Numerical Solutions of Stochastic Differential Equations with Piecewise Continuous Arguments under Khasminskii-Type Conditions

Abstract

The main purpose of this paper is to investigate the convergence of the Euler method to stochastic differential equations with piecewise continuous arguments (SEPCAs). The classical Khasminskii-type theorem gives a powerful tool to examine the global existence of solutions for stochastic differential equations (SDEs) without the linear growth condition by the use of the Lyapunov functions. However, there is no such result for SEPCAs. Firstly, this paper shows SEPCAs which have nonexplosion global solutions under local Lipschitz condition without the linear growth condition. Then the convergence in probability of numerical solutions to SEPCAs under the same conditions is established. Finally, an example is provided to illustrate our theory.

1. Introduction

Stochastic modeling has come to play an important role in many branches of science and industry. Such models have been used with great success in a variety of application areas, including biology, epidemiology, mechanics, economics, and finance. Most stochastic differential equations are nonlinear and cannot be solved explicitly, but it is very important to research the existence and uniqueness of solution of stochastic differential equations. Many authors have studied the problem of SDEs. The classical existence-and-uniqueness theorem requires the coefficients 𝑓(𝑥(𝑡)) and 𝑔(𝑥(𝑡)) to satisfy the local Lipschitz condition and the linear growth condition (see [1]). However, there are many SDEs that do not satisfy the linear growth condition, so more general conditions have been introduced to replace theirs. Khasminskii [2] has studied Khasminskii's test for SDEs which are the most powerful conditions. Similarly, the classical existence-and-uniqueness theorem for stochastic differential delay equations (SDDEs) requires the coefficients 𝑓(𝑥(𝑡),𝑥(𝑡−𝜏)) and 𝑔(𝑥(𝑡),𝑥(𝑡−𝜏)) to satisfy the local Lipschitz condition and the linear growth condition (see [3–6]). Mao [7] has proved Khasminskii-type theorem, and this is a natural generalization of the classical Khasminskii test.

In recent years, differential equations with piecewise continuous arguments (EPCAs) had attracted much attention, and many useful conclusions were obtained. These systems have applications in certain biomedical models, control systems with feedback delay in the work of L. Cooke and J. Wiener [8]. The general theory and basic results for EPCAs have by now been thoroughly investigated in the book of Wiener [9]. 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. In this paper, we give the Khasminskii-type theorems for SEPCAs, which shows that SEPCAs have nonexplosion global solutions under local Lipschitz condition without the linear growth condition.

On the other hand, there is in general no explicit solution to an SEPCA, whence numerical solutions are required in practice. Numerical solutions to SDEs have been discussed under the local Lipschitz condition and the linear growth condition by many authors (see [5]). Mao [10] gives the convergence in probability of numerical solutions to SDDEs under Khasminskii-Type conditions. Dai and Liu [11] give the mean-square stability of the numerical solutions of linear stochastic differential equations with piecewise continuous arguments. However, SEPCAs do not have the convergence results. The other 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 Euler method. In Section 3, we obtain the existence and uniqueness of solution to stochastic differential equations with piecewise continuous arguments under Khasminskii-type conditions. Then the convergence in probability of numerical solutions to stochastic differential equations with piecewise continuous arguments under the same conditions is established. Finally, an example is provided to illustrate our theory.

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<∞ almost surely and ∫^𝑇0|𝑓(𝑡)|²𝑑𝑡<∞ almost surely, respectively. For any 𝑎,𝑏∈𝑅 with 𝑎<𝑏, denote 𝐶([𝑎,𝑏];𝑅^𝑛) as the family of continuous functions 𝜙 from [𝑎,𝑏] to 𝑅^𝑛 with the norm ‖𝜙‖=sup_{𝑎≤𝜃≤𝑏}|𝜙(𝜃)|. Denote 𝐶^𝑏ℱ_𝑡([𝑎,𝑏];𝑅^𝑛) as 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. Let 𝐶²(𝑅^𝑛;𝑅₊) denote the family of all continuous nonnegative functions 𝑉(𝑥) defined on 𝑅^𝑛 such that they are continuously twice differentiable in 𝑥. Given 𝑉∈𝐶²(𝑅^𝑛;𝑅₊), we define the operator 𝐿𝑉∶𝑅^𝑛×𝑅^𝑛→𝑅 by 𝐿𝑉(𝑥,𝑦)=𝑉_𝑥(𝑥)𝑓(𝑥,𝑦)+12trace𝑔^𝑇(𝑥,𝑦)𝑉_𝑥𝑥(𝑥)𝑔(𝑥,𝑦),(2.1) where 𝑉_𝑥(𝑥)=𝜕𝑉(𝑥)𝜕𝑥₁,…,𝜕𝑉(𝑥)𝜕𝑥_𝑛,𝑉^𝑥𝑥(𝑥)=𝜕²𝑉(𝑥)𝜕𝑥_𝑖𝜕𝑥_𝑗_𝑛×𝑛.(2.2) Let us emphasize that 𝐿𝑉 is defined on 𝑅^𝑛×𝑅^𝑛, while 𝑉 is only defined on 𝑅^𝑛.

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,(2.3) with initial data 𝑥(0)=𝑐₀, where 𝑓∶𝑅^𝑛×𝑅^𝑛→𝑅^𝑛,𝑔∶𝑅^𝑛×𝑅^𝑛→𝑅^𝑛×𝑑,𝑐₀ 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.(2.4) Moreover, we also require the coefficients 𝑓(𝑥(𝑡),𝑥([𝑡])) and 𝑔(𝑥(𝑡),𝑥([𝑡])) 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 |||𝑓(𝑥,𝑦)−𝑓⁻𝑥,𝑦⁻|||²∨|||𝑔(𝑥,𝑦)−𝑔𝑥⁻,𝑦⁻|||²≤𝐿_𝑖|||𝑥−⁻𝑥|||²+|||𝑦−𝑦⁻|||²,(2.5) 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 ||𝑓(𝑥,𝑦)||²∨||𝑔(𝑥,𝑦)||²≤𝐾1+|𝑥|²+||𝑦||²,(2.6) for all (𝑥,𝑦)∈𝑅^𝑛×𝑅^𝑛.(H3) There are a function 𝑉∈𝐶²(𝑅^𝑛;𝑅₊) and a positive constant 𝛼 such that 𝐿𝑉(𝑥,𝑦li)m≤in𝛼f_|𝑥|→∞(1𝑉+(𝑥𝑉)=(𝑥)∞+,𝑉(𝑦)),((22..78)) for all (𝑥,𝑦)∈𝑅^𝑛×𝑅^𝑛.

Let us first give the definition of the solution.

Definition 2.1 (see [11]). An 𝑅^𝑛-valued stochastic process {𝑥(𝑡)} is called a solution of (2.3) 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 (2.4) 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.(2.9)Let ℎ=1/𝑚 be a given stepsize with integer 𝑚≥1, and let the gridpoints 𝑡_𝑛 be defined by 𝑡_𝑛=𝑛ℎ(𝑛=0,1,2,…). For simplicity, we assume that 𝑇=𝑁ℎ. We consider the Euler-Maruyama method to (2.3), 𝑦_𝑛+1=𝑦_𝑛+𝑓𝑦_𝑛,𝑦^ℎ([𝑛ℎ])ℎ+𝑔𝑦_𝑛,𝑦^ℎ([𝑛ℎ])Δ𝐵_𝑛,(2.10) 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 (2.3) leads to a numerical process of the following type: 𝑦_{𝑘𝑚+𝑙+1}=𝑦_{𝑘𝑚+𝑙}+𝑓𝑦_{𝑘𝑚+𝑙},𝑦_𝑘𝑚ℎ+𝑔𝑦_{𝑘𝑚+𝑙},𝑦_𝑘𝑚Δ𝐵_{𝑘𝑚+𝑙},(2.11) 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),(2.12) 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}.

3. Convergence in Probability of the Euler-Maruyama Method

In this section, we concentrate on (2.3) under the local Lipschitz condition (H1) without the linear growth condition (H2) to establish the generalized existence and uniqueness theorem for stochastic differential equations with piecewise continuous arguments. We then give the convergence in probability of the EM method to (2.3) under the local Lipschitz condition (H1) and some additional conditions (H3).

Theorem 3.1. Under the conditions (H1) and (H3), there is a unique global solution 𝑥(𝑡) to (2.3) with initial data 𝑥(0)=𝑐₀ on 𝑡∈[0,∞). Moreover, the solution has the property that 𝐸𝑉(𝑥(𝑡))<∞,forany𝑡≥0.(3.1)

Proof. Applying the standard truncation technique to (2.3), we obtain the unique maximal local solution 𝑥(𝑡) exists on [0,𝜂_𝑒) under the local Lipschitz condition in a similar way as the proof of [10, Theorem 3.15, page 91], where 𝜂_𝑒 is the explosion time. For each integer 𝑖≥|𝑐₀|, define the stopping time 𝜂_𝑖=inf𝑡∈0,𝜂_𝑒∶||𝑥(𝑡)||≥𝑖.(3.2) Clearly, 𝜂_𝑖 is increasing as 𝑖→∞. We denote that 𝜂_∞=lim_𝑖→∞𝜂_𝑖 and inf∅=∞. Hence, 𝜂_∞≤𝜂_𝑒  almost surely. If we can obtain 𝜂_∞=∞  almost surely, then 𝜂_𝑒=∞  almost surely.
In what follows, we will prove 𝜂_∞=∞  almost surely and assertion (3.1). By the Itô formula and condition (2.8), we derive that 𝑑𝑉(𝑥(𝑡))=≤𝐿𝛼𝑉[1(+𝑥𝑉(𝑡)(,𝑥𝑥((𝑡[)𝑡)]+))𝑑𝑉𝑡(+𝑥𝑉([_𝑥𝑡]()𝑥)](𝑑𝑡)𝑡)+𝑔(𝑉𝑥_𝑥(𝑡),𝑥([𝑡]))𝑑𝐵(𝑡)(𝑥(𝑡))𝑔(𝑥(𝑡),𝑥([𝑡]))𝑑𝐵(𝑡),(3.3) for 0≤𝑡<𝜂_∞. Now, for 𝑡₁∈[0,1), we can integrate both sides of (3.3) from 0 to 𝜂_𝑖∧𝑡₁, 𝑉𝑥𝜂_𝑖∧𝑡₁≤𝑉+(𝛼𝑥(_{𝜂𝑖∧𝑡10}0))+𝛼[1+𝑉(𝑥(0))]𝑉(𝑥(𝑡))𝑑𝑡+^{𝜂𝑖∧𝑡1}₀𝑉_𝑥(𝑥(𝑡))𝑔(𝑥(𝑡),𝑥([𝑡]))𝑑𝐵(𝑡).(3.4) We take the expectations in both sides of (3.4), 𝐸𝑉𝑥𝜂_𝑖∧𝑡₁≤≤𝑉𝛽₁(𝑥(0))+𝛼[1+𝑉(𝑥(0))]+𝛼𝐸^{𝜂𝑖∧𝑡1}₀𝑉(𝑥(𝑡))𝑑𝑡+𝛼𝐸^{𝜂𝑖∧𝑡1}₀𝑉(𝑥(𝑡))𝑑𝑡,(3.5) where 𝛽₁=𝑉(𝑥(0))+𝛼[1+𝑉(𝑥(0))]=(1+𝛼)𝑉𝑐₀+𝛼<∞.(3.6) It is easy to compute 𝐸𝑉𝑥𝜂_𝑖∧𝑡₁≤=𝛽𝛽₁₁+𝛼𝐸^{𝜂𝑖∧𝑡1}₀𝑉(𝑥(𝑡))𝑑𝑡+𝛼^𝑡1₀𝐸𝑉𝑥𝜂_𝑖∧𝑡𝑑𝑡.(3.7) Now the Gronwall inequality yields that 𝐸𝑉𝑥𝜂_𝑖∧𝑡₁≤𝛽₁𝑒^𝛼𝑡1≤𝛽₁𝑒^𝛼,0≤𝑡₁<1.(3.8) So we have 𝐸𝑉𝑥𝜂_𝑖∧1=lim_𝑡1→1𝐸𝑉𝑥𝜂_𝑖∧𝑡₁≤𝛽₁𝑒^𝛼.(3.9) Defining 𝛾_𝑖=inf_{|𝑥|≥𝑖}𝑉(𝑥),∀𝑖≥||𝑐₀||,(3.10) denoting 𝐼_𝐴 as the indicator function of a set 𝐴, we compute 𝛽₁𝑒^𝛼≥𝐸𝑉𝑥𝜂_𝑖∧1≥𝐸𝑉𝑥𝜂_𝑖𝐼_{{𝜂𝑖≤1}}≥𝛾_𝑖𝑃𝜂_𝑖≤1.(3.11) Letting 𝑖→∞, we have that 𝑃(𝜂_∞≤1)=0, namely, 𝑃𝜂_∞>1=1.(3.12) By (3.8) and (3.12), 𝐸𝑉𝑥𝑡₁≤𝛽₁𝑒^𝛼,0≤𝑡₁≤1.(3.13) Now let us prove 𝜂_∞>2, for 𝑡₂∈[1,2), and we can integrate both sides of (3.3) from 1 to 𝜂_𝑖∧𝑡₂ and take the expectations 𝐸𝑉𝑥𝜂_𝑖∧𝑡₂≤≤𝐸𝛽₂𝑉(𝑥(1))+𝛼[1+𝐸𝑉(𝑥(1))]+𝛼𝐸^{𝜂𝑖∧𝑡2}₁𝑉(𝑥(𝑡))𝑑𝑡+𝛼^𝑡2₁𝐸𝑉𝑥𝜂_𝑖∧𝑡𝑑𝑡,(3.14) where 𝛽₂≤𝛽₁𝑒^𝛼+𝛼1+𝛽₁𝑒^𝛼<∞.(3.15) Now the Gronwall inequality yields that 𝐸𝑉𝑥𝜂_𝑖∧𝑡₂≤𝛽₂𝑒^{𝛼(𝑡2−1)}≤𝛽₂𝑒^𝛼,1≤𝑡₂<2.(3.16) Hence, we have 𝐸𝑉𝑥𝜂_𝑖∧2=lim_𝑡1→2𝐸𝑉𝑥𝜂_𝑖∧𝑡₂≤𝛽₂𝑒^𝛼.(3.17) By (3.10) and (3.17), we compute 𝛽₂𝑒^𝛼≥𝐸𝑉𝑥𝜂_𝑖∧2≥𝐸𝑉𝑥𝜂_𝑖𝐼_{{𝜂𝑖≤2}}≥𝛾_𝑖𝑃𝜂_𝑖≤2.(3.18) Letting 𝑖→∞, we have that 𝑃(𝜂_∞≤2)=0, namely, 𝑃𝜂_∞>2=1.(3.19) From (3.16) and (3.19), we yield 𝐸𝑉𝑥𝑡₂≤𝛽₂𝑒^𝛼,1≤𝑡₂≤2.(3.20) Repeating this procedure, we can show that, for any integer 𝑗≥1,𝜂_∞>𝑗  almost surely, 𝐸𝑉𝑥𝑡_𝑗≤𝛽_𝑗𝑒^𝛼,𝑗−1≤𝑡_𝑗≤𝑗,(3.21) where 𝛽_𝑗≤𝛽_𝑗−1𝑒^𝛼+𝛼1+𝛽_𝑗−1𝑒^𝛼<∞.(3.22) By (3.13), (3.20), and (3.21), we obtain 𝐸𝑉(𝑥(𝑡))≤𝛽_𝑗𝑒^𝛼<∞,0≤𝑡≤𝑗.(3.23) Therefore, we must have 𝜂_∞=∞ almost surely as well as the required assertion (3.1). The proof is completed.

Theorem 3.2. Under the conditions (H1) and (H3), if 𝜀∈(0,1) and 𝑇>0, then there exists a sufficiently large integer 𝑖̂, dependent on 𝜀 and 𝑇 such that 𝑃𝜂_𝑖≤𝑇≤𝜀,∀𝑖≥𝑖.̂(3.24)

Proof. By Theorem 3.1, we have 𝐸𝑉𝑥𝜂_𝑖∧𝑡≤𝛽_𝑗𝑒^𝛼<∞,0≤𝑡≤𝑗.(3.25) Choose 𝑗 large enough for 𝑗>𝑇. From (3.25), we get 𝐸𝑉𝑥𝜂_𝑖∧([𝑇]+1)≤𝛽_[𝑇]+1𝑒^𝛼<∞.(3.26) It follows from (3.10) and (3.26) that 𝛽_[𝑇]+1𝑒^𝛼≥≥𝐸𝐸𝑉𝑉𝑥𝑥𝜂_𝑖𝜂∧_𝑖([𝑇]+1)≥𝛾_𝑖𝐼_{{𝜂𝑖≤[𝑇]+1}}𝑃𝜂_𝑖≤[𝑇]+1,(3.27) while by (H3), 𝛾_𝑖→∞ as 𝑖→∞. Thus, there is a sufficiently large integer 𝑖̂ such that 𝛾_𝑖≥𝛽^[𝑇]+1𝑒^𝛼𝜀,∀𝑖≥𝑖.̂(3.28) Therefore, we get that 𝑃𝜂_𝑖≤𝑇≤𝑃𝜂_𝑖≤[𝑇]+1≤𝛽^[𝑇]+1𝑒^𝛼𝛾_𝑖<𝜀.(3.29) The proof is completed.

The following lemma shows that both 𝑦(𝑡) and 𝑧(𝑡) are close to each other.

Lemma 3.3. Under the condition (H1), let 𝑇>0 be arbitrary. Then 𝐸sup_{0≤𝑡≤𝑇∧𝜃𝑖}||𝑦(𝑡)−𝑧(𝑡)||²≤𝐶¹(𝑖)ℎ^1/2,(3.30) where 𝐶₁(𝑖)=4(2𝑖²𝐿_𝑖+|𝑓(0,0)|²∨|𝑔(0,0)|²)(1+(16√3/3)𝑑(𝑇+1)^1/2).

Proof. For 𝑡∈[0,𝑇∧𝜃_𝑖), there are two integers 𝑘 and 𝑙 such that 𝑡∈[𝑡_{𝑘𝑚+𝑙},𝑡_{𝑘𝑚+𝑙+1}). So we compute ||𝑦(𝑡)−𝑧(𝑡)||²=||||^𝑡𝑡_{𝑘𝑚+𝑙}𝑓(𝑧(𝑠),𝑧([𝑠]))𝑑𝑠+^𝑡𝑡_{𝑘𝑚+𝑙}𝑔(𝑧(𝑠),𝑧([𝑠]))𝑑𝐵(𝑠)||||²=||||^𝑡𝑡_{𝑘𝑚+𝑙}𝑓𝑦_{𝑘𝑚+𝑙},𝑦_𝑘𝑚𝑑𝑠+^𝑡𝑡_{𝑘𝑚+𝑙}𝑔𝑦_{𝑘𝑚+𝑙},𝑦_𝑘𝑚𝑑𝐵(𝑠)||||²=||𝑓𝑦_{𝑘𝑚+𝑙},𝑦_𝑘𝑚𝑡−𝑡_{𝑘𝑚+𝑙}+𝑔𝑦_{𝑘𝑚+𝑙},𝑦_𝑘𝑚𝐵(𝑡)−𝐵𝑡_{𝑘𝑚+𝑙}||²≤2||𝑓𝑦_{𝑘𝑚+𝑙},𝑦_𝑘𝑚||²ℎ²+2||𝑔𝑦_{𝑘𝑚+𝑙},𝑦_𝑘𝑚||²||𝐵(𝑡)−𝐵𝑡_{𝑘𝑚+𝑙}||²,(3.31) since ||𝑓𝑦_{𝑘𝑚+𝑙},𝑦_𝑘𝑚||²≤2||𝑓𝑦_{𝑘𝑚+𝑙},𝑦_𝑘𝑚−𝑓(0,0)||²+2||𝑓(0,0)||²≤2𝐿_𝑖||𝑦^{𝑘𝑚+𝑙}||²+||𝑦_𝑘𝑚||²+2||𝑓(0,0)||²≤22𝑖²𝐿_𝑖+||𝑓(0,0)||².(3.32) Similarly, we obtain that ||𝑔𝑦_{𝑘𝑚+𝑙},𝑦_𝑘𝑚||²≤22𝑖²𝐿_𝑖+||𝑔(0,0)||².(3.33) Substituting (3.32) and (3.33) into (3.31) gives ||𝑦(𝑡)−𝑧(𝑡)||²≤𝐶ℎ²+||𝐵(𝑡)−𝐵𝑡_{𝑘𝑚+𝑙}||²,(3.34) where 𝐶=4(2𝑖²𝐿_𝑖+|𝑓(0,0)|²∨|𝑔(0,0)|²). Let 𝑛_𝑡=𝑘𝑚+𝑙 for 𝑡∈[𝑡_{𝑘𝑚+𝑙},𝑡_{𝑘𝑚+𝑙+1}), then we have that 𝐸sup_{0≤𝑡≤𝑇∧𝜃𝑖}||𝐵(𝑡)−𝐵𝑡_𝑛𝑡||²≤^𝑑_𝑖=1𝐸sup_{0≤𝑡≤𝑇∧𝜃𝑖}||𝐵_𝑖(𝑡)−𝐵_𝑖𝑡_𝑛𝑡||²≤_𝑑_𝑖=1𝐸sup_{𝑢=0,1,2,…,𝑁}sup_{𝑡𝑢≤𝑡≤𝑡𝑢+1∧𝑇}||𝐵_𝑖(𝑡)−𝐵_𝑖𝑡_𝑢||²≤_𝑑_𝑖=1𝐸sup_{𝑢=0,1,2,…,𝑁}sup_{𝑡𝑢≤𝑡≤𝑡𝑢+1∧𝑇}||𝐵_𝑖(𝑡)−𝐵_𝑖𝑡_𝑢||⁴^1/2,(3.35) while by the Doob martingale inequality, we have 𝐸sup_{𝑢=0,1,2,…,𝑁}sup_{𝑡𝑢≤𝑡≤𝑡𝑢+1∧𝑇}||𝐵_𝑖(𝑡)−𝐵_𝑖𝑡_𝑢||⁴≤^𝑁_𝑢=0𝐸sup_{𝑡𝑢≤𝑡≤𝑡𝑢+1∧𝑇}||𝐵_𝑖(𝑡)−𝐵_𝑖𝑡_𝑢||⁴≤43^4𝑁_𝑢=0𝐸||𝐵_𝑖𝑡_𝑢+1∧𝑇−𝐵_𝑖𝑡_𝑢||⁴≤25627^𝑁_𝑢=0ℎ²≤25627(𝑇+1)ℎ.(3.36) Substituting (3.36) into (3.35) yields 𝐸sup_{0≤𝑡≤𝑇∧𝜃𝑖}||𝐵(𝑡)−𝐵𝑡_𝑛𝑡||²≤^𝑑_𝑖=122576(𝑇+1)ℎ^1/2=16√39𝑑(𝑇+1)^1/2ℎ^1/2.(3.37) Thus, we obtain 𝐸sup_{0≤𝑡≤𝑇∧𝜃𝑖}||𝑦(𝑡)−𝑧(𝑡)||²≤𝐶ℎ²+𝐶16√39𝑑(𝑇+1)^1/2ℎ^1/2≤𝐶1+16√33𝑑(𝑇+1)^1/2ℎ^1/2≤𝐶₁(𝑖)ℎ^1/2,(3.38) where 𝐶₁(𝑖)=4(2𝑖²𝐿_𝑖+|𝑓(0,0)|²∨|𝑔(0,0)|²)(1+(16√3/3)𝑑(𝑇+1)^1/2). The proof is completed.

Lemma 3.4. Under the condition (H1), for any 𝑇>0, there exists a positive constant 𝐶₂(𝑖) dependent on 𝑖 and independent of ℎ such that 𝐸sup_{0≤𝑡≤𝑇}||𝑥𝑡∧𝜂_𝑖∧𝜃_𝑖−𝑦𝑡∧𝜂_𝑖∧𝜃_𝑖||²≤𝐶²(𝑖)ℎ^1/2,(3.39) where 𝐶₂(𝑖)=8𝑇(𝑇+4)𝐿_𝑖𝐶₁(𝑖)𝑒^{8𝑇(𝑇+4)𝐿𝑖}.

Proof. It follows from (2.4) and (2.12) that ||𝑥𝑡∧𝜂_𝑖∧𝜃_𝑖−𝑦𝑡∧𝜂_𝑖∧𝜃_𝑖||²≤2||||^{𝑡∧𝜂𝑖∧𝜃𝑖}₀𝑓(𝑥(𝑠),𝑥([𝑠]))−𝑓(𝑧(𝑠),𝑧([𝑠]))𝑑𝑠||||²+2||||^{𝑡∧𝜂𝑖∧𝜃𝑖}₀𝑔(𝑥(𝑠),𝑥([𝑠]))−𝑔(𝑧(𝑠),𝑧([𝑠]))𝑑𝐵(𝑠)||||².(3.40) By the Hölder inequality, we obtain ||𝑥𝑡∧𝜂_𝑖∧𝜃_𝑖−𝑦𝑡∧𝜂_𝑖∧𝜃_𝑖||²≤+2𝑇2||||^𝑡0_𝑡0||𝑓𝑥𝑠∧𝜂_𝑖∧𝜃_𝑖,𝑥𝑠∧𝜂_𝑖∧𝜃_𝑖−𝑓𝑧𝑠∧𝜂_𝑖∧𝜃_𝑖,𝑧𝑠∧𝜂_𝑖∧𝜃_𝑖||²𝑑𝑠𝑔𝑥𝑠∧𝜂_𝑖∧𝜃_𝑖,𝑥𝑠∧𝜂_𝑖∧𝜃_𝑖−𝑔𝑧𝑠∧𝜂_𝑖∧𝜃_𝑖,𝑧𝑠∧𝜂_𝑖∧𝜃_𝑖𝑑𝐵(𝑠)||||².(3.41) This implies that for any 0≤𝑡₁≤𝑇, 𝐸su≤p_{0≤𝑡≤𝑡1}2𝑇||𝑥𝐸s𝑡u∧p𝜂^𝑖_{0≤𝑡≤𝑡1}∧𝜃_𝑖−𝑦𝑡∧𝜂_𝑖∧𝜃_𝑖||²^𝑡0−||𝑓𝑓𝑥𝑧𝑠𝑠∧∧𝜂𝜂_𝑖𝑖∧𝜃_𝑖,𝑥𝑠∧𝜂_𝑖∧𝜃_𝑖+2𝐸sup_{0≤𝑡≤𝑡1}∧𝜃_𝑖,𝑧𝑠∧𝜂_𝑖∧𝜃_𝑖||²𝑑𝑠||||^𝑡0−𝑔𝑔𝑧𝑥𝑠𝑠∧∧𝜂_𝑖𝜂_𝑖∧𝜃_𝑖,𝑥𝑠∧𝜂_𝑖∧𝜃_𝑖∧𝜃_𝑖,𝑧𝑠∧𝜂_𝑖∧𝜃_𝑖𝑑𝐵(𝑠)||||².(3.42) By Doob martingale inequality, it is not difficult to show that 𝐸su≤p_{0≤𝑡≤𝑡1}2𝑇||𝑥𝐸𝑡_𝑡10∧𝜂^𝑖∧𝜃_𝑖−𝑦𝑡∧𝜂_𝑖∧𝜃_𝑖||²+8𝐸_𝑡10||𝑓𝑥𝑠∧𝜂_𝑖∧𝜃_𝑖,𝑥𝑠∧𝜂_𝑖∧𝜃_𝑖−𝑓𝑧𝑠∧𝜂_𝑖∧𝜃_𝑖,𝑧𝑠∧𝜂_𝑖∧𝜃_𝑖||²𝑑𝑠||𝑔𝑥𝑠∧𝜂_𝑖∧𝜃_𝑖,𝑥𝑠∧𝜂_𝑖∧𝜃_𝑖−𝑔𝑧𝑠∧𝜂_𝑖∧𝜃_𝑖,𝑧𝑠∧𝜂_𝑖∧𝜃_𝑖||²𝑑𝑠.(3.43) Note from (H1) and Lemma 3.3 that 𝐸^𝑡1₀≤||𝑓𝐿_𝑖𝑥𝑠∧𝜂_𝑖∧𝜃_𝑖,𝑥𝑠∧𝜂_𝑖∧𝜃_𝑖−𝑓𝑧𝑠∧𝜂_𝑖∧𝜃_𝑖,𝑧𝑠∧𝜂_𝑖∧𝜃_𝑖||²𝑑𝑠≤2𝐿𝐸_𝑖^𝑡1₀||𝑥𝑠∧𝜂_𝑖∧𝜃_𝑖−𝑧𝑠∧𝜂_𝑖∧𝜃_𝑖||²+||𝑥𝑠∧𝜂_𝑖∧𝜃_𝑖−𝑧𝑠∧𝜂_𝑖∧𝜃_𝑖||²𝑑𝑠+2𝐿𝐸_𝑖^𝑡1₀||𝑥𝑠∧𝜂_𝑖∧𝜃_𝑖−𝑦𝑠∧𝜂_𝑖∧𝜃_𝑖||²𝑑𝑠+2𝐿_𝑖𝐸^𝑡1₀||𝑦𝑠∧𝜂_𝑖∧𝜃_𝑖−𝑧𝑠∧𝜂_𝑖∧𝜃_𝑖||²𝑑𝑠+2𝐿_𝑖𝐸^𝑡1₀||𝑥𝑠∧𝜂_𝑖∧𝜃_𝑖−𝑦𝑠∧𝜂_𝑖∧𝜃_𝑖||²𝑑𝑠≤4𝐿_𝑖𝐸^𝑡1₀||𝑦𝑠∧𝜂_𝑖∧𝜃_𝑖−𝑧𝑠∧𝜂_𝑖∧𝜃_𝑖||²𝑑𝑠^𝑡1₀𝐸sup_{0≤𝑡≤𝑠∧𝜂𝑖∧𝜃𝑖}||𝑥(𝑡)−𝑦(𝑡)||²𝑑𝑠+4𝐿_𝑖𝑇𝐶₁(𝑖)ℎ^1/2.(3.44) Similarly, we obtain that 𝐸^𝑡1₀≤||𝑔4𝐿𝑥_𝑖𝑠∧𝜂_𝑖∧𝜃_𝑖,𝑥𝑠∧𝜂_𝑖∧𝜃_𝑖−𝑔𝑧𝑠∧𝜂_𝑖∧𝜃_𝑖,𝑧𝑠∧𝜂_𝑖∧𝜃_𝑖||²𝑑𝑠^𝑡1₀𝐸sup_{0≤𝑡≤𝑠∧𝜂𝑖∧𝜃𝑖}||𝑥(𝑡)−𝑦(𝑡)||²𝑑𝑠+4𝐿_𝑖𝑇𝐶₁(𝑖)ℎ^1/2.(3.45) Substituting (3.44) and (3.45) into (3.43) gives 𝐸su≤p_{0≤𝑡≤𝑡1}8(||𝑥𝑇+𝑡∧4)𝜂𝐿^𝑖_𝑖∧𝜃_𝑖−𝑦𝑡∧𝜂_𝑖∧𝜃_𝑖||²^𝑡1₀𝐸sup_{0≤𝑡≤𝑠∧𝜂𝑖∧𝜃𝑖}||𝑥(𝑡)−𝑦(𝑡)||²𝑑𝑠+8(𝑇+4)𝐿_𝑖𝑇𝐶₁(𝑖)ℎ^1/2.(3.46) By the Gronwall inequality, we must get 𝐸sup_{0≤𝑡≤𝑇}||𝑥𝑡∧𝜂^𝑖∧𝜃_𝑖−𝑦𝑡∧𝜂_𝑖∧𝜃_𝑖||²≤𝐶²(𝑖)ℎ^1/2,(3.47) where 𝐶₂(𝑖)=8𝑇(𝑇+4)𝐿_𝑖𝐶(1+4𝑑)𝑒^{8𝑇(𝑇+4)𝐿𝑖}.

Lemma 3.5. Under the conditions (H1) and (H3) if 𝜀∈(0,1) and 𝑇>0, then there exists a sufficiently large integer 𝑖̂ (dependent on 𝜀 and 𝑇) and sufficiently small ℎ such that 𝑃𝜃𝑖̂≤𝑇≤𝜀∀ℎ≤ℎ.(3.48)

Proof. By Itô formula, we have 𝑑𝑉(𝑦(𝑡))=𝑉_𝑦(𝑦(𝑡))𝑓(𝑧(𝑡),𝑧([𝑡]))+12trace𝑔^𝑇+𝑉_𝑦(𝑧(𝑡),𝑧([𝑡]))𝑉_𝑦𝑦(𝑦(𝑡))𝑔(𝑧(𝑡),𝑧([𝑡]))𝑑𝑡=𝐿𝑉((𝑦𝑦((𝑡𝑡))),𝑔𝑦((𝑧[𝑡(]𝑡))),+𝑧𝑉([_𝑦𝑡]))𝑑𝐵(𝑡)+1(𝑦(𝑡))[𝑓(𝑧(𝑡),𝑧([𝑡]))−𝑓(𝑦(𝑡),𝑦([𝑡]))]2trace𝑔^𝑇−𝑔(_𝑇𝑧(𝑡),𝑧([𝑡]))𝑉_𝑦𝑦(𝑦(𝑡))𝑔(𝑧(𝑡),𝑧([𝑡]))+𝑉_𝑦(𝑦(𝑡),𝑦([𝑡]))𝑉_𝑦𝑦(𝑦(𝑡))𝑔(𝑦(𝑡),𝑦([𝑡]))𝑑𝑡(𝑦(𝑡))𝑔(𝑧(𝑡),𝑧([𝑡]))𝑑𝐵(𝑡).(3.49) By condition (H1), 𝑉_𝑦+(𝑦1(𝑡))[𝑓(𝑧(𝑡),𝑧([𝑡]))−𝑓(𝑦(𝑡),𝑦([𝑡]))]2trace𝑔^𝑇=𝑉(𝑧(𝑡),𝑧([𝑡]))𝑉_𝑦𝑦(𝑦(𝑡))𝑔(𝑧(𝑡),𝑧([𝑡]))−𝑔^𝑇(𝑦(𝑡),𝑦([𝑡]))𝑉_𝑦𝑦(𝑦(𝑡))𝑔(𝑦(𝑡),𝑦([𝑡]))_𝑦+1(𝑦(𝑠))[𝑓(𝑧(𝑠),𝑧([𝑠]))−𝑓(𝑦(𝑠),𝑦([𝑠]))]2trace𝑔^𝑇+1(𝑧(𝑠),𝑧([𝑠]))−𝑔^𝑇(𝑦(𝑠),𝑦([𝑠]))𝑉_𝑦𝑦(𝑦(𝑠))𝑔(𝑧(𝑠),𝑧([𝑠]))2trace𝑔^𝑇≤𝑐_𝑖(𝑦(𝑠),𝑦([𝑠]))𝑉_𝑦𝑦(𝑦(𝑠))[𝑔(𝑧(𝑠),𝑧([𝑠]))−𝑔(𝑦(𝑠),𝑦([𝑠]))]||𝑦(𝑡)−𝑧(𝑡)||+||𝑦([𝑡])−𝑧([𝑡])||,(3.50) where 𝑐_𝑖 denotes a positive constant independent of ℎ. Substituting (3.50) into (3.49), we obtain that, for 0≤𝑡≤𝜃_𝑖, 𝑑𝑉(𝑦(𝑡))≤+𝐿𝑉𝑉_𝑦(𝑦(𝑡),𝑦([𝑡]))+𝑐_𝑖||𝑦(𝑡)−𝑧(𝑡)||+||𝑦([𝑡])−𝑧([𝑡])||𝑑𝑡(𝑦(𝑡))𝑔(𝑧(𝑡),𝑧([𝑡]))𝑑𝐵(𝑡).(3.51) Hence, for 𝑡∈[𝑛,𝑛+1), we can integrate both sides of (3.51) from 𝑛 to 𝑡∧𝜃_𝑖 and take the expectations 𝐸𝑉𝑦𝑡∧𝜃_𝑖≤𝐸+𝑉𝐸(𝑦_{𝑡∧𝜃𝑖𝑛}(𝑛))+𝐸^{𝑡∧𝜃𝑖}_𝑛𝐿𝑉(𝑦(𝑠),𝑦([𝑠]))𝑑𝑠𝑐_𝑖||𝑦(𝑡)−𝑧(𝑡)||+||𝑦([𝑡])−𝑧([𝑡])||𝑑𝑠,(3.52) while 𝑐_𝑖𝐸=^𝑡𝑛𝑐||𝑦_𝑖𝑠∧𝜃_𝑖−𝑧𝑠∧𝜃_𝑖||+||𝑦𝑠∧𝜃_𝑖−𝑧𝑠∧𝜃_𝑖||𝑑𝑠^𝑡𝑛𝐸||𝑦𝑠∧𝜃_𝑖−𝑧𝑠∧𝜃_𝑖||𝑑𝑠+𝑐_𝑖^𝑡𝑛≤𝑐_𝑖𝐸||𝑦𝑠∧𝜃_𝑖−𝑧𝑠∧𝜃_𝑖||𝑑𝑠^𝑡𝑛𝐸||𝑦𝑠∧𝜃^𝑖−𝑧𝑠∧𝜃_𝑖||²^1/2𝑑𝑠+𝑐_𝑖^𝑡𝑛𝐸||𝑦𝑠∧𝜃^𝑖−𝑧𝑠∧𝜃_𝑖||²^1/2≤2𝑐_𝑖𝑑𝑠^𝑡𝑛𝐸sup_{0≤𝑢≤𝑠∧𝜃𝑖}||𝑦(𝑢)−𝑧(𝑢)||²^1/2≤2𝑐_𝑖𝑑𝑠^𝑡𝑛𝐶₁≤2𝑐_𝑖(𝑖)ℎ^1/2^1/2𝑑𝑠𝑇𝐶₁(𝑖)ℎ^1/2^1/2≤𝐶₃(𝑖)ℎ^1/4,(3.53) where 𝐶₃(𝑖)=2𝑐_𝑖𝑇(𝐶₁(𝑖))^1/2. Substituting this into (3.52) yields that 𝐸𝑉𝑦𝑡∧𝜃_𝑖≤𝐸𝑉(𝑦(𝑛))+𝛽̃+𝐸^{𝑡∧𝜃𝑖}_𝑛(𝐿𝑉(𝑦(𝑠),𝑦([𝑠])))𝑑𝑠,(3.54) where 𝛽̃=𝐶₃(𝑖)ℎ^1/4. For 𝑡∈[0,1), by condition (H3), we obtain that 𝐸𝑉𝑦𝑡∧𝜃_𝑖≤≤𝑉𝑉((𝑦𝑦((00))))++𝛼𝛼𝐸[1+^{𝑡∧𝜃𝑖}₀𝑉[(1𝑦+(0𝑉))](𝑦+(𝛽𝑠)̃+)+𝛼𝐸𝑉(𝑦_{𝑡∧𝜃𝑖0}([𝑠]))]𝑑𝑠+𝛽̃≤𝛽̃+𝛽̂₁𝑉(𝑦(𝑠))𝑑𝑠+𝛼^𝑡0𝐸𝑉𝑦𝑠∧𝜃_𝑖𝑑𝑠,(3.55) where 𝛽̂₁=𝑉(𝑦(0))+𝛼[1+𝑉(𝑦(0))]=(1+𝛼)𝑉𝑐₀+𝛼<∞.(3.56) Hence, by the Gronwall inequality, 𝐸𝑉𝑦𝑡∧𝜃_𝑖≤≤𝛽̃𝛽+̃+𝛽̂₁𝛽̂+₁𝛼^𝑡0𝐸𝑉𝑦𝑠∧𝜃_𝑖𝑑𝑠𝑒^𝛼𝑡≤𝛽̃+𝛽̂₁𝑒^𝛼<∞,(3.57) for 0≤𝑡<1. Consequently, 𝐸𝑉𝑦1∧𝜃_𝑖=lim_𝑡→1𝐸𝑉𝑦𝑡∧𝜃_𝑖≤𝛽̃+𝛽̂₁𝑒^𝛼<∞.(3.58) Define 𝛾_𝑖=inf_{||𝑦||≥𝑖}𝑉(𝑦),∀𝑖≥||𝑐₀||,(3.59) and denote 𝐼_𝐴 as the indicator function of a set 𝐴, then we have 𝛽̃+𝛽̂₁𝑒^𝛼≥𝐸𝑉𝑦𝜃_𝑖∧1≥𝐸𝑉𝑦𝜃_𝑖𝐼_{{𝜃𝑖≤1}}≥𝛾_𝑖𝑃𝜃_𝑖≤1.(3.60) Letting 𝑖→∞, we have that 𝑃(𝜃_∞≤1)=0, namely, 𝑃𝜃_∞>1=1.(3.61) By (3.57) and (3.61), 𝐸𝑉(𝑦(𝑡))≤𝛽̃+𝛽̂₁𝑒^𝛼<∞,0≤𝑡≤1.(3.62) For 𝑡∈[1,2), by (3.54), we have 𝐸𝑉𝑦𝑡∧𝜃_𝑖≤≤𝐸𝛽̃𝑉+(𝛽𝑦̂₂(1))+𝛼[1+𝐸𝑉(𝑦(1))]+𝛽̃+𝛼^𝑡1𝐸𝑉𝑦𝑠∧𝜃_𝑖𝑑𝑠+𝛼^𝑡1𝐸𝑉𝑦𝑠∧𝜃_𝑖𝑑𝑠,(3.63) where 𝛽̂₂≤(1+𝛼)𝛽̃+𝛽̂₁𝑒^𝛼+𝛼<∞.(3.64) Hence, by the Gronwall inequality, 𝐸𝑉𝑦𝑡∧𝜃_𝑖≤≤𝛽̃𝛽+̃+𝛽̂₂𝛽̂+₂𝛼^𝑡1𝐸𝑉𝑦𝑠∧𝜃_𝑖𝑑𝑠𝑒^{𝛼(𝑡−1)}≤𝛽̃+𝛽̂₂𝑒^𝛼<∞,(3.65) for 1≤𝑡<2. Consequently, we can obtain that 𝐸𝑉𝑦2∧𝜃_𝑖=lim_𝑡→2𝐸𝑉𝑦𝑡∧𝜃_𝑖≤𝛽̃+𝛽̂₂𝑒^𝛼<∞.(3.66) In the same way, we have 𝐸𝑉(𝑦(𝑡))≤𝛽̃+𝛽̂₂𝑒^𝛼<∞,1≤𝑡≤2.(3.67) Repeating this procedure, for 𝑡∈[𝑁−1,𝑇), we can show that 𝐸𝑉𝑦𝑡∧𝜃_𝑖≤𝛽̃+𝛽̂_𝑇𝑒^𝛼<∞,(3.68) where 𝛽̂_𝑇≤(1+𝛼)𝛽̃+𝛽̂_𝑁−1𝑒^𝛼+𝛼<∞.(3.69) Consequently, we can obtain that 𝐸𝑉𝑦𝑇∧𝜃_𝑖=lim_𝑡→𝑇𝐸𝑉𝑦𝑡∧𝜃_𝑖≤𝛽̃+𝛽̂_𝑇𝑒^𝛼.(3.70) We compute 𝛽̃+𝛽̂_𝑇𝑒^𝛼≥𝐸𝑉𝑦𝑇∧𝜃_𝑖≥𝐸𝑉𝑦𝜃_𝑖𝐼_{{𝜃𝑖≤𝑇}}≥𝛾_𝑖𝑃𝜃_𝑖≤𝑇.(3.71) Then we have 𝑃𝜃_𝑖≤𝑇≤𝛽̃+𝛽̂^𝑇𝑒^𝛼𝛾_𝑖=𝐶³(𝑖)ℎ^1/4+𝛽̂_𝑇𝑒^𝛼𝛾_𝑖.(3.72) Now, for any 𝜀∈(0,1), choose 𝑖=𝑖̂ sufficiently large for 𝛽̂_𝑇𝑒^𝛼𝛾𝑖̂≤2𝜀,(3.73) and then choose ℎ sufficiently small for 𝐶₃𝑖̂ℎ^1/4𝑒^𝛼𝛾𝑖̂≤2𝜀.(3.74) Hence, 𝑃𝜃𝑖̂≤𝑇≤𝜀∀ℎ≤ℎ.(3.75)

The following theorems describe the convergence in probability of the EM method to (2.3) under the local Lipschitz condition (H1) and some additional conditions (H3).

Theorem 3.6. Under the conditions (H1) and (H3), for arbitrarily small 𝜎∈(0,1), lim_ℎ→0𝑃𝜔∶sup_{0≤𝑡≤𝑇}||𝑥(𝑡)−𝑦(𝑡)||>𝜎=0,(3.76) for any 𝑇>0.

Proof. For arbitrarily small 𝜎,𝜀∈(0,1). We set Ω=⁻𝜔∶sup_{0≤𝑡≤𝑇}||𝑥(𝑡)−𝑦(𝑡)||>𝜎.(3.77) By Theorem 3.2 and Lemma 3.5, there exists a pair of 𝑖̂ and ℎ such that 𝑃𝜂𝑖̂≤𝑇≤𝜀𝑃𝜃𝑖̂≤𝑇≤𝜀3,3,∀ℎ≤ℎ.(3.78) For ℎ≤ℎ, 𝑃Ω⁻≤≤𝑃𝑃Ω⁻₋∩𝜂𝑖̂∧𝜃𝑖̂>𝑇+𝑃𝜂𝑖̂∧𝜃𝑖̂≤𝑇≤𝑃Ω₋∩𝜂𝑖̂∧𝜃𝑖̂>𝑇+𝑃𝜂𝑖̂≤𝑇+𝑃𝜃𝑖̂≤𝑇Ω∩𝜂𝑖̂∧𝜃𝑖̂>𝑇+2𝜀3.(3.79) By Lemma 3.4, we get 𝜎²𝑃Ω⁻∩𝜂𝑖̂∧𝜃𝑖̂>𝑇≤𝐸sup_{0≤𝑡≤𝑇}|||𝑥𝑡∧𝜂𝑖̂∧𝜃𝑖̂−𝑦𝑡∧𝜂𝑖̂∧𝜃𝑖̂|||²𝐼_{𝜂≤𝐸sup_{0≤𝑡≤𝑇}𝑖̂^∧𝜃𝑖̂^>𝑇}|||𝑥𝑡∧𝜂𝑖̂∧𝜃𝑖̂−𝑦𝑡∧𝜂𝑖̂∧𝜃𝑖̂|||²≤𝐶₂𝑖̂ℎ^1/2.(3.80) Hence, 𝑃Ω⁻∩𝜂𝑖̂∧𝜃𝑖̂>𝑇≤𝐶²𝑖̂ℎ^1/2𝜎².(3.81) For all sufficiently small ℎ, we obtain 𝑃Ω⁻∩𝜂𝑖̂∧𝜃𝑖̂>𝑇≤𝜀3.(3.82) From (3.79) and (3.82), we see that for all sufficiently small ℎ, 𝑃Ω⁻≤𝜀,(3.83) which proves the theorem.

Of course, 𝑧(𝑡) is computable but 𝑦(𝑡) is not, so the following theorem is much more useful in practice.

Theorem 3.7. Under the conditions (H1) and (H3), for arbitrarily small 𝜎∈(0,1), lim_ℎ→0𝑃𝜔∶sup_{0≤𝑡≤𝑇}||𝑥(𝑡)−𝑧(𝑡)||>𝜎=0,(3.84) for any 𝑇>0.

Proof. For arbitrarily small 𝜎,𝜀∈(0,1). We denote Ω=𝜔∶sup_{0≤𝑡≤𝑇}||𝑥(𝑡)−𝑧(𝑡)||>𝜎.(3.85) In the same way as Theorem 3.6, we can see that 𝑃Ω≤𝑃Ω∩𝜂𝑖̂∧𝜃𝑖̂>𝑇+2𝜀3.(3.86) But by Lemma 3.3, we get 𝜎²𝑃Ω∩𝜂𝑖̂∧𝜃𝑖̂>𝑇≤𝐸sup_{0≤𝑡≤𝑇}|||𝑥𝑡∧𝜂𝑖̂∧𝜃𝑖̂−𝑧𝑡∧𝜂𝑖̂∧𝜃𝑖̂|||²𝐼_{𝜂≤𝐸sup_{0≤𝑡≤𝑇}𝑖̂^∧𝜃𝑖̂^>𝑇}|||𝑥𝑡∧𝜂𝑖̂∧𝜃𝑖̂−𝑧𝑡∧𝜂𝑖̂∧𝜃𝑖̂|||²≤2𝐸sup_{0≤𝑡≤𝑇}|||𝑥𝑡∧𝜂𝑖̂∧𝜃𝑖̂−𝑦𝑡∧𝜂𝑖̂∧𝜃𝑖̂|||²+2𝐸sup_{0≤𝑡≤𝑇}|||𝑦𝑡∧𝜂𝑖̂∧𝜃𝑖̂−𝑧𝑡∧𝜂𝑖̂∧𝜃𝑖̂|||²≤2𝐸sup_{0≤𝑡≤𝑇}|||𝑥𝑡∧𝜂𝑖̂∧𝜃𝑖̂−𝑦𝑡∧𝜂𝑖̂∧𝜃𝑖̂|||²+2𝐸sup_{0≤𝑡≤𝑇}|||𝑦𝑡∧𝜃𝑖̂−𝑧𝑡∧𝜃𝑖̂|||²≤2𝐶₂𝑖̂+𝐶₁𝑖̂ℎ^1/2.(3.87) therefore, 𝑃Ω∩𝜂𝑖̂∧𝜃𝑖̂>𝑇≤2𝐶²𝑖̂+𝐶₁𝑖̂ℎ^1/2𝜎².(3.88) For all sufficiently small ℎ, we obtain 𝑃Ω∩𝜂𝑖̂∧𝜃𝑖̂>𝑇≤𝜀3.(3.89) From (3.86) and (3.89), we see that for all sufficiently small ℎ, 𝑃Ω≤𝜀,(3.90) which proves the assertion (3.84).

4. Numerical Example

Let us now discuss a numerical example to demonstrate the results which we obtain.

Example 4.1. Let us consider the stochastic differential equations with piecewise continuous arguments 𝑑𝑥(𝑡)=−𝑥³(𝑡)+𝑥([𝑡])𝑑𝑡+sin𝑥²(𝑡)+𝑥([𝑡])𝑑𝐵(𝑡)∀𝑡≥0.(4.1) Defining 𝑉(𝑥)=𝑥², we have 𝐿𝑉(𝑥,𝑦)=2𝑥−𝑥³+𝑦+sin𝑥²+𝑦²≤−2𝑥⁴+2𝑥𝑦+2sin𝑥²²+2𝑦²≤31+𝑥²+𝑦²,(4.2) where 𝛼=3. In other words, the equation satisfies condition (H3). By Theorem 3.1, we can conclude that the SEPCA (4.1) has a unique global solution 𝑥(𝑡) on 𝑡∈[0,∞). Moreover, the EM method can be applied to approximate the solution of the SEPCA (4.1). Given the stepsize ℎ=1/𝑚, by (2.10), (2.11), and (2.12), the Euler method to (4.1) leads to a numerical process of the following type: 𝑦_{𝑘𝑚+𝑙+1}=𝑦_{𝑘𝑚+𝑙}+−𝑦^{3𝑘𝑚+𝑙}+𝑦_𝑘𝑚ℎ+sin𝑦^{2𝑘𝑚+𝑙}+𝑦_𝑘𝑚Δ𝐵_{𝑘𝑚+𝑙}.(4.3) The continuous Euler-Maruyama approximate solution is defined by 𝑦(𝑡)=𝑦(0)+^𝑡0−𝑧³(𝑠)+𝑧([𝑠])𝑑𝑠+^𝑡0sin𝑧²(𝑠)+𝑧([𝑠])𝑑𝐵(𝑠),(4.4) where 𝑧(𝑡)=𝑦_{𝑘𝑚+𝑙} and 𝑧([𝑡])=𝑦_𝑘𝑚 for 𝑡∈[𝑡_{𝑘𝑚+𝑙},𝑡_{𝑘𝑚+𝑙+1}). By Theorems 3.6 and 3.7, we also have the convergence in probability of the EM method to (4.1) under the local Lipschitz condition (H1) and some additional conditions (H3).

Acknowledgment

The financial support from the National Natural Science Foundation of China (no. 11071050) is gratefully acknowledged.