Numerical Solutions of Stochastic Differential Delay Equations with Poisson Random Measure under the Generalized Khasminskii-Type Conditions

Abstract

The Euler method is introduced for stochastic differential delay equations (SDDEs) with Poisson random measure under the generalized Khasminskii-type conditions which cover more classes of such equations than before. The main aims of this paper are to prove the existence of global solutions to such equations and then to investigate the convergence of the Euler method in probability under the generalized Khasminskii-type conditions. Numerical example is given to indicate our results.

1. Introduction

To take into consideration stochastic effects such as corporate defaults, operational failures, market crashes or central bank announcements in financial market, the research on stochastic differential equations (SDEs) with Poisson random measure (see [1, 2]) is important, since Merton initiated the model of such equations in 1976 (see [3]). Due to the rate of change of financial dynamics system depending on its past history, SDDE with Poisson random measure (see [4, 5]), the case we propose and consider in this work, is meaningful.

Since there is no explicit solution for an SDDE with Poisson random measure, one needs, in general, numerical methods which can be classified into strong approximations and weak approximations (see [6–8]).

We here give an overview of the results on the strong approximations of differential equation driven by Wiener process and Poisson random measure. Platen [9] presented a convergence theorem with order 𝛾∈{0.5,1,1.5,…} and originally introduced the jump-adapted methods which are based on all the jump times. Moreover, Bruti-Liberati and Platen (see [10]) get the jump-adapted order 1.5 scheme, and they also construct the derivative free or implicit jump-adapted schemes with desired order of strong convergence. In [11], for a class of pure jump systems, the order of Taylor schemes is given under weaker conditions than the current literature. In [7, 10], Bruti-Liberati and Platen present the drift-implicit schemes which have order 𝛾∈{0.5,1}. Recently, [8] develops adaptive time stepping algorithms based on a jump augmented Monte Carlo Euler-Maruyama method, which achieve a prescribed precision. Mao [4] presents the convergence of numerical solutions for variable delay differential equations with Poisson random measure. In [12], the improved Runge-Kutta methods are presented to improve the accuracy behaviour of problems with small noise for SDEs driven by Poisson random measure. Clearly, the results above require that the SDEs with poisson random measure satisfy the global Lipschitz conditions and the linear growth conditions. In [5], the Euler scheme is proved to converge to the analytic solution for SDDEs with Wiener process and Poisson random measure under weaker conditions than the global Lipschitz condition and the linear growth condition.

However, there are many SDDEs with Poisson random measure, especially highly nonlinear equations, which do not satisfy the above-mentioned conditions and classical Khasminskii-type conditions (see [13–15]). And in Section 5, we give such highly nonlinear equation. Our work is motivated by [16] in which the generalized Khasminskii-type conditions are applied to SDDEs with Wiener process. The main contribution in our paper is to present Euler method for SDDEs with Poisson random measure under the generalized Khasminskii-type conditions which cover more classes of these equations than all the mentioned classical conditions above.

Our work is organized as follows. In Section 2, the properties of SDDEs with Poisson random measure are given under the generalized Khasminskii-type conditions. In Section 3, Euler method is analyzed under such conditions. In Section 4, we present the convergence in probability of the Euler method. In Section 5, an example is given.

2. The Generalized Khasminskii-Type Conditions for SDDEs with Poisson Random Measure

2.1. Problem's Setting

Throughout this paper, unless otherwise specified, we use the following notations. Let |⋅| be the Euclidean norm in 𝐑^𝑑,𝑑∈𝐍. Let 𝑢₁∨𝑢₂=max{𝑢₁,𝑢₂} and 𝑢₁∧𝑢₂=min{𝑢₁,𝑢₂}. If 𝐴 is a vector or matrix, its transpose is denoted by 𝐴^𝑇. If 𝐴 is a matrix, its trace norm is denoted by |𝐴|=√trace(𝐴^𝑇𝐴). Let 𝜏>0 and 𝐑₊=[0,∞). Let 𝐶([−𝜏,0];𝐑^𝑑) denote the family of continuous functions from [−𝜏,0] to 𝐑^𝑑 with the norm |𝜑|=sup_{−𝜏≤𝜃≤0}|𝜑(𝜃)|. Denote by 𝐶(𝐑^𝑑;𝐑₊) the family of continuous functions from 𝐑^𝑑 to 𝐑₊. Let 𝐶²(𝐑^𝑑;𝐑₊) denote the family of continuously two times differentiable 𝐑₊-valued functions from 𝐑^𝑑 to 𝐑₊. [𝑧] denotes the largest integer which is less than or equal to 𝑧 in 𝐑. 𝐼_𝒜 denotes the indicator function of a set 𝒜.

The following 𝑑-dimensional SDDE with Poisson random measure is considered in our paper: 𝑑𝑥(𝑡)=𝑎+(𝑥(𝑡⁻),𝑥((𝑡−𝜏)⁻))𝑑𝑡+𝑏(𝑥(𝑡⁻),𝑥((𝑡−𝜏)⁻))𝑑𝑊(𝑡)_𝜀𝑐(𝑥(𝑡⁻),𝑥((𝑡−𝜏)⁻),𝑣)𝑝_𝜙̃(𝑑𝑣×𝑑𝑡),(2.1) for 𝑡>0, where 𝑝_𝜙̃(𝑑𝑣×𝑑𝑡)∶=𝑝_𝜙(𝑑𝑣×𝑑𝑡)−𝜙(𝑑𝑣)𝑑𝑡. Here 𝑥(𝑡⁻) denotes lim_𝑠↑𝑡𝑥(𝑠). The initial data of (2.1) is given by {𝑥(𝑡)∶−𝜏≤𝑡≤0}=𝜉(𝑡)∈𝐶[−𝜏,0];𝐑^𝑑,(2.2) where 𝑥(−𝜏⁻)=𝑥(−𝜏).

The drift coefficient 𝑎∶𝐑^𝑑×𝐑^𝑑→𝐑^𝑑, the diffusion coefficient 𝑏∶𝐑^𝑑×𝐑^𝑑→𝐑^𝑑×𝑚0, and the jump coefficient 𝑐∶𝐑^𝑑×𝐑^𝑑×𝜀→𝐑^𝑑 are assumed to be Borel measurable functions and the coefficients are sufficiently smooth.

The randomness in (2.1) is generated by the following (see [8]). An 𝑚₀-dimensional Wiener process 𝑊={𝑊(𝑡)=(𝑊¹(𝑡),…,𝑊^𝑚0(𝑡))^𝑇} with independent scalar components is defined on a filtered probability space (Ω^𝑊,ℱ^𝑊,(ℱ^𝑊𝑡)_𝑡≥0,𝐏^𝑊). A Poisson random measure 𝑝_𝜙(𝜔,𝑑𝑣×𝑑𝑡) is on Ω^𝐽×𝜀×[0,∞), where 𝜀⊆𝐑^𝑟⧵{0} with 𝑟∈𝐍, and its deterministic compensated measure 𝜙(𝑑𝑣)𝑑𝑡=𝜆𝑓(𝑣)𝑑𝑣𝑑𝑡.𝑓(𝑣) is a probability density, and we require finite intensity 𝜆=𝜙(𝜀)<∞. The Poisson random measure is defined on a filtered probability space (Ω^𝐽,ℱ^𝐽,(ℱ^𝐽𝑡)_𝑡≥0,𝐏^𝐽). The process 𝑥(𝑡) is thus defined on a product space (Ω,ℱ,(ℱ_𝑡)_𝑡≥0,𝐏), where Ω=Ω^𝑊×Ω^𝐽,ℱ=ℱ^𝑊×ℱ^𝐽,(ℱ_𝑡)_𝑡≥0=(ℱ^𝑊𝑡)_𝑡≥0×(ℱ^𝐽𝑡)_𝑡≥0, 𝐏=𝐏^𝑊×𝐏^𝐽, and ℱ₀ contains all 𝐏-null sets. The Wiener process and the Poisson random measure are mutually independent.

To state the generalized Khasminskii-type conditions, we define the operator 𝐿𝑉∶𝐑^𝑑×𝐑^𝑑→𝐑 by 𝐿𝑉(𝑥,𝑦)=𝑉_𝑥(𝑥)𝑎(𝑥,𝑦)+12trace𝑏^𝚃+(𝑥,𝑦)𝑉_𝑥𝑥(𝑥)𝑏(𝑥,𝑦)_𝜀𝑉(𝑥+𝑐(𝑥,𝑦,𝑣))−𝑉(𝑥)−𝑉_𝑥(𝑥)𝑐(𝑥,𝑦,𝑣)𝜙(𝑑𝑣),(2.3) where 𝑉∈𝐶²𝐑^𝑑;𝐑₊,𝑉_𝑥=𝜕𝑉(𝑥)𝜕𝑥₁,…,𝜕𝑉(𝑥)𝜕𝑥_𝑑,𝑉^𝑥𝑥=𝜕²𝑉(𝑥)𝜕𝑥_𝑖𝜕𝑥_𝑗_𝑑×𝑑.(2.4)

Now the generalized Khasminskii-type conditions are given by the following assumptions.

Assumption 2.1. For each integer 𝑘≥1, there exists a positive constant 𝐶_𝑘, dependent on 𝑘, such that ||𝑎(𝑥,𝑦)−𝑎𝑥,𝑦||²∨||𝑏(𝑥,𝑦)−𝑏𝑥,𝑦||²≤𝐶_𝑘||𝑥−𝑦||²+||𝑥−𝑦||²,(2.5) for 𝑥,𝑦∈𝐑^𝑑 with |𝑥|∨|𝑦|≤𝑘. And there exists a positive constant 𝐶 such that _𝜀||𝑐(𝑥,𝑦,𝑣)−𝑐𝑥,𝑦,𝑣||²𝜙(𝑑𝑣)≤𝐶||𝑥−𝑦||²+||𝑥−𝑦||²,(2.6) for 𝑥,𝑦∈𝐑^𝑑.

Assumption 2.2. There are two functions 𝑉∈𝐶²(𝐑^𝑑;𝐑₊) and 𝑈∈𝐶(𝐑^𝑑;𝐑₊) as well as two positive constants 𝜇₁ and 𝜇₂ such that 𝐿𝑉(𝑥,𝑦)≤𝜇₁lim_|𝑥|→∞𝑉(𝑥)=∞,(2.7)(1+𝑉(𝑥)+𝑉(𝑦)+𝑈(𝑦))−𝜇₂𝑈(𝑥),(2.8) for all (𝑥,𝑦)∈𝐑^𝑑×𝐑^𝑑.

Assumption 2.3. There exists a positive constant 𝐶 such that _𝜀||𝑐(0,0,𝑣)||²𝜙(𝑑𝑣)≤𝐶.(2.9)

Assumption 2.4. There exists a positive constant 𝐿 such that the initial data (2.2) satisfies ||𝜉(𝑡)−𝜉(𝑠)||≤𝐿|𝑡−𝑠|^1/2,for−𝜏≤𝑡,𝑠≤0.(2.10)

2.2. The Existence of Global Solutions

In this section, we analyze the existence and the property of the global solution to (2.1) under Assumptions 2.1, 2.2, and 2.4.

In order to demonstrate the existence of the global solution to (2.1), we redefine the following concepts mainly according to [17, 18].

Definition 2.5. Let {𝑥(𝑡)}_{𝑡≥−𝜏} be an 𝐑^𝑑-valued stochastic process. The process is said to be càdlàg if it is right continuous and for almost all 𝜔∈Ω the left limit lim_𝑠↑𝑡𝑥(𝑠) exists and is finite for all 𝑡>−𝜏.

Definition 2.6. Let 𝜎_∞ be a stopping time such that 0≤𝜎_∞≤𝑇 a.s. An 𝐑^𝑑-valued, ℱ_𝑡-adapted, and càdlàg process {𝑥(𝑡)∶−𝜏≤𝑡<𝜎_∞} is called a local solution of (2.1) if 𝑥(𝑡)=𝜉(𝑡) on 𝑡∈[−𝜏,0] and, moreover, there is a nondecreasing sequence {𝜎_𝑘}_𝑘≥1 of stopping times such that 0≤𝜎_𝑘↑𝜎_∞ a.s. and 𝑥𝑡∧𝜎_𝑘=𝑥+(0)+^{𝑡∧𝜎𝑘}₀𝑎(𝑥(𝑠⁻),𝑥((𝑠−𝜏)⁻))𝑑𝑠+^{𝑡∧𝜎𝑘}₀𝑏(𝑥(𝑠⁻),𝑥((𝑠−𝜏)⁻))𝑑𝑊(𝑠)_{𝑡∧𝜎𝑘0}_𝜀𝑐(𝑥(𝑠⁻),𝑥((𝑠−𝜏)⁻),𝑣)𝑝_𝜙̃(𝑑𝑣×𝑑𝑠)(2.11) holds for any 𝑡∈[0,𝑇] and 𝑘≥1 with probability 1. If, furthermore, lim_{𝑡→𝜎∞}sup||𝑥(𝑡)||=∞whenever𝜎_∞<𝑇,(2.12) then it is called a maximal local solution of (2.1) and 𝜎_∞ is called the explosion time. A local solution {𝑥(𝑡)∶−𝜏≤𝑡<𝜎_∞} to (2.1) is called a global solution if 𝜎_∞=∞.

Lemma 2.7. Under Assumptions 2.1 and 2.4, for any given initial data (2.2), there is a unique maximal local solution to (2.1).

Proof. From Assumption 2.4, for the initial data (2.2), we have max_{−𝜏≤𝑡≤0}||𝜉(𝑡)||≤max_{−𝜏≤𝑡≤0}||𝜉(𝑡)−𝜉(0)||+||𝜉(0)||≤𝐿√𝜏+||𝜉(0)||.(2.13) For each integer 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1, we define 𝑧^[𝑘]=|𝑧|∧𝑘|𝑧|𝑧,0^[𝑘]=0,(2.14) for 𝑧∈𝐑^𝑑. And then we define the truncation functions 𝑎_𝑘(𝑥,𝑦)=𝑎𝑥^[𝑘],𝑦^[𝑘],𝑏_𝑘(𝑥,𝑦)=𝑏𝑥^[𝑘],𝑦^[𝑘],𝑐_𝑘(𝑥,𝑦,𝑣)=𝑐(𝑥,𝑦,𝑣),(2.15) for 𝑥,𝑦∈𝐑^𝑑 and each 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1. Moreover, we define the following equation: 𝑑𝑥_𝑘(𝑡)=𝑎+_𝑘𝑥_𝑘(𝑡⁻),𝑥_𝑘((𝑡−𝜏)⁻)𝑑𝑡+𝑏_𝑘𝑥_𝑘(𝑡⁻),𝑥_𝑘((𝑡−𝜏)⁻)𝑑𝑊(𝑡)_𝜀𝑐_𝑘𝑥_𝑘(𝑡⁻),𝑥_𝑘((𝑡−𝜏)⁻),𝑣𝑝_𝜙̃(𝑑𝑣×𝑑𝑡),(2.16) on 𝑡∈[0,𝑇] with initial data 𝑥_𝑘(𝑡)=𝜉(𝑡) on 𝑡∈[−𝜏,0]. Obviously, the equation satisfies the global Lipschitz conditions and the linear growth conditions. Therefore according to [4], there is a unique global solution 𝑥_𝑘(𝑡) to (2.16) and its solution is a càdlàg process (see [17]). We define the stopping time 𝜎_𝑘=𝑇∧inf𝑡∈[0,𝑇]∶||𝑥_𝑘(𝑡)||≥𝑘,(2.17) for 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1, and 𝜎₁=⋯=𝜎_{√[𝐿𝜏+|𝜉(0)|]}=𝜎_{√[𝐿𝜏+|𝜉(0)|]+1},(2.18) where we set inf𝜙=∞ (as usual 𝜙 denotes the empty set) throughout our paper. We can easily get 𝑥_𝑘(𝑡)=𝑥_𝑘+1(𝑡),−𝜏≤𝑡≤𝜎_𝑘,(2.19) which means {𝜎_𝑘}_𝑘≥1 is a nondecreasing sequence and then let lim_𝑘→∞𝜎_𝑘=𝜎_∞ a.s. Now, we define {𝑥(𝑡)∶−𝜏≤𝑡<𝜎_∞} with 𝑥(𝑡)=𝜉(𝑡) on 𝑡∈[−𝜏,0] and 𝑥(𝑡)=𝑥_𝑘(𝑡),𝑡∈𝜎_𝑘−1,𝜎_𝑘,𝑘≥1,(2.20) where 𝜎₀=0. And from (2.16) and (2.19), we can also obtain 𝑥𝑡∧𝜎_𝑘==𝑥𝑥_𝑘0𝑡∧𝜎_𝑘++^{𝑡∧𝜎𝑘}₀𝑎(𝑥(𝑠⁻),𝑥((𝑠−𝜏)⁻))𝑑𝑠+^{𝑡∧𝜎𝑘}₀𝑏(𝑥(𝑠⁻),𝑥((𝑠−𝜏)⁻))𝑑𝑊(𝑠)_{𝑡∧𝜎𝑘0}_𝜀𝑐(𝑥(𝑠⁻),𝑥((𝑠−𝜏)⁻),𝑣)𝑝_𝜙̃(𝑑𝑣×𝑑𝑠),(2.21) for any 𝑡∈[0,𝑇] and 𝑘≥1 with probability 1. Moreover, if 𝜎_∞<𝑇, then lim_{𝑡→𝜎∞}sup||𝑥(𝑡)||≥lim_𝑘→∞sup||𝑥𝜎_𝑘||=lim_𝑘→∞sup||𝑥_𝑘𝜎_𝑘||=∞.(2.22) Hence {𝑥(𝑡)∶−𝜏≤𝑡<𝜎_∞} is a maximal local solution to (2.1).
To show the uniqueness of the solution to (2.1), let {𝑥(𝑡)∶−𝜏≤𝑡<𝜎_∞} be another maximal local solution. As the same proof as Theorem 2.8 in [17], we infer that 𝐏𝑥(𝑡,𝜔)=𝑥(𝑡,𝜔),(𝑡,𝜔)∈−𝜏,𝜎_𝑘∧𝜎_𝑘×Ω=1,𝑘≥1.(2.23) Hence by 𝑘→∞, we get 𝐏𝑥(𝑡,𝜔)=𝑥(𝑡,𝜔),(𝑡,𝜔)∈−𝜏,𝜎_∞∧𝜎_∞×Ω=1.(2.24)
Therefore 𝑥(𝑡) is a unique local solution and then it is a unique maximal local solution to (2.1).
So we complete the whole proof.

Now, the existence of the global solution to (2.1) is shown in the following theorem.

Theorem 2.8. Under Assumptions 2.1, 2.2, and 2.4, for any given initial data (2.2), there is a unique global solution 𝑥(𝑡) to (2.1) on 𝑡∈[−𝜏,∞).

Proof. According to Lemma 2.7, there exists a unique maximal local solution to (2.1) on [−𝜏,𝜎_∞). Hence in order to show that this local solution is a global one, we only need to demonstrate 𝜎_∞=∞ a.s. Using Itô's formula (see [1]) to 𝑉(𝑥(𝑡)), we have 𝑑𝑉(𝑥(𝑡))=𝑉+_𝑥1(𝑥(𝑡⁻))𝑎(𝑥(𝑡⁻),𝑥((𝑡−𝜏)⁻))2trace𝑏^𝚃+(𝑥(𝑡⁻),𝑥((𝑡−𝜏)⁻))𝑉_𝑥𝑥(𝑥(𝑡⁻))𝑏(𝑥(𝑡⁻),𝑥((𝑡−𝜏)⁻))𝑑𝑡_𝜀−𝑉(𝑉(𝑥((𝑥𝑡₋(𝑡⁻)+𝑐(𝑥(𝑡⁻),𝑥((𝑡−𝜏)⁻),𝑣))+𝑉_𝑥))−𝑉_𝑥(𝑥(𝑡⁻))𝑐(𝑥(𝑡⁻),𝑥((𝑡−𝜏)⁻),𝑣)𝜙(𝑑𝑣)𝑑𝑡+(𝑥(𝑡⁻))𝑏(𝑥(𝑡⁻),𝑥((𝑡−𝜏)⁻))𝑑𝑊(𝑡)_𝜀=𝐿𝑉(𝑥(𝑉(𝑡₋(𝑥(𝑡⁻)+𝑐(𝑥(𝑡⁻),𝑥((𝑡−𝜏)⁻),𝑣))−𝑉(𝑥(𝑡⁻)))𝑝_𝜙̃(𝑑𝑣×𝑑𝑡)+),𝑥((𝑡−𝜏)⁻))𝑑𝑡+𝑉_𝑥(𝑥(𝑡⁻))𝑏(𝑥(𝑡⁻),𝑥((𝑡−𝜏)⁻))𝑑𝑊(𝑡)_𝜀(𝑉(𝑥(𝑡⁻)+𝑐(𝑥(𝑡⁻),𝑥((𝑡−𝜏)⁻),𝑣))−𝑉(𝑥(𝑡⁻)))𝑝_𝜙̃(𝑑𝑣×𝑑𝑡),(2.25) for 𝑡∈[0,𝜎_∞).
Our proof is divided into the following steps.
Step 1. For any integer 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1 and 0≤𝑡≤𝜏, by taking integration and expectations and using Assumption 2.2 to (2.25), we get 𝐄𝑉≤𝐄𝑥𝑡_{𝑡∧𝜎𝑘0}∧𝜎_𝑘−𝐄𝑉(𝑥(0))𝜇₁(1+𝑉(𝑥(𝑠⁻))+𝑉(𝑥((𝑠−𝜏)⁻))+𝑈(𝑥((𝑠−𝜏)⁻)))−𝜇₂𝑈(𝑥(𝑠⁻))𝑑𝑠,(2.26) which means 𝐄𝑉𝑥𝑡∧𝜎_𝑘≤𝐶₁+𝜇₁𝐄^{𝑡∧𝜎𝑘}₀𝑉(𝑥(𝑠⁻))𝑑𝑠−𝜇₂𝐄^{𝑡∧𝜎𝑘}₀𝑈(𝑥(𝑠⁻))𝑑𝑠,(2.27) where 𝐶₁=𝑉(𝑥(0))+𝜇₁𝜏+𝜇₁^0−𝜏𝑉(𝜉(𝑠))𝑑𝑠+𝜇₁^0−𝜏𝑈(𝜉(𝑠))𝑑𝑠<∞.(2.28) From (2.27), we obtain 𝐄𝑉𝑥𝑡∧𝜎_𝑘≤≤𝐶𝐶₁₁+𝜇₁𝐄^{𝑡∧𝜎𝑘}₀𝑉(𝑥(𝑠⁻))𝑑𝑠+𝜇₁^𝑡0𝐄𝑉𝑥𝑠∧𝜎^−𝑘𝑑𝑠,(2.29) by the Gronwall inequality (see [18]), which leads to 𝐄𝑉𝑥𝑡∧𝜎_𝑘≤𝐶₁𝑒^𝜇1𝜏,(2.30) for 0≤𝑡≤𝜏 and 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1. Let 𝜍_𝑘=inf_{|𝑥|≥𝑘,0≤𝑡<∞}𝑉(𝑥),for𝑘≥𝐿√𝜏+||𝜉(0)||+1.(2.31) Therefore, from (2.30), we have 𝜍_𝑘𝐏𝜎_𝑘≤𝜏≤𝐄𝑉𝑥𝜎_𝑘𝐼_{{𝜎𝑘≤𝜏}}≤𝐄𝑉𝑥𝜏∧𝜎_𝑘≤𝐶₁𝑒^𝜇1𝜏,(2.32) by taking 𝑘→∞, which gives 𝐏𝜎_∞≤𝜏=0.(2.33) Hence we get 𝐏𝜎_∞>𝜏=1.(2.34) It thus follows from (2.30) and (2.34) that 𝐄𝑉(𝑥(𝑡))≤𝐶₁𝑒^𝜇1𝜏,0≤𝑡≤𝜏,(2.35) by taking 𝑘→∞.
Moreover, from (2.27), we get 𝐄^{𝜏∧𝜎𝑘}₀𝑈(𝑥(𝑠⁻))𝑑𝑠≤𝜇²⁻¹𝐶¹≤𝜇₂₋₁+𝜇₁𝐄^{𝜏∧𝜎𝑘}₀𝑉(𝑥(𝑠⁻))𝑑𝑠𝐶¹+𝜇₁^𝜏0𝐄𝑉𝑥𝑠∧𝜎^−𝑘𝑑𝑠,(2.36) by taking 𝑘→∞, which gives 𝐄^𝜏0𝑈(𝑥(𝑠))𝑑𝑠≤𝜇²⁻¹𝐶₁+𝜏𝜇₁𝐶₁𝑒^𝜇1𝜏<∞,(2.37) where (2.34) and (2.35) are used.
Step 2. For any integer 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1 and 0≤𝑡≤2𝜏, the similar analysis as above gives 𝐄𝑉𝑥𝑡∧𝜎_𝑘≤𝐶₂+𝜇₁𝐄^{𝑡∧𝜎𝑘}₀𝑉(𝑥(𝑠⁻))𝑑𝑠−𝜇₂𝐄^{𝑡∧𝜎𝑘}₀𝑈(𝑥(𝑠⁻))𝑑𝑠,(2.38) where 𝐶₂=𝑉(𝑥(0))+2𝜇₁𝜏+𝜇₁^0−𝜏𝑉(𝜉(𝑠))𝑑𝑠+𝜇₁^0−𝜏+𝜇₁𝑈(𝜉(𝑠))𝑑𝑠𝐄^𝜏0𝑉(𝑥(𝑠))𝑑𝑠+𝜇₁𝐄^𝜏0𝑈(𝑥(𝑠))𝑑𝑠<∞,(2.39) from (2.35) and (2.37).
Thus, 𝐄𝑉𝑥𝑡∧𝜎_𝑘≤𝐶₂+𝜇₁𝐄^𝑡0𝑉𝑥𝑠∧𝜎^−𝑘𝑑𝑠,(2.40) which gives 𝐄𝑉𝑥𝑡∧𝜎_𝑘≤𝐶₂𝑒^2𝜇1𝜏,(2.41) for 0≤𝑡≤2𝜏 and 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1. Hence we get 𝜍_𝑘𝐏𝜎_𝑘≤2𝜏≤𝐄𝑉𝑥2𝜏∧𝜎_𝑘≤𝐶₂𝑒^2𝜇1𝜏,(2.42) by taking 𝑘→∞, which implies 𝐏𝜎_∞≤2𝜏=0,(2.43) that is, 𝐏𝜎_∞>2𝜏=1.(2.44) Moreover, by taking 𝑘→∞ to (2.41), we then get 𝐄𝑉(𝑥(𝑡))≤𝐶₂𝑒^2𝜇1𝜏,0≤𝑡≤2𝜏.(2.45) Therefore, from (2.38), (2.44), and (2.45), we have 𝐄^02𝜏𝑈(𝑥(𝑠))𝑑𝑠≤𝜇²⁻¹𝐶₂+2𝜏𝜇₁𝐶₁𝑒^2𝜇1𝜏<∞.(2.46)
Step 3. So for any 𝑖∈𝐍, we repeat the similar analysis as above and then obtain 𝐄𝑉(𝑥(𝑡𝐏))≤𝜎_∞𝐶_𝑖>𝑖𝜏=1,𝐄𝑒^{𝑖𝜇1𝜏},0≤𝑡≤𝑖𝜏,_𝜏0𝑈(𝑥(𝑠))𝑑𝑠≤𝜇²⁻¹𝐶_𝑖+𝑖𝜏𝜇₁𝐶_𝑖𝑒^{𝑖𝜇1𝜏}<∞,(2.47) where 𝐶_𝑖=𝑉(𝑥(0))+𝜇₁𝐄^{(𝑖−1)𝜏−𝜏}(1+𝑉(𝑥(𝑠))+𝑈(𝑥(𝑠)))𝑑𝑠<∞.(2.48)
So we can get 𝐏(𝜎_∞=∞)=1 and the required result follows.

In the following lemma, we show that the solution of (2.1) remains in a compact set with a large probability.

Lemma 2.9. Under Assumptions 2.1, 2.2, and 2.4, for any pair of 𝜖∈(0,1) and 𝑇>0, there exists a sufficiently large integer 𝑘^∗, dependent on 𝜖 and 𝑇, such that 𝐏𝜎_𝑘≤𝑇≤𝜖,∀𝑘≥𝑘^∗,(2.49) where 𝜎_𝑘 is defined in Lemma 2.7.

Proof. According to Theorem 2.8, we can get 𝐄𝑉𝑥𝑇∧𝜎_𝑘≤𝐶_𝑖𝑒^{𝑖𝜇1𝜏},(2.50) for 𝑖 large enough to 𝑖𝜏≥𝑇 and 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1. Therefore, we have 𝜍_𝑘𝐏𝜎_𝑘≤𝑇≤𝐄𝑉𝑥𝜎_𝑘𝐼_{{𝜎𝑘≤𝑇}}≤𝐄𝑉𝑥𝑇∧𝜎_𝑘≤𝐶_𝑖𝑒^{𝑖𝜇1𝜏},(2.51) where 𝜍_𝑘=inf_{|𝑥|≥𝑘}𝑉(𝑥)for𝑘≥𝐿√𝜏+||𝜉(0)||+1.(2.52) Under (2.7) in Assumption 2.2, there exists a sufficiently large integer 𝑘^∗ such that 𝐏𝜎_𝑘≤𝑇≤𝐶^𝑖𝑒^{𝑖𝜇1𝜏}𝜍_𝑘≤𝜖,∀𝑘≥𝑘^∗.(2.53) So we complete the proof.

3. The Euler Method

In this section, we introduce the Euler method to (2.1) under Assumptions 2.1, 2.2, 2.3, and 2.4.

Given a step size Δ𝑡=𝜏/𝑚∈(0,1),𝑚∈𝐍, the Euler method applied to (2.1) computes approximation 𝑋_𝑛≈𝑥(𝑡_𝑛), where 𝑡_𝑛=𝑛Δ𝑡 for 𝑛=−𝑚,−(𝑚−1),…,−1,0,1,…, by setting 𝑋_𝑛=𝜉(𝑛Δ𝑡)for𝑛=−𝑚,−(𝑚−1),…,−1,0,(3.1) and forming 𝑋_𝑛+1=𝑋_𝑛+𝑎𝑋_𝑛,𝑋_𝑛−𝑚Δ𝑡+𝑏𝑋_𝑛,𝑋_𝑛−𝑚Δ𝑊_𝑛+^𝑡𝑛+1_𝑡𝑛_𝜀𝑐𝑋_𝑛,𝑋_𝑛−𝑚,𝑣𝑝_𝜙̃(𝑑𝑣×𝑑𝑡),(3.2) for 𝑛=0,1,…, where Δ𝑊_𝑛=𝑊(𝑡_𝑛+1)−𝑊(𝑡_𝑛).

The continuous-time Euler method 𝑋(𝑡) on 𝑡∈[−𝜏,∞) is then defined by 𝑋(𝑡)∶=𝜉(𝑡)for𝑡∈[−𝜏,0],(3.3)𝑋(𝑡)∶=𝑋+₀+^𝑡0𝑎(𝑍(𝑠),𝑍(𝑠−𝜏))𝑑𝑠+^𝑡0𝑏(𝑍(𝑠),𝑍(𝑠−𝜏))𝑑𝑊(𝑠)_𝑡0_𝜀𝑐(𝑍(𝑠),𝑍(𝑠−𝜏),𝑣)𝑝_𝜙̃(𝑑𝑣×𝑑𝑠),(3.4) for 𝑡≥0, where 𝑍(𝑡)=^∞_𝑛_=−𝑚𝑋_𝑛𝐼_{[𝑛Δ𝑡,(𝑛+1)Δ𝑡)}(𝑡)for𝑡∈[−𝜏,∞).(3.5)

Actually, we can see in [11] that 𝑝_𝜙={𝑝_𝜙(𝑡)∶=𝑝_𝜙(𝜀×[0,𝑡])} is a process that counts the number of jumps until some given time. The Poisson random measure 𝑝_𝜙(𝑑𝑣×𝑑𝑡) generates a sequence of pairs {(𝜄_𝑖,𝜉_𝑖),𝑖∈{1,2,…,𝑝_𝜙(𝑇)}} for a given finite positive constant 𝑇 if 𝜆<∞. Here {𝜄_𝑖∶Ω→𝐑₊,𝑖∈{1,2,…,𝑝_𝜙(𝑇)}} is a sequence of increasing nonnegative random variables representing the jump times of a standard Poisson process with intensity 𝜆, and {𝜉_𝑖∶Ω→𝜀,𝑖∈{1,2,…,𝑝_𝜙(𝑇)}} is a sequence of independent identically distributed random variables, where 𝜉_𝑖 is distributed according to 𝜙(𝑑𝑣)/𝜙(𝜀). Then (3.2) can equivalently be of the following form: 𝑋_𝑛+1=𝑋+_𝑛𝑏+𝑋_𝑛𝑎𝑋_𝑛,𝑋_𝑛−𝑚−_𝜀𝑐𝑋_𝑛,𝑋_𝑛−𝑚,𝑣𝜙(𝑑𝑣)Δ𝑡,𝑋_𝑛−𝑚Δ𝑊_𝑛+^{𝑝𝜙(𝑡𝑛+1)}_{𝑖=𝑝𝜙}_{𝑡𝑛+1}𝑐𝑋_𝑛,𝑋_𝑛−𝑚,𝜉_𝑖.(3.6)

In order to analyze the Euler method, we will give two lemmas.

The first lemma shows the close relation between the continuous-time Euler solution (3.4) and its step function 𝑍(𝑡).

Lemma 3.1. Suppose Assumptions 2.1 and 2.3 hold. Then for any 𝑇>0, there exists a positive constant 𝐾₁(𝑘), dependent on integer 𝑘 and independent of Δ𝑡, such that for all Δ𝑡∈(0,1) the continuous-time Euler method (3.4) satisfies 𝐄||𝑋(𝑡)−𝑍(𝑡)||²≤𝐾₁(𝑘)Δ𝑡,(3.7) for 0≤𝑡≤𝑇∧𝜎_𝑘∧𝜌_𝑘 and 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1, where 𝜎_𝑘 is defined in Lemma 2.7 and 𝜌_𝑘=inf{𝑡≥0∶|𝑋(𝑡)|≥𝑘}.

Proof. For 0≤𝑡≤𝑇∧𝜎_𝑘∧𝜌_𝑘 and 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1, there is an integer 𝑛 such that 𝑡∈[𝑡_𝑛,𝑡_𝑛+1). Thus it follows from (3.2) that 𝑋(𝑡)−𝑍(𝑡)=+𝑋_𝑛+^𝑡𝑡_𝑛𝑎(𝑍(𝑠),𝑍(𝑠−𝜏))𝑑𝑠+^𝑡𝑡_𝑛𝑏(𝑍(𝑠),𝑍(𝑠−𝜏))𝑑𝑊(𝑠)_𝑡𝑡𝑛_𝜀𝑐(𝑍(𝑠),𝑍(𝑠−𝜏),𝑣)𝑝_𝜙̃(𝑑𝑣×𝑑𝑠)−𝑋_𝑛.(3.8)
Therefore, by taking expectations and the Cauchy-Schwarz inequality and using the martingale properties of 𝑑𝑊(𝑡) and 𝑝_𝜙̃(𝑑𝑣×𝑑𝑡), we get 𝐄||𝑋(𝑡)−𝑍(𝑡)||²≤3𝐄||||^𝑡𝑡_𝑛𝑎(𝑍(𝑠),𝑍(𝑠−𝜏))𝑑𝑠||||²+3𝐄||||^𝑡𝑡_𝑛𝑏(𝑍(𝑠),𝑍(𝑠−𝜏))𝑑𝑊(𝑠)||||²+3𝐄||||^𝑡𝑡_𝑛_𝜀𝑐(𝑍(𝑠),𝑍(𝑠−𝜏),𝑣)𝑝_𝜙̃(𝑑𝑣×𝑑𝑠)||||²≤+3Δ3𝐄𝑡𝐄^𝑡𝑡_{𝑛𝑡𝑡𝑛}||𝑎(𝑍(𝑠),𝑍(𝑠−𝜏))||²𝑑𝑠+3𝐄^𝑡𝑡_𝑛||𝑏(𝑍(𝑠),𝑍(𝑠−𝜏))||²𝑑𝑠_𝜀||𝑐(𝑍(𝑠),𝑍(𝑠−𝜏),𝑣)||²𝜙(𝑑𝑣)𝑑𝑠,(3.9) where the inequality |𝑢₁+𝑢₂+𝑢₃|²≤3|𝑢₁|²+3|𝑢₂|²+3|𝑢₃|² for 𝑢₁,𝑢₂,𝑢₃∈𝐑^𝑑 is used. Therefore, by applying Assumption 2.1, we get 𝐄^𝑡𝑡_𝑛≤||𝑎2(𝑍𝐄(𝑠_𝑡𝑡𝑛),𝑍(𝑠−𝜏))||²𝑑𝑠≤2𝐶_𝑘||𝑎(𝑍(𝑠),𝑍(𝑠−𝜏))−𝑎(0,0)||²𝑑𝑠+2𝐄^𝑡𝑡_𝑛||𝑎(0,0)||²𝑑𝑠≤4𝑘₂𝐄^𝑡𝑡_𝑛||𝑍(𝑠)||²+||𝑍(𝑠−𝜏)||²𝑑𝑠+2||𝑎(0,0)||²Δ𝑡𝐄𝐶_𝑘Δ𝑡+2||𝑎(0,0)||²Δ𝑡,_𝑡𝑡𝑛𝐄||𝑏(𝑍(𝑠),𝑍(𝑠−𝜏))||²𝑑𝑠≤4𝑘²𝐶_𝑘Δ𝑡+2||𝑏(0,0)||²Δ𝑡,_𝑡𝑡𝑛_𝜀||𝑐(𝑍(𝑠),𝑍(𝑠−𝜏),𝑣)||²𝜙(𝑑𝑣)𝑑𝑠≤4𝑘²𝐶Δ𝑡+2Δ𝑡_𝜀||𝑐(0,0,𝑣)||²𝜙(𝑑𝑣).(3.10) Hence by substituting (3.10) into (3.9), we get 𝐄||𝑋(𝑡)−𝑍(𝑡)||²≤Δ𝑡24𝑘²𝐶_𝑘+12𝑘²𝐶+6||𝑎(0,0)||²+6||𝑏(0,0)||²+6_𝜀||𝑐(0,0,𝑣)||²𝜙(𝑑𝑣),(3.11) for 0≤𝑡≤𝑇∧𝜎_𝑘∧𝜌_𝑘 and 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1.
So from Assumption 2.3, we can get the result (3.7) by choosing 𝐾₁(𝑘)=24𝑘²𝐶_𝑘+12𝑘²𝐶+6||𝑎(0,0)||²+6||𝑏(0,0)||²+6_𝜀||𝑐(0,0,𝑣)||²𝜙(𝑑𝑣).(3.12)

In the following lemma, we demonstrate that the solution of continuous-time Euler method (3.4) remains in a compact set with a large probability.

Lemma 3.2. Under Assumptions 2.1, 2.2, 2.3, and 2.4, for any pair of 𝜖∈(0,1) and 𝑇>0, there exist a sufficiently large integer 𝑘^∗ and a sufficiently small Δ𝑡^∗1 such that 𝐏𝜌_𝑘^∗≤𝑇≤𝜖,∀Δ𝑡≤Δ𝑡^∗1,(3.13) where 𝜌_𝑘^∗ is defined in Lemma 3.1.

Proof. Our proof is completed by the following steps.
Step 1. Using Itô's formula (see [1]) to 𝑉(𝑋(𝑡)), for 𝑡≥0, we have 𝑑𝑉𝑋(𝑡)=𝑉+_𝑥1𝑋(𝑡)𝑎(𝑍(𝑡),𝑍(𝑡−𝜏))2trace𝑏^𝚃+(𝑍(𝑡),𝑍(𝑡−𝜏))𝑉_𝑥𝑥𝑋(𝑡)𝑏(𝑍(𝑡),𝑍(𝑡−𝜏))𝑑𝑡_𝜀−𝑉𝑉𝑋(𝑡)+𝑐(𝑍(𝑡),𝑍(𝑡−𝜏),𝑣)+𝑉_𝑥𝑋(𝑡)−𝑉_𝑥𝑋(𝑡)𝑐(𝑍(𝑡),𝑍(𝑡−𝜏),𝑣)𝜙(𝑑𝑣)𝑑𝑡+𝑋(𝑡)𝑏(𝑍(𝑡),𝑍(𝑡−𝜏))𝑑𝑊(𝑡)_𝜀=𝐿𝑉𝑉𝑋(𝑡)+𝑐(𝑍(𝑡),𝑍(𝑡−𝜏),𝑣)−𝑉𝑋(𝑡)𝑝_𝜙̃(𝑑𝑣×𝑑𝑡)+𝑉_𝑥𝑋(𝑡),𝑋(𝑡−𝜏)𝑑𝑡+𝑓𝑋(𝑡),𝑋(𝑡−𝜏),𝑍(𝑡),𝑍(𝑡−𝜏)𝑑𝑡+𝑋(𝑡)𝑏(𝑍(𝑡),𝑍(𝑡−𝜏))𝑑𝑊(𝑡)_𝜀𝑉𝑋(𝑡)+𝑐(𝑍(𝑡),𝑍(𝑡−𝜏),𝑣)−𝑉𝑋(𝑡)𝑝_𝜙̃(𝑑𝑣×𝑑𝑡),(3.14) where 𝑓(𝑥,𝑦,𝑍₁,𝑍₂)∶𝐑^𝑑×𝐑^𝑑×𝐑^𝑑×𝐑^𝑑→𝐑 is defined by 𝑓=𝑥,𝑉𝑦_𝑥,𝑍₁,𝑍₂(𝑥)𝑎𝑍₁,𝑍₂−𝑎(𝑥,𝑦)+12trace𝑏^𝚃𝑍₁−1,𝑍₂𝑉_𝑥𝑥(𝑥)𝑏𝑍₁,𝑍₂2trace𝑏^𝚃+(𝑥,𝑦)𝑉_𝑥𝑥(𝑥)𝑏(𝑥,𝑦)_𝜀+𝑉_𝑥(𝑥)𝑐(𝑥,𝑦,𝑣)−𝑉_𝑥(𝑥)𝑐𝑍₁,𝑍₂,𝑣𝜙(𝑑𝑣)_𝜀𝑉𝑥+𝑐𝑍₁,𝑍₂,𝑣−𝑉(𝑥+𝑐(𝑥,𝑦,𝑣))𝜙(𝑑𝑣).(3.15) Moreover, for (𝑥,𝑦,𝑍₁,𝑍₂)∈𝐑^𝑑×𝐑^𝑑×𝐑^𝑑×𝐑^𝑑 with |𝑥|∨|𝑦|∨|𝑍₁|∨|𝑍₂|≤𝑘, we have 𝑓𝑥,𝑦,𝑍₁,𝑍₂=+𝑉_𝑥1(𝑥)𝑎𝑍₁,𝑍₂−𝑎(𝑥,𝑦)2trace𝑏^𝚃𝑍₁+1,𝑍₂−𝑏^𝚃(𝑥,𝑦)𝑉_𝑥𝑥(𝑥)𝑏𝑍₁,𝑍₂2trace𝑏^𝚃+(𝑥,𝑦)𝑉_𝑥𝑥(𝑥)𝑏𝑍₁,𝑍₂−𝑏(𝑥,𝑦)_𝜀+𝑉_𝑥(𝑥)𝑐(𝑥,𝑦,𝑣)−𝑐𝑍₁,𝑍₂,𝑣𝜙(𝑑𝑣)_𝜀≤𝐿_𝑘𝑉𝑥+𝑐𝑍₁,𝑍₂,𝑣−𝑉(𝑥+𝑐(𝑥,𝑦,𝑣))𝜙(𝑑𝑣)||𝑥−𝑍₁||+||𝑦−𝑍₂||,(3.16) where Assumptions 2.1 and 2.2 are used and 𝐿_𝑘 is a positive constant dependent on integer 𝑘, intensity 𝜆 and independent of Δ𝑡. Therefore, from (3.16), Assumption 2.4, and (3.7) in Lemma 3.1, we obtain 𝐄≤^{𝑡∧𝜌𝑘}₀𝐿_𝑘𝑓𝑋(𝑠),𝑋(𝑠−𝜏),𝑍(𝑠),𝑍(𝑠−𝜏)𝑑𝑠≤2𝐿𝐄_𝑘^{𝑡∧𝜌𝑘}₀||𝑋(𝑠)−𝑍(𝑠)||𝑑𝑠+𝐿_𝑘𝐄^{𝑡∧𝜌𝑘}₀||𝑋(𝑠−𝜏)−𝑍(𝑠−𝜏)||𝑑𝑠𝐄^{𝑡∧𝜌𝑘}₀||𝑋(𝑠)−𝑍(𝑠)||𝑑𝑠+𝐿_𝑘^0−𝜏≤2𝐿_𝑘||𝜉(𝑠)−𝑍(𝑠)||𝑑𝑠^𝑡0𝐄||𝑋𝑠∧𝜌_𝑘−𝑍𝑠∧𝜌_𝑘||²^1/2𝑑𝑠+𝐿_𝑘−1𝑛_=−𝑚^𝑡𝑛+1_𝑡𝑛≤2𝐿_𝑘||𝜉(𝑠)−𝜉𝑡_𝑛||𝑑𝑠𝑇√𝐾₁(𝑘)Δ𝑡+𝐿_𝑘𝐿𝜏√Δ𝑡,(3.17) for 0≤𝑡≤𝑇 and 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1. Hence by taking expectations and integration to (3.14), applying the martingale properties of 𝑑𝑊(𝑡) and 𝑝_𝜙̃(𝑑𝑣×𝑑𝑡), and then using (3.17) and Assumption 2.2, we obtain 𝐄𝑉=𝑉𝑋𝑋𝑡∧₀𝜌_𝑘+𝐄_{𝑡∧𝜌𝑘0}+𝐄^{𝑡∧𝜌𝑘}₀𝐿𝑉𝑋(𝑠),𝑋(𝑠−𝜏)𝑑𝑠≤𝑉𝑋₀𝑓𝑋(𝑠),𝑋(𝑠−𝜏),𝑍(𝑠),𝑍(𝑠−𝜏)𝑑𝑠+𝐄_{𝑡∧𝜌𝑘0}+2𝐿_𝑘𝑇√𝐾₁(𝑘)Δ𝑡+𝜏𝐿_𝑘𝐿√Δ𝑡𝜇¹1+𝑉𝑋(𝑠)+𝑉𝑋(𝑠−𝜏)+𝑈𝑋(𝑠−𝜏)−𝜇₂𝑈𝑋(𝑠)𝑑𝑠,(3.18) for 0≤𝑡≤𝑇 and 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1.
Step 2. For 0≤𝑡≤𝜏 and 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1, it follows from (3.18) that 𝐄𝑉𝑋𝑡∧𝜌_𝑘≤𝑉𝑋₀+2𝐿_𝑘𝑇√𝐾₁(𝑘)Δ𝑡+𝜏𝐿_𝑘𝐿√Δ𝑡+𝛼₁+𝜇₁𝐄^{𝑡∧𝜌𝑘}₀𝑉𝑋(𝑠)𝑑𝑠−𝜇₂𝐄^{𝑡∧𝜌𝑘}₀𝑈𝑋(𝑠)𝑑𝑠,(3.19) where 𝛼₁=𝜇₁∫^0−𝜏(1+𝑉(𝜉(𝑠))+𝑈(𝜉(𝑠)))𝑑𝑠. Thus from (3.19), we get 𝐄𝑉𝑋𝑡∧𝜌_𝑘≤+𝑉𝜏𝐿𝑋_0𝑘+2𝐿_𝑘𝑇√𝐾₁(𝑘)Δ𝑡𝐿√Δ𝑡+𝛼₁+𝜇₁𝐄^𝑡0𝑉𝑋𝑠∧𝜌_𝑘𝑑𝑠,(3.20) by the Gronwall inequality (see [18]), which gives 𝐄𝑉𝑋𝑡∧𝜌_𝑘≤𝑉𝑋₀+2𝐿_𝑘𝑇√𝐾₁(𝑘)Δ𝑡+𝜏𝐿_𝑘𝐿√Δ𝑡+𝛼₁𝑒^𝜇1𝑡,(3.21) for 0≤𝑡≤𝜏 and 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1. Moreover, from (3.19) and (3.21), we have 𝐄≤^{𝜏∧𝜌𝑘}₀𝜇𝑈₂₋₁𝑋(𝑠)𝑑𝑠𝑉𝑋⁰+2𝐿_𝑘𝑇√𝐾₁(𝑘)Δ𝑡+𝜏𝐿_𝑘𝐿√Δ𝑡+𝛼₁+𝜇₁^𝜏0≤𝑉𝑋𝐄𝑉𝑋𝑠∧𝜌_𝑘𝑑𝑠₀+2𝐿_𝑘𝑇√𝐾₁(𝑘)Δ𝑡+𝜏𝐿_𝑘𝐿√Δ𝑡+𝛼₁𝑒^𝜇1𝜏𝜇²⁻¹,(3.22) for 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1.
Step 3. For 0≤𝑡≤2𝜏 and 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1, it follows from (3.18) that 𝐄𝑉≤𝑉𝑋𝑋𝑡∧₀𝜌_𝑘+2𝐿_𝑘𝑇√𝐾₁(𝑘)Δ𝑡+𝜏𝐿_𝑘𝐿√Δ𝑡+𝛼₂+𝜇₁𝐄^{𝑡∧𝜌𝑘}₀𝑉𝑋(𝑠)𝑑𝑠−𝜇₂𝐄^{𝑡∧𝜌𝑘}₀𝑈𝑋(𝑠)𝑑𝑠.(3.23) As the same way as Step 2, we can obtain 𝐄𝑉≤𝑋𝑉𝑡𝑋∧₀𝜌_𝑘𝐄+2𝐿_𝑘𝑇√𝐾₁(𝑘)Δ𝑡+𝜏𝐿_𝑘𝐿√Δ𝑡+𝛼₂𝑒^𝜇1𝑡,0≤𝑡≤2𝜏,(3.24)_{2𝜏∧𝜌𝑘0}≤𝑉𝑈𝑋𝑋(𝑠)𝑑𝑠₀+2𝐿_𝑘𝑇√𝐾₁(𝑘)Δ𝑡+𝜏𝐿_𝑘𝐿√Δ𝑡+𝛼₂𝑒^2𝜇1𝜏𝜇²⁻¹,(3.25) where 𝛼₂==𝜇𝜇₁₁𝐄^{2𝜏∧𝜌𝑘}₀1+𝑉𝑋(𝑠−𝜏)+𝑈𝑋(𝑠−𝜏)𝑑𝑠≤𝜇₁𝐄^{2𝜏∧𝜌𝑘−𝜏−𝜏}1+𝑉𝑋(𝑠)+𝑈𝑋(𝑠)𝑑𝑠=𝛼₁𝐄^{𝜏∧𝜌𝑘}_−𝜏1+𝑉𝑋(𝑠)+𝑈𝑋(𝑠)𝑑𝑠≤𝛼₁+𝜇₁𝐄^{𝜏∧𝜌𝑘}₀1+𝑉𝑋(𝑠)+𝑈𝑋(𝑠)𝑑𝑠+𝜇₁𝜏+𝜇₁^𝜏0≤𝜇₁𝐄𝑉𝑋𝑠∧𝜌_𝑘𝑑𝑠+𝜇₁𝐄^{𝜏∧𝜌𝑘}₀𝑈𝑋(𝑠)𝑑𝑠𝜏+𝑉𝑋₀+2𝐿_𝑘𝑇√𝐾₁(𝑘)Δ𝑡+𝜏𝐿_𝑘𝐿√Δ𝑡+𝛼₁𝑒^𝜇1𝜏+𝑒^𝜇1𝜏𝜇₁𝜇²⁻¹,(3.26) from (3.21) and (3.22). So (3.24) becomes 𝐄𝑉𝑋𝑡∧𝜌_𝑘≤𝑉𝑋₀+2𝐿_𝑘𝑇√𝐾₁(𝑘)Δ𝑡+𝜏𝐿_𝑘𝐿√Δ𝑡+𝛼₁𝛽^1,0+𝛽_2,0,(3.27) for 0≤𝑡≤2𝜏 and 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1, where 𝛽_1,0=1+𝑒^𝜇1𝜏+𝑒^𝜇1𝜏𝜇₁𝜇²⁻¹𝑒^2𝜇1𝜏,𝛽_2,0=𝜇₁𝜏−𝛼₁𝑒^2𝜇1𝜏.(3.28)
Step 4. By repeating the same way in Steps 2 and 3, we get 𝐄𝑉𝑋𝑇∧𝜌_𝑘≤𝑉𝑋₀+2𝐿_𝑘𝑇√𝐾₁(𝑘)Δ𝑡+𝜏𝐿_𝑘𝐿√Δ𝑡+𝛼₁𝛽¹+𝛽₂,(3.29) for 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1, where 𝛽₁ and 𝛽₂ are two constants dependent on 𝜇₁,𝜇₂,𝜏,𝑇 and independent of 𝑘 and Δ𝑡. Therefore, we have 𝐏𝜌_𝑘≤𝑇≤𝑉𝑋⁰𝛽₁+𝛼₁𝛽₁+𝛽₂+2𝛽₁𝐿_𝑘𝑇√𝐾₁(𝑘)Δ𝑡+𝛽₁𝜏𝐿_𝑘𝐿√Δ𝑡𝜗_𝑘,(3.30) where 𝜗_𝑘=inf_{||𝑋||≥𝑘}𝑉𝑋,𝑘≥𝐿√𝜏+||𝜉(0)||+1.(3.31) Now, for any 𝜖∈(0,1), we can choose sufficiently large integer 𝑘^∗ such that 𝑉𝑋₀𝛽₁+𝛼₁𝛽₁+𝛽₂𝜗_𝑘^∗≤𝜖2(3.32) and sufficiently small Δ𝑡^∗1 such that 2𝛽₁𝐿_𝑘^∗𝑇𝐾₁𝑘^∗Δ𝑡^∗1+𝛽₁𝜏𝐿_𝑘^∗𝐿√Δ𝑡^∗1𝜗_𝑘^∗≤𝜖2.(3.33) So from (3.30), we can obtain 𝐏𝜌_𝑘^∗≤𝑇≤𝜖,∀Δ𝑡≤Δ𝑡^∗1.(3.34)

4. Convergence in Probability

In this section, we show the convergence in probability of the Euler method to (2.1) over a finite time interval [0,𝑇], which is based on the following lemma.

Lemma 4.1. Under Assumptions 2.1, 2.3, and 2.4, for any 𝑇>0, there exists a positive constant 𝐾₂(𝑘), dependent on 𝑘 and independent of Δ𝑡, such that for all Δ𝑡∈(0,1) the solution of (2.1) and the continuous-time Euler method (3.4) satisfy 𝐄sup_{0≤𝑡≤𝑇}||𝑥𝑡∧𝜎_𝑘∧𝜌_𝑘−𝑋𝑡∧𝜎_𝑘∧𝜌_𝑘||²≤𝐾²(𝑘)Δ𝑡,(4.1) where 𝜎_𝑘 and 𝜌_𝑘 are defined in Lemmas 2.7 and 3.1, respectively, and 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1.

Proof. From (2.1) and (3.4), for any 0≤𝑡^≤𝑇 and 𝑘≥[𝐿√𝜏+|𝜉(0)|]+1, we have 𝐄sup_{0≤𝑡≤𝑡′}||𝑥𝑡∧𝜎_𝑘∧𝜌_𝑘−𝑋𝑡∧𝜎_𝑘∧𝜌_𝑘||²≤3𝐄sup_{0≤𝑡≤𝑡′}||||^𝑡∧𝜎_𝑘^∧𝜌_𝑘0(𝑎(𝑥(𝑠⁻),𝑥((𝑠−𝜏)⁻))−𝑎(𝑍(𝑠),𝑍(𝑠−𝜏)))𝑑𝑠||||^𝟐+3𝐄sup_{0≤𝑡≤𝑡′}||||^𝑡∧𝜎_𝑘^∧𝜌_𝑘0(𝑏(𝑥(𝑠⁻),𝑥((𝑠−𝜏)⁻))−𝑏(𝑍(𝑠),𝑍(𝑠−𝜏)))𝑑𝑊(𝑠)||||^𝟐+3𝐄sup_{0≤𝑡≤𝑡′}||||^𝑡∧𝜎_𝑘^∧𝜌_𝑘0_𝜀(𝑐(𝑥(𝑠⁻),𝑥((𝑠−𝜏)⁻),𝑣)−𝑐(𝑍(𝑠),𝑍(𝑠−𝜏),𝑣))𝑝_𝜙̃(𝑑𝑣×𝑑𝑠)||||^𝟐,(4.2) where the inequality |𝑢₁+𝑢₂+𝑢₃|²≤3|𝑢₁|²+3|𝑢₂|²+3|𝑢₃|² for 𝑢₁,𝑢₂,𝑢₃∈𝐑^𝑑 is used. Therefore, by using the Cauchy-Schwarz inequality, Assumptions 2.1 and 2.4, Fubini's Theorem, and Lemma 3.1, we obtain 𝐄sup_{0≤𝑡≤𝑡′}||||^𝑡∧𝜎_𝑘^∧𝜌_𝑘0(𝑎(𝑥(𝑠⁻),𝑥((𝑠−𝜏)⁻))−𝑎(𝑍(𝑠),𝑍(𝑠−𝜏)))𝑑𝑠||||²≤𝐄sup_{0≤𝑡≤𝑡′}^{𝑡∧𝜎𝑘∧𝜌𝑘}₀≤𝑇𝐄_{𝑡′∧𝜎𝑘∧𝜌𝑘0}1²𝑑𝑠^{𝑡∧𝜎𝑘∧𝜌𝑘}₀||𝑎(𝑥(𝑠⁻),𝑥((𝑠−𝜏)⁻))−𝑎(𝑍(𝑠),𝑍(𝑠−𝜏))||²𝑑𝑠≤𝑇𝐶_𝑘||𝑎(𝑥(𝑠⁻),𝑥((𝑠−𝜏)⁻))−𝑎(𝑍(𝑠),𝑍(𝑠−𝜏))||²𝑑𝑠≤2𝑇𝐶𝐄_𝑘^{𝑡′∧𝜎𝑘∧𝜌𝑘}₀||𝑥(𝑠⁻)−𝑍(𝑠)||²𝑑𝑠+𝑇𝐶_𝑘𝐄^{𝑡′∧𝜎𝑘∧𝜌𝑘}₀||𝑥((𝑠−𝜏)⁻)−𝑍(𝑠−𝜏)||²𝑑𝑠𝐄^{𝑡′∧𝜎𝑘∧𝜌𝑘}₀||𝑥(𝑠⁻)−𝑍(𝑠)||²𝑑𝑠+𝑇𝐶_𝑘^0−𝜏≤4𝑇𝐶_𝑘||𝜉(𝑠)−𝑍(𝑠)||²𝑑𝑠+4𝑇𝐶𝐄_𝑘^{𝑡′∧𝜎𝑘∧𝜌𝑘}₀||𝑋(𝑠)−𝑍(𝑠)||²𝑑𝑠+𝑇𝐶𝐄_𝑘−1^{𝑡′∧𝜎𝑘∧𝜌𝑘}₀||𝑥(𝑠⁻)−𝑋(𝑠)||²𝑑𝑠_𝑛_=−𝑚^𝑡𝑛+1_𝑡𝑛≤4𝑇𝐶_𝑘||𝜉(𝑠)−𝜉𝑡_𝑛||²𝑑𝑠^𝑡′₀+4𝑇𝐶_𝑘𝐄||𝑋𝑠∧𝜎_𝑘∧𝜌_𝑘−𝑍𝑠∧𝜎_𝑘∧𝜌_𝑘||²𝑑𝑠^𝑡′₀𝐄sup_{0≤𝑢≤𝑠}||𝑥𝑢∧𝜎_𝑘∧𝜌^−𝑘−𝑋𝑢∧𝜎_𝑘∧𝜌_𝑘||²𝑑𝑠+𝑇𝐶^𝑘≤4𝑇₂𝐿²𝜏Δ𝑡+4𝑇𝐶𝐶_𝑘𝑘𝐾₁(𝑘)+𝑇𝐶_𝑘𝐿²𝜏Δ𝑡^𝑡′₀𝐄sup_{0≤𝑢≤𝑠}||𝑥𝑢∧𝜎_𝑘∧𝜌^−𝑘−𝑋𝑢∧𝜎_𝑘∧𝜌_𝑘||²𝑑𝑠.(4.3) Moreover, by using the martingale properties of 𝑑𝑊(𝑡) and 𝑝_𝜙̃(𝑑𝑣×𝑑𝑡), Assumptions 2.1 and 2.4, Fubini's Theorem, and Lemma 3.1, we have 𝐄sup_{0≤𝑡≤𝑡′}||||^𝑡∧𝜎_𝑘^∧𝜌_𝑘0(𝑏(𝑥(𝑠⁻),𝑥((𝑠−𝜏)⁻))−𝑏(𝑍(𝑠),𝑍(𝑠−𝜏)))𝑑𝑊(𝑠)||||²≤4𝐄_{𝑡′∧𝜎𝑘∧𝜌𝑘0}≤4𝐶_𝑘||𝑏(𝑥(𝑠⁻),𝑥((𝑠−𝜏)⁻))−𝑏(𝑍(𝑠),𝑍(𝑠−𝜏))||²𝑑𝑠≤8𝐶_𝑘𝐄^{𝑡′∧𝜎𝑘∧𝜌𝑘}₀||𝑥(𝑠⁻)−𝑍(𝑠)||²𝑑𝑠+4𝐶_𝑘𝐄^{𝑡′∧𝜎𝑘∧𝜌𝑘}₀||𝑥((𝑠−𝜏)⁻)−𝑍(𝑠−𝜏)||²𝑑𝑠𝐄^{𝑡′∧𝜎𝑘∧𝜌𝑘}₀||𝑥(𝑠⁻)−𝑍(𝑠)||²𝑑𝑠+4𝐶_𝑘^0−𝜏≤16𝐶_𝑘||𝜉(𝑠)−𝑍(𝑠)||²𝑑𝑠+4𝐶𝐄_𝑘−1^{𝑡′∧𝜎𝑘∧𝜌𝑘}₀||𝑋(𝑠)−𝑍(𝑠)||²𝑑𝑠+16𝐶_𝑘𝐄^{𝑡′∧𝜎𝑘∧𝜌𝑘}₀||𝑥(𝑠⁻)−𝑋(𝑠)||²𝑑𝑠_𝑛_=−𝑚^𝑡𝑛+1_𝑡𝑛≤16𝑇𝐶_𝑘||𝜉(𝑠)−𝜉𝑡_𝑛||²𝑑𝑠+16𝐶_𝑘𝐾₁(𝑘)+4𝐶_𝑘𝐿²𝜏Δ𝑡^𝑡′₀𝐄sup_{0≤𝑢≤𝑠}||𝑥𝑢∧𝜎_𝑘∧𝜌^−𝑘−𝑋𝑢∧𝜎_𝑘∧𝜌_𝑘||²𝐄sup_{0≤𝑡≤𝑡′}𝑑𝑠,||||^𝑡∧𝜎_𝑘^∧𝜌_𝑘0_𝜀(𝑐(𝑥(𝑠⁻),𝑥((𝑠−𝜏)⁻),𝑣)−𝑐(𝑍(𝑠),𝑍(𝑠−𝜏),𝑣))𝑝_𝜙̃(𝑑𝑣×𝑑𝑠)||||²≤4𝐄||||_{𝑡′∧𝜎𝑘∧𝜌𝑘0}_𝜀(𝑐(𝑥(𝑠⁻),𝑥((𝑠−𝜏)⁻),𝑣)−𝑐(𝑍(𝑠),𝑍(𝑠−𝜏),𝑣))𝑝_𝜙̃(𝑑𝑣×𝑑𝑠)||||²=4𝐄^{𝑡′∧𝜎𝑘∧𝜌𝑘}₀_𝜀≤16𝑇𝐶𝐾₁||𝑐(𝑥(𝑠⁻),𝑥((𝑠−𝜏)⁻),𝑣)−𝑐(𝑍(𝑠),𝑍(𝑠−𝜏),𝑣)||²𝜙(𝑑𝑣)𝑑𝑠+16𝐶_𝑡′0(𝑘)+4𝐶𝐿²𝜏Δ𝑡𝐄sup_{0≤𝑢≤𝑠}||𝑥𝑢∧𝜎_𝑘∧𝜌^−𝑘−𝑋𝑢∧𝜎_𝑘∧𝜌_𝑘||²𝑑𝑠.(4.4) Therefore by substituting (4.3) and (4.4) into (4.2), we get 𝐄sup_{0≤𝑡≤𝑡′}||𝑥𝑡∧𝜎_𝑘∧𝜌_𝑘−𝑋𝑡∧𝜎_𝑘∧𝜌_𝑘||²≤Δ𝑡12𝑇₂+12𝑇𝐶_𝑘𝐶_𝑘𝐾₁(𝑘)+3𝑇𝐶_𝑘𝐿²𝜏+48𝑇𝐶_𝑘𝐾₁(𝑘)+48𝑇𝐶𝐾₁(𝑘)+12𝐶_𝑘𝐿²𝜏+12𝐶𝐿²𝜏+48𝐶_𝑘+48𝐶^𝑡′₀𝐄sup_{0≤𝑢≤𝑠}||𝑥𝑢∧𝜎_𝑘∧𝜌^−𝑘−𝑋𝑢∧𝜎_𝑘∧𝜌_𝑘||²𝑑𝑠.(4.5) So by using the Gronwall inequality (see [18]), we have the result (4.1) by choosing 𝐾₂(𝑘)=1+21𝑇2𝐶²𝐶_𝑘𝑘𝐾₁(𝑘)+3𝑇𝐶_𝑘𝐿²𝜏+48𝑇𝐶_𝑘𝐾₁(𝑘)+48𝑇𝐶𝐾₁(𝑘)𝐿²𝜏+12𝐶𝐿²𝜏exp12𝑇²𝐶_𝑘+48𝑇𝐶_𝑘+48𝑇𝐶.(4.6)

Now, we state our main theorem which shows the convergence in probability of the continuous-time Euler method (3.4).

Theorem 4.2. Under Assumptions 2.1, 2.2, 2.3, and 2.4, for sufficiently small 𝜖,𝜍∈(0,1), there is a Δ𝑡^∗ such that for all Δ𝑡<Δ𝑡^∗𝐏sup_{0≤𝑡≤𝑇}||𝑥(𝑡)−𝑋(𝑡)||²≥𝜍≤𝜖,(4.7) for any 𝑇>0.

Proof. For sufficiently small 𝜖,𝜍∈(0,1), we define Ω=𝜔∶sup_{0≤𝑡≤𝑇}||𝑥(𝑡)−𝑋(𝑡)||²≥𝜍.(4.8) By Lemmas 2.9 and 3.2, there exists a pair of 𝑘^⋆ and Δ𝑡^⋆1 such that 𝐏𝜎_𝑘^⋆≤𝑇≤𝜖𝐏𝜌_𝑘⋆3,≤𝑇≤𝜖3,∀Δ𝑡≤Δ𝑡^⋆1.(4.9) We then have 𝐏Ω≤≤𝐏𝐏Ω∩𝜎_𝑘^⋆∧𝜌_𝑘^⋆>𝑇+𝐏𝜎_𝑘^⋆∧𝜎_𝑘^⋆≤𝑇≤𝐏Ω∩𝜎_𝑘^⋆∧𝜎_𝑘^⋆>𝑇+𝐏𝜎_𝑘^⋆≤𝑇+𝐏𝜌_𝑘^⋆≤𝑇Ω∩𝜎_𝑘^⋆∧𝜌_𝑘^⋆>𝑇+2𝜖3,(4.10) for Δ𝑡≤Δ𝑡^⋆1. Moreover, from Lemma 4.1, we have 𝜍𝐏Ω∩𝜎_𝑘^⋆∧𝜌_𝑘^⋆>𝑇≤𝐄𝐼_{{𝜎𝑘⋆∧𝜌𝑘⋆>𝑇}}sup_{0≤𝑡≤𝑇}||𝑥(𝑡)−𝑋(𝑡)||²≤𝐄sup_{0≤𝑡≤𝑇}||𝑥𝑡∧𝜎_𝑘⋆∧𝜌_𝑘^⋆−𝑋𝑡∧𝜎_𝑘^⋆∧𝜌_𝑘^⋆||²≤𝐾₂𝑘^⋆Δ𝑡,(4.11) which gives 𝐏Ω∩𝜎_𝑘^⋆∧𝜌_𝑘^⋆>𝑇≤𝜖3,(4.12) for Δ𝑡≤Δ𝑡^⋆2. Hence, from the inequalities above, we get 𝐏Ω≤𝜖,(4.13) for Δ𝑡≤Δ𝑡^⋆, where Δ𝑡^⋆=min{Δ𝑡^⋆1,Δ𝑡^⋆2}.
We remark that the continuous-time Euler solution 𝑋(𝑡) cannot be computed, since it requires knowledge of the entire Brownian motion and Poisson random measure paths. So the last theorem demonstrates the convergence in probability of the discrete Euler solution (3.2).

Theorem 4.3. Under Assumptions 2.1, 2.2, 2.3, and 2.4, for sufficiently small 𝜖,𝜍∈(0,1), there is a Δ𝑡^∗ such that for all Δ𝑡<Δ𝑡^∗𝐏||𝑥(𝑡)−𝑍(𝑡)||²≥𝜍,0≤𝑡≤𝑇≤𝜖,(4.14) for any 𝑇>0.

Proof. For sufficiently small 𝜖,𝜍∈(0,1), we define Ω=𝜔∶||𝑥(𝑡)−𝑍(𝑡)||²≥𝜍,0≤𝑡≤𝑇.(4.15) As the same analysis as the proof in Theorem 4.2, we have 𝐏Ω≤𝐏Ω∩𝜎_𝑘^⋆∧𝜌_𝑘^⋆>𝑇+2𝜖3.(4.16)
Recalling that 𝜍𝐏Ω∩𝜎_𝑘^⋆∧𝜌_𝑘^⋆>𝑇≤≤𝐄𝐄||𝑥||𝑥𝑇(𝑇∧)𝜎−_𝑘⋆𝑍(𝑇)||²𝐼_{{𝜎𝑘⋆∧𝜌𝑘⋆>𝑇}}∧𝜌_𝑘^⋆−𝑍𝑇∧𝜎_𝑘^⋆∧𝜌_𝑘^⋆||²≤2𝐄sup_{0≤𝑡≤𝑇}||𝑥𝑡∧𝜎_𝑘⋆∧𝜌_𝑘^⋆−𝑋𝑡∧𝜎_𝑘^⋆∧𝜌_𝑘^⋆||²+2𝐄||𝑋𝑇∧𝜎_𝑘^⋆∧𝜌_𝑘^⋆−𝑍𝑇∧𝜎_𝑘^⋆∧𝜌_𝑘^⋆||²≤2𝐾₁𝑘^⋆Δ𝑡+2𝐾₂𝑘^⋆Δ𝑡(4.17) and using Lemmas 3.1 and 4.1, we obtain 𝐏Ω∩𝜎_𝑘^⋆∧𝜌_𝑘^⋆>𝑇≤𝜖3,(4.18) for sufficiently small Δ𝑡. So the inequalities above demonstrate 𝐏Ω≤𝜖,(4.19) for all sufficiently small Δ𝑡.
Hence we complete the result (4.14).

5. Numerical Example

In this section, a numerical example is analyzed under Assumptions 2.1, 2.2, 2.3, and 2.4 which cover many highly nonlinear SDDEs driven by Poisson random measure.

Now, we consider the following equation: 𝑑𝑥(𝑡)=+2𝑥²((𝑡−0.05)⁻)−4𝑥³(𝑡⁻)𝑑𝑡+3𝑥²((𝑡−0.05)⁻)𝑑𝑊(𝑡)_𝜀𝑣𝑥²((𝑡−0.05)⁻)𝑝_𝜙̃(𝑑𝑣×𝑑𝑡),𝑡>0,(5.1) with 𝑥(𝑡)=𝑡²,𝑡∈[−0.05,0], where 𝑑=𝑚=𝑟=1. The compensated measure of the Poisson random measure 𝑝_𝜙(𝑑𝑣×𝑑𝑡) is given by 𝜙(𝑑𝑣)𝑑𝑡=𝜆𝑓(𝑣)𝑑𝑣𝑑𝑡, where 𝜆=5 and 𝑓(𝑣)=1√2𝜋𝑣exp−(ln2𝑣)²,0≤𝑣<∞,(5.2) is the density function of a lognormal random variable.

Clearly, the equation cannot satisfy the global Lipschitz conditions, the linear growth conditions and the classical Khasminskii-type conditions. But, the local Lipschitz conditions are satisfied. On the other hand, for 𝑉(𝑥)=|𝑥|², we have 𝐿𝑉(𝑥,𝑦)=2𝑥2𝑦²−4𝑥³+9𝑦⁴+_𝜀𝑣²𝑦⁴𝜆1≤2𝑥₂√2𝜋𝑣exp−(ln2𝑣)²𝑑𝑣+60𝑦⁴−8𝑥⁴=601+𝑥²+𝑦²+𝑦⁴−8𝑥⁴,(5.3) where 𝑈(𝑥)=𝑥⁴,𝜇₁=60,𝜇₂=8. In other words, the equation satisfies Assumptions 2.1, 2.2, 2.3, and 2.4.

So according to Theorem 2.8, (5.1) has a unique global solution 𝑥(𝑡) on 𝑡∈[−0.05,∞).

Given the stepsize Δ𝑡=0.0025, we can have the Euler method to (5.1) 𝑋𝑋_𝑛=(𝑛Δ𝑡)²,for𝑛=−20,−19,…,−1,0,_𝑛+1=𝑋_𝑛+2𝑋^2𝑛−20−4𝑋^3𝑛Δ𝑡+3𝑋^2𝑛−20Δ𝑊_𝑛+^𝑡𝑛+1_𝑡𝑛_𝜀𝑣𝑋^2𝑛−20𝑝_𝜙̃(𝑑𝑣×𝑑𝑡),(5.4) for 𝑛=0,1,…, where Δ𝑊_𝑛=𝑊(𝑡_𝑛+1)−𝑊(𝑡_𝑛).

And in Matlab experiment, we actually obtain the discrete Euler 𝑋_𝑛+1=+𝑋3^𝑛𝑋+2_2𝑛−20𝑋^2𝑛−20−4𝑋^3𝑛−5√𝑒𝑋^2𝑛−20Δ𝑡Δ𝑊_𝑛+𝑋^{2𝑝𝑛−20𝜙(𝑡𝑛+1)}_{𝑖=𝑝𝜙}_{𝑡𝑛+1}𝜉_𝑖,(5.5) where 𝜉_𝑖 is distributed according to 𝑓(𝑣). Subsequently, we can get the result in Theorems 4.2 and 4.3.

Acknowledgment

This work is supported by the NSF of China (no. 11071050).