝑝𝑑𝑠, (4.9) where 𝐶_𝑝 is a constant. Note from the linear growth conditions that 𝐸sup_{0≤𝑡≤𝑡1}𝑦(𝑡)^𝑝≤3^𝑝−1𝑦(0)^𝑝+𝑇^𝑝−1𝐸^𝑡1₀𝐾^𝑝/21+|𝑧(𝑠)|²+|𝑧([𝑠])|^2𝑝/2𝑑𝑠+𝐶_𝑝𝑇^𝑝/2−1𝐸^𝑡1₀𝐾^𝑝/21+|𝑧(𝑠)|²+|𝑧([𝑠])|^2𝑝/2𝑑𝑠≤3^𝑝−1𝑦(0)^𝑝+3^{(3𝑝/2)−2}𝐾^𝑝/2𝑇^𝑝−1+𝐶_𝑝𝑇^𝑝/2−1×𝐸^𝑡1₀1+|𝑧(𝑠)|^𝑝+|𝑧([𝑠])|^𝑝𝑑𝑠≤3^𝑝−1𝑦(0)^𝑝+3^{(3𝑝/2)−2}𝐾^𝑝/2𝑇^𝑝+𝐶_𝑝𝑇^𝑝/2+2×3^{(3𝑝/2)−2}𝐾^𝑝/2𝑇^𝑝−1+𝐶_𝑝𝑇^𝑝/2−1𝐸sup_{0≤𝑢≤𝑠}𝑦(𝑢)^𝑝𝑑𝑠. (4.10) By the Gronwall inequality, we must get 𝐸sup_{0≤𝑡≤𝑇}𝑦(𝑡)^𝑝≤𝐶₄1+𝑥₀^𝑝, (4.11) where 𝐶₄=𝐶₄(𝑝,𝑇,𝐾) is a constant independent of ℎ.

According to Theorem 3.2, we have the following theorem.

Theorem 4.3. Under the conditions (H1) and (H2), the EM approximate solution converges to the exact solution of (2.1) in the sense that lim_ℎ→0𝐸sup_{0≤𝑡≤𝑇}𝑦(𝑡)−𝑥(𝑡)²=0. (4.12)

5. A Motivating Example

In the above section, we give the strong convergence numerical solution of SEPCAs under the local Lipschitz condition (H1) and the linear growth condition (H2). However, there are many SEPCAs that do not satisfy the linear growth condition, consider the following SEPCA: 𝑑𝑥(𝑡)=−𝑥³(𝑡)+𝑥([𝑡])𝑑𝑡+sin𝑥²(𝑡)+𝑥([𝑡])𝑑𝐵(𝑡)∀𝑡≥0. (5.1) Clearly, the equation do not satisfy the linear growth condition (H2). But the example is analyzed under condition (H3) which covers many nonlinear SEPCAs. On the other hand, we have 𝑥−𝑥³+1+𝑦2sin𝑥²+𝑦²≤−𝑥⁴+𝑥𝑦+sin𝑥²²+𝑦²≤21+𝑥²+𝑦². (5.2) In other words, the equation satisfies condition (H3). Moreover, we also have 2𝑥^𝑇𝑓𝑥,𝑦+𝑔𝑥,𝑦²≤|𝑥|²+2𝐾1+|𝑥|²+𝑦²≤(1+2𝐾)1+|𝑥|²+𝑦². (5.3) We see clearly that (H3) follows from (H2). Therefore, The following result is more general than Theorem 3.2. Let us now turn to establish the convergence of the Euler-Maruyama method to (2.1) under the conditions (H1) and (H3).

6. Convergence of the Euler-Maruyama Method under Monotone Condition

In this section, we give the convergence of the EM method to (2.1) under the local Lipschitz condition (H1) and the monotone condition (H3). We prove the bounded 𝑝th moment property of the EM approximate solution and the exact solution to (2.1) under the monotone condition (H3).

Lemma 6.1. Under the monotone condition (H3), there exists a positive constant 𝐶₅ such that the solution of (2.1) satisfies 𝐸sup_{0≤𝑡≤𝑇}|𝑥(𝑡)|^𝑝∨𝐸sup_{0≤𝑡≤𝑇}𝑦(𝑡)^𝑝≤𝐶₅, (6.1) where 𝐶₅=𝐶₅(𝑝,𝑇,𝐾₁,𝑥₀) is a constant independent of ℎ.

Proof. By Itô formula, for all 𝑡≥0, we have 1+|𝑥(𝑡)|^2𝑝/2=1+|𝑥(0)|^2𝑝/2+^𝑡₀𝑝1+|𝑥(𝑡)|^{2(𝑝−2)/2}×𝑥(𝑡)^𝑇𝑓(𝑥(𝑠),𝑥([𝑠]))+𝑝−12𝑔(𝑥(𝑠),𝑥([𝑠]))²+𝑑𝑠_𝑡0𝑝1+|𝑥(𝑡)|^{2(𝑝−2)/2}𝑥(𝑠)^𝑇𝑔(𝑥(𝑠)𝑠(𝑠)≤|𝑥(0,𝑥([]))𝑑𝐵1+)|^2𝑝/2+^𝑡₀𝑝1+|𝑥(𝑡)|^{2(𝑝−2)/2}𝐾₁1+|𝑥(𝑠)|²+|𝑥([𝑠])|²+𝑑𝑠_𝑡0𝑝1+|𝑥(𝑡)|^{2(𝑝−2)/2}𝑥(𝑠)^𝑇𝑔(𝑥(𝑠),𝑥([𝑠]))𝑑𝐵(𝑠). (6.2) For any 𝑡₁∈[0,𝑇], we have 𝐸sup_{0≤𝑡≤𝑡1}1+|𝑥(𝑡)|^2𝑝/2≤1+|𝑥(0)|^2𝑝/2+𝐸sup_{0≤𝑡≤𝑡1}^𝑡₀𝑝1+|𝑥(𝑡)|^{2(𝑝−2)/2}𝐾₁1+|𝑥(𝑠)|²+|𝑥([𝑠])|²𝑑𝑠+𝐸sup_{0≤𝑡≤𝑡1}^𝑡₀𝑝1+|𝑥(𝑡)|^{2(𝑝−2)/2}𝑥(𝑠)^𝑇𝑔(𝑥(𝑠),𝑥([𝑠]))𝑑𝐵(𝑠). (6.3) By the Burkholder-Davis-Gundy inequality and Exercise 2.5 in [8], it is not difficult to show that 𝐸sup_{0≤𝑡≤𝑡1}^𝑡₀𝑝1+|𝑥(𝑡)|^{2(𝑝−2)/2}𝑥(𝑠)^𝑇𝑔(𝑥(𝑠),𝑥([𝑠]))𝑑𝐵(𝑠)≤3𝐸^𝑡1₀𝑝²1+|𝑥(𝑡)|^2𝑝−2|𝑥(𝑠)|²𝑔(𝑥(𝑠),𝑥([𝑠]))²𝑑𝑠^1/2≤3𝐸sup_{0≤𝑡≤𝑡1}1+|𝑥(𝑡)|^2𝑝/2𝑡1₀𝑝²1+|𝑥(𝑡)|^{2(𝑝−2)/2}𝑔(𝑥(𝑠),𝑥([𝑠]))²𝑑𝑠^1/2≤3𝐸sup_{0≤𝑡≤𝑡1}1+|𝑥(𝑡)|^2𝑝/2𝑡1₀𝑝²𝐾₁1+|𝑥(𝑡)|^{2(𝑝−2)/2}1+|𝑥(𝑠)|²+|𝑥([𝑠])|²𝑑𝑠^1/2≤0.5𝐸sup_{0≤𝑡≤𝑡1}1+|𝑥(𝑡)|^2𝑝/2+4.5𝐸^𝑡1₀𝑝²𝐾₁1+|𝑥(𝑡)|^{2(𝑝−2)/2}1+|𝑥(𝑠)|²+|𝑥([𝑠])|²𝑑𝑠. (6.4) Substituting (6.4) into (6.3) and using the Hölder inequality 𝐸sup_{0≤𝑡≤𝑡1}1+|𝑥(𝑡)|^2𝑝/2≤21+|𝑥(0)|^2𝑝/2+2𝑝𝐾₁1+4.5𝑝𝐸^𝑡1₀1+|𝑥(𝑡)|^{2(𝑝−2)/2}1+|𝑥(𝑠)|²+|𝑥([𝑠])|²|𝑥(0𝑑𝑠≤21+)|^2𝑝/2+2𝑝𝐾₁1+4.5𝑝𝐸^𝑡1₀1+𝑥(𝑡)|^2𝑝/2𝑑𝑠^{(𝑝−2)/𝑝}×𝐸^𝑡1₀1+|𝑥(𝑠)|²+|𝑥([𝑠])|^2𝑝/2𝑑𝑠^2/𝑝≤21+|𝑥(0)|^2𝑝/2+2𝑝𝐾₁1+4.5𝑝𝑇𝐸sup_{0≤𝑡≤𝑡1}1+|𝑥(𝑡)|^{2𝑝/2(𝑝−2)/𝑝}×𝐸^𝑡1₀1+|𝑥(𝑠)|²+|𝑥([𝑠])|^2𝑝/2𝑑𝑠^2/𝑝. (6.5) So we obtain 𝐸sup_{0≤𝑡≤𝑡1}1+|𝑥(𝑡)|^2𝑝/2≤21+|𝑥(0)|^2𝑝/2+2𝑝𝐾₁1+4.5𝑝𝑇^{(𝑝−2)/𝑝}(𝐸^𝑡1₀1+|𝑥(𝑠)|²+|𝑥([𝑠])|²)^𝑝/2𝑑𝑠^2/𝑝𝑝/2≤2^(𝑝/2)−121+|𝑥(0)|^{2𝑝/2𝑝/2}+2^(𝑝/2)−12𝑝𝐾₁1+4.5𝑝𝑇^{(𝑝−2)/𝑝}𝐸^𝑡1₀1+|𝑥(𝑠)|²+|𝑥([𝑠])|^2𝑝/2𝑑𝑠^2/𝑝𝑝/2≤2^𝑝−11+|𝑥(0)|^2𝑝2/4+2^𝑝−1𝑝𝐾₁1+4.5𝑝𝑇^{(𝑝−2)/𝑝𝑝/2}𝐸^𝑡1₀1+|𝑥(𝑠)|²+|𝑥([𝑠])|^2𝑝/2𝑑𝑠≤2^𝑝−11+|𝑥(0)|^2𝑝2/4+2^{(3𝑝/2)−1}𝑝𝐾₁1+4.5𝑝𝑇^{(𝑝−2)/𝑝𝑝/2𝑡1}₀𝐸sup_{0≤𝑢≤𝑠}1+|𝑥(𝑢)|^2𝑝/2𝑑𝑠. (6.6) By the Gronwall inequality, we must get 𝐸sup_{0≤𝑡≤𝑇}|𝑥(𝑡)|^𝑝≤𝐶₅, (6.7) where 𝐶₅=𝐶₅(𝑝,𝑇,𝐾,𝑥₀) is a constant independent of ℎ. Similarly, we can show that 𝐸sup_{0≤𝑡≤𝑇}𝑦(𝑡)^𝑝≤𝐶₅. (6.8) The proof is completed.

According to Theorem 3.2, we obtain the following.

Theorem 6.2. Under the conditions (H1) and (H3), the EM approximate solution converges to the exact solution of (2.1) in the sense that lim_ℎ→0𝐸sup_{0≤𝑡≤𝑇}𝑦(𝑡)−𝑥(𝑡)²=0. (6.9)

Acknowledgments

The financial support from the National Natural Science Foundation of China (no. 11071050) and the Youths’ Key Projects of Heilongjiang Provincial Education Department of P. R. China (no. 1155G001) is gratefully acknowledged.