Approximate Solutions of Hybrid Stochastic Pantograph Equations with Levy Jumps

Abstract

We investigate a class of stochastic pantograph differential equations with Markovian switching and Levy jumps. We prove that the approximate solutions converge to the true solutions in 𝐿² sense as well as in probability under local Lipschitz condition and generalize the results obtained by Fan et al. (2007), Milošević and Jovanović (2011), and Marion et al. (2002) to cover a class of more general stochastic pantograph differential equations with jumps. Finally, an illustrative example is given to demonstrate our established theory.

1. Introduction

Stochastic delay differential equations (SDDEs) have 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. Similar to SDEs, an explicit solution can rarely be obtained for SDDEs. It is necessary to develop numerical methods and to study the properties of these methods. There are many results for the numerical solutions of SDDEs [1–12].

Recently, as a special case of SDDEs, a class of stochastic pantograph delay equations (SPEs) has been received a great deal of attention and various studies have been carried out on the convergence of SPEs [13–16]. However, all equations of the above-mentioned works are driven by white noise perturbations with continuous initial data, and white noise perturbations are not always appropriate to interpret real data in a reasonable way. In real phenomena, the state of stochastic pantograph delay system may be perturbed by abrupt pulses or extreme events. A more natural mathematical framework for these phenomena takes into account other than purely Brownian perturbations. In particular, we incorporate the Levy perturbations with jumps into stochastic pantograph delay system to model abrupt changes.

The study of the convergence of the numerical solutions to SDDEs with jumps is in its infancy [17–20], and there is no research on the numerical solutions to SPEs with Markovian switching and Levy jumps (SPEwMsLJs). In this paper, we study the strong convergence of the Euler method for a class of SPEs with Markovian switching and Levy jumps (SPEwMsLJs). SPEwMsLJs may be regarded as an extension of SPEs with Markovian switching and SPEs with Levy jumps. The main aim is to prove that the Euler approximate solutions converges to the true solutions for SPEwMsLJs in 𝐿² sense. On the other hand, we study the convergence in probability of the Euler approximate solutions to the true solutions under local Lipschitz condition and some additional conditions in term of Lyapunov-type functions. It should be pointed out that the proof for SPEwMsLJs is certainly not a straightforward generalization of that for SPEs and SPEwMs without Levy jumps. Although the way of analysis follows the ideas of [21], we need to develop several new techniques to deal with Levy jumps. Some known results in Fan et al. [14], Milošević and Jovanović [16], and Marion et al. [21] are generalized to cover a class of more general SPEwMsLJs.

The paper is organized as follows. In Section 2, we introduce some notations and hypotheses concerning (4), and the Euler methods is used to produce a numerical solutions. In Section 3, we establish some useful lemmas and prove that the approximate solutions converge to the true solutions of SPEwMsLJs in 𝐿² sense. By applying Theorem 4, we study the convergence in probability of the approximate solutions to the true solutions in Section 4. Finally, we give an illustrative example in Section 5.

2. Preliminaries and the Approximate Solution

Let (Ω,ℱ,𝑃) be a complete probability space with a filtration (ℱ_𝑡)_𝑡≥0 satisfying the usual condition; that is, the filtration (ℱ_𝑡) is continuous on the right and (ℱ₀) contains all 𝑃-null sets. Let {𝑊(𝑡), 𝑡≥0} be a 𝑑-dimensional Wiener process defined on the probability space (Ω,ℱ,𝑃) adapted to the filtration (ℱ_𝑡)_𝑡≥0. Let 𝐷([0,𝑇],𝑅^𝑛) denote the family of function 𝑓 from [0,𝑇]→𝑅^𝑛 that are right continuous and have limits on the left. Also 𝐷([0,𝑇],𝑅^𝑛) is equipped with the norm ‖𝑓‖=sup_{0≤𝑡≤𝑇}|𝑓(𝑡)|, where |⋅| is the Euclidean norm in 𝑅^𝑛; that is, |𝑥|=√𝑥^⊤𝑥(𝑥∈𝑅^𝑛). Let 𝑇>0, 𝑃≥2, and ℒ^𝑃([0,𝑇];𝑅^𝑛) denote the family of all 𝑅^𝑛-valued measurable (ℱ_𝑡)-adapted processes 𝑓={𝑓(𝑡)}_{0≤𝑡≤𝑇} such that 𝐸sup_{0≤𝑡≤𝑇}|𝑓(𝑡)|^𝑃<∞. Let (𝑅^𝑛,ℬ(𝑅^𝑛)) be a measurable space and 𝜋(𝑑𝑢) a 𝜎-finite measure on it. Let 𝑝=𝑝(𝑡), and let 𝑡∈𝐷_𝑝 be a stationary ℱ_𝑡-Poisson point process on 𝑅^𝑛 with characteristic measure 𝜋. Denote by 𝑁(𝑑𝑡,𝑑𝑢) the Poisson counting measure associated with 𝑝; that is, 𝑁(𝑡,𝐴)=_{𝑠∈𝐷𝑝,𝑠≤𝑡}𝐼_𝐴𝑝(𝑠). (1) We refer to Mao [3] for the properties of a Wiener process and SDDEs and to Ikeda and Watanabe [22] for the details on Poisson point process.

Let 𝑟(𝑡), 𝑡≥0, be a right-continuous Markov chain on the probability space (Ω,ℱ,𝑃) taking values in a finite state space 𝑆={1,2,…,𝑁} with generator Γ=(𝛾_𝑖𝑗)_𝑁×𝑁 given by 𝑃𝑟()(𝑡)=𝛾𝑡+Δ=𝑗∣𝑟=𝑖_𝑖𝑗Δ+∘(Δ),if𝑖≠𝑗,1+𝛾_𝑖𝑗Δ+∘(Δ),if𝑖=𝑗, (2) where Δ>0. Here 𝛾_𝑖𝑗≥0 is the transition rate from 𝑖 to 𝑗, 𝑖≠𝑗, while 𝛾_𝑖𝑖=−_𝑗≠𝑖𝛾_𝑖𝑗. (3) We assume that Markov chain 𝑟(⋅) is independent of the Brownian motion 𝑊(⋅) and compensated Poisson random measure 𝑁(⋅,⋅). It is known that almost every sample path of 𝑟(⋅) is right-continuous step function with a finite number of simple jumps in any finite subinterval of 𝑅₊.

In this paper, we study the following hybrid stochastic pantograph equations with Levy jump: (𝑡)𝑋(𝑡)(𝑡)𝑋(𝑡)(𝑡)(𝑡)+𝑑𝑋=𝑓,𝑋𝑞𝑡,𝑟𝑑𝑡+𝑔,𝑋𝑞𝑡,𝑟𝑑𝑊_𝑅𝑛ℎ𝑋(𝑡)(𝑡)𝑁(),𝑋(0),𝑋𝑞𝑡,𝑟,𝑢𝑑𝑡,𝑑𝑢=𝑋₀, (4) where 0<𝑞<1 and 𝑓∶𝑅⁺×𝑅^𝑛×𝑆→𝑅^𝑛,𝑔∶𝑅⁺×𝑅^𝑛×𝑆→𝑅^𝑛×𝑚,ℎ∶𝑅⁺×𝑅^𝑛×𝑆×𝑅^𝑛→𝑅^𝑛, (5)𝑊(𝑡) is a standard 𝑚-dimensional Brownian motion, and 𝑁(𝑑𝑡,𝑑𝑢) is the compensated Poisson random measure given by 𝑁(𝑑𝑡,𝑑𝑢)=𝑁(𝑑𝑡,𝑑𝑢)−𝜋(𝑑𝑢)𝑑𝑡. (6) Here 𝜋(𝑑𝑢) is the Levy measure associated to 𝑁.

Let 𝐶²(𝑅^𝑛×𝑆,𝑅₊) denote the family of all nonnegative functions 𝑉(𝑥,𝑖) on 𝑅^𝑛×𝑆 which are continuously twice differentiable in 𝑥. For each 𝑉∈𝐶²(𝑅^𝑛×𝑆,𝑅₊), define an operator 𝐿𝑉 from 𝑅^𝑛×𝑅^𝑛×𝑆 to 𝑅 by 𝐿𝑉𝑥,𝑦,𝑖≡𝑉_𝑥()𝑓+1𝑥,𝑖𝑥,𝑦,𝑖2trace𝑔^⊤𝑥,𝑦,𝑖𝑉_𝑥𝑥()𝑔+𝑥,𝑖𝑥,𝑦,𝑖_𝑅𝑛𝑉𝑥+ℎ𝑥,𝑦,𝑢,𝑖−𝑉(𝑥,𝑖)−𝑉_𝑥()ℎ𝜋()+𝑥,𝑖𝑥,𝑦,𝑢𝑑𝑢_𝑁𝑗=1𝛾_𝑖𝑗𝑉𝑥,𝑗, (7) where 𝑉_𝑥()=()𝑥,𝑖𝜕𝑉𝑥,𝑖𝜕𝑥₁(),…,𝜕𝑉𝑥,𝑖𝜕𝑥_𝑛,𝑉_𝑥𝑥()=𝜕𝑥,𝑖²𝑉(𝑥,𝑖)𝜕𝑥_𝑖𝜕𝑥_{𝑗𝑛×𝑛}. (8)

In order to define the Euler approximate solution of (4), we need the property of embedded discrete Markov chain. The following lemma [23] describes this property.

Lemma 1. For ℎ>0 and 𝑛≥0, then {𝑟^ℎ_𝑛,𝑛=0,1,2,…} is a discrete Markov chain with the one-step transition probability matrix 𝑃(ℎ)=𝑃_𝑖𝑗(ℎ)_𝑁×𝑁=𝑒^ℎΓ. (9)

Given a step size ℎ>0, the discrete Markov chain {𝑟^ℎ_𝑛,𝑛=0,1,2,…} can be simulated as follows (see Mao and Yuan [24]). Let 𝑟(0)=𝑖₀ and generate a random number 𝜁₁ which is uniformly distributed in [0,1]. If 𝜁₁=1, then let 𝑟^ℎ₁=𝑖₁=𝑁 or otherwise find the unique integer 𝑖₁∈𝑆 for ^𝑖1−1_𝑗=1𝑃_𝑖0,𝑗(ℎ)≤𝜁₁<^𝑖1_𝑗=1𝑃_𝑖0,𝑗(ℎ) (10) and let 𝑟^ℎ₁=𝑖₁, where we set ∑⁰_𝑖=1𝑃_𝑖0,𝑗(ℎ)=0 as usual. Generate independently a new random number 𝜁₂ which is again uniformly distributed in [0,1]. If 𝜁₂=1 then let 𝑟^ℎ₂=𝑖₂=𝑁 or otherwise find the unique integer 𝑖₂∈𝑆 for ^𝑖2−1_𝑗=1𝑃_𝑖1,𝑗(ℎ)≤𝜁₂<^𝑖2_𝑗=1𝑃_𝑖1,𝑗(ℎ) (11) and let 𝑟^ℎ₂=𝑖₂. Repeating this procedure, a trajectory of {𝑟^ℎ_𝑛,𝑛=0,1,2,…} can be generated. This procedure can be carried out independently to obtain more trajectories.

Now we define the Euler approximate solution to (4) with discrete Markov chain {𝑟^ℎ_𝑛=𝑟(𝑛ℎ),𝑛=0,1,2,…}. For system (4), the discrete approximation is given by the iterative scheme 𝑌_𝑛+1=𝑌_𝑛+𝑓𝑌_𝑛,𝑌_[𝑞𝑛],𝑟^ℎ_𝑛ℎ+𝑔𝑌_𝑛,𝑌_[𝑞𝑛],𝑟^ℎ_𝑛Δ𝑊_𝑛+_𝑅𝑛ℎ𝑌_𝑛,𝑌_[𝑞𝑛],𝑟^ℎ_𝑛,𝑢𝑁(ℎ,𝑑𝑢), (12) with initial value 𝑌₀=𝑋(0), and [𝑢] represents the integer part of 𝑢. Here 𝑡_𝑛=𝑛ℎ for 𝑛≥0. We have 𝑌_𝑛≈𝑋(𝑡_𝑛), 𝑌_[𝑞𝑛]≈𝑋(𝑞𝑡_𝑛), Δ𝑊_𝑛=𝑊(𝑡_𝑛+1)−  𝑊(𝑡_𝑛), and 𝑁(ℎ,𝑑𝑢)=𝑁(𝑡_𝑛+1,𝑑𝑢)−𝑁(𝑡_𝑛,𝑑𝑢).

Let us introduce the following notations: 𝑍₁(𝑡)=𝑌_𝑛,𝑍₂(𝑡)=𝑌_[𝑞𝑛],𝑟(𝑡)=𝑟^ℎ_𝑛 (13) for 𝑡∈[𝑡_𝑛,𝑡_𝑛+1). Then we define the continuous Euler approximate solution as follows: 𝑌(𝑡)=𝑌(0)+^𝑡₀𝑓𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)+𝑑𝑠_𝑡0𝑔𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)(𝑠)+𝑑𝑊_{𝑡0𝑅𝑛}ℎ𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠),𝑢𝑁(𝑑𝑠,𝑑𝑢),0≤𝑡≤𝑇, (14) which interpolates the discrete approximation (7).

In order to establish the strong convergence theorem, we suppose the following assumptions are satisfied.

Assumption 2. For each 𝑖∈𝑆 and 𝑢∈𝑅^𝑛, 𝑓(0,0,𝑖)²=𝑔(0,0,𝑖)²=_𝑅𝑛|ℎ(0,0,𝑖,𝑢)|²𝜋(𝑑𝑢)=0. (15)

Assumption 3. For every 𝑑≥1, there exists a positive constant 𝐾_𝑑 such that for all 𝑥₁,𝑦₁,𝑥₂,𝑦₂,𝑢∈𝑅^𝑛 and 𝑖∈𝑆, |𝑥₁|∨|𝑦₁|∨|𝑥₂|∨|𝑦₂|≤𝑑, 𝑓𝑥₁,𝑦₁,𝑖−𝑓𝑥₂,𝑦₂,𝑖²∨𝑔𝑥₁,𝑦₁,𝑖−𝑔𝑥₂,𝑦₂,𝑖²∨_𝑅𝑛ℎ𝑥₁,𝑦₁,𝑖,𝑢−ℎ𝑥₂,𝑦₂,𝑖,𝑢²𝜋(𝑑𝑢)≤𝐾_𝑑𝑥₁−𝑥₂²+𝑦₁−𝑦₂². (16)

3. Strong Convergence of Numerical Solutions

In this section, we will prove that the Euler approximate solutions converge to the true solutions in 𝐿² sense under the local Lipschitz condition.

Theorem 4. If Assumptions 2 and 3 hold, then the Euler approximate solutions converge to the true solutions of (4) in 𝐿² sense with order 1/2; that is, there exists a positive constant 𝐶_𝑑 such that 𝐸sup_{0≤𝑡≤𝑇∧𝜏𝑑}|𝑋(𝑡)−𝑌(𝑡)|²≤𝐶_𝑑[ℎ+∘(ℎ)], (17) where 𝜃_𝑑=inf{𝑡∈[0,𝑇]∶|𝑋(𝑡)|≥𝑑} and 𝜌_𝑑=inf{𝑡∈[0,𝑇]∶|𝑌(𝑡)|≥𝑑}, and let 𝜏_𝑑=𝜃_𝑑∧𝜌_𝑑.

The proof of Theorem 4 is very technical, so we present some useful lemmas.

Lemma 5. Under Assumptions 2 and 3, for any 𝑡∈[0,𝑇] and 𝑃≥2, there exists a positive constant 𝐶₁(𝑑) such that 𝐸^𝑡₀𝑌𝑠∧𝜏_𝑑−𝑍₁𝑠∧𝜏_𝑑^𝑃𝑑𝑠≤𝐶₁(𝑑)ℎ^𝑃/2𝑇, (18) where 𝐶₁(𝑑) is a positive constant independent of the step size ℎ.

Proof. For any 𝑡∈[0,𝑇∧𝜏_𝑑], there exists an integer 𝑛 such that 𝑡∈[𝑛ℎ,(𝑛+1)ℎ). Then 𝑌(𝑡)−𝑍₁(𝑡)=𝑌(𝑡)−𝑌_𝑛=^𝑡_𝑛ℎ𝑓𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠+)𝑑𝑠_𝑡𝑛ℎ𝑔𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)(𝑠)+𝑑𝑊_{𝑡𝑛ℎ𝑅𝑛}ℎ𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠),𝑢𝑁(𝑑𝑠,𝑑𝑢). (19) Using the inequality |𝑎+𝑏+𝑐|^𝑃≤3^𝑃−1[|𝑎|^𝑃+3|𝑏|^𝑃+3|𝑐|^𝑃], we get 𝐸sup_{()ℎ𝑛ℎ≤𝑡≤𝑛+1}𝑌(𝑡)−𝑍₁(𝑡)^𝑃≤3^𝑃−1𝐸sup_{()ℎ𝑛ℎ≤𝑡≤𝑛+1}^𝑡_𝑛ℎ𝑓𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)𝑑𝑠^𝑃+3^𝑃−1𝐸×sup_{()ℎ𝑛ℎ≤𝑡≤𝑛+1}^𝑡_𝑛ℎ𝑔𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)𝑑𝑊(𝑠)^𝑃+3^𝑃−1𝐸sup_{()ℎ𝑛ℎ≤𝑡≤𝑛+1}^𝑡_{𝑛ℎ𝑅𝑛}ℎ𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)×𝑁(,𝑢𝑑𝑠,𝑑𝑢)^𝑃. (20) By the Hölder inequality and Assumptions 2 and 3, we have 𝐸sup_{()ℎ𝑛ℎ≤𝑡≤𝑛+1}^𝑡_𝑛ℎ𝑓𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)𝑑𝑠^𝑃≤𝐸^(𝑛+1)ℎ_𝑛ℎ𝑓𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)𝑑𝑠^𝑃≤ℎ^𝑃−1𝐸^(𝑛+1)ℎ_𝑛ℎ𝑓𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)^𝑃𝑑𝑠≤ℎ^𝑃−1𝐸^(𝑛+1)ℎ_𝑛ℎ𝑓𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)^2𝑃/2𝑑𝑠≤ℎ^𝑃−1𝐸^(𝑛+1)ℎ_𝑛ℎ2𝑓𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)−𝑓(0,0),𝑟(𝑠)²+2𝑓(0,0),𝑟(𝑠)^2𝑃/2𝑑𝑠≤ℎ^𝑃−1𝐸^(𝑛+1)ℎ_𝑛ℎ2𝐾_𝑑𝑍₁(𝑠)²+𝑍₂(𝑠)^2𝑃/2𝑑𝑠≤ℎ^𝑃−12^𝑃−1𝐾_𝑑^𝑃/2𝐸^(𝑛+1)ℎ_𝑛ℎ𝑍₁(𝑠)^𝑃+𝑍₂(𝑠)^𝑃𝑑𝑠. (21) By the definition of 𝜏_𝑑, we have |𝑌(𝑡)|<𝑑, 𝑡∈[0,𝜏_𝑑∧𝑇]. so we get that |𝑍₁(𝑡)|^𝑃<𝑑^𝑃, |𝑍₂(𝑡)|^𝑃<𝑑^𝑃, and 𝐸sup_{()ℎ𝑛ℎ≤𝑡≤𝑛+1}^𝑡_𝑛ℎ𝑓𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)𝑑𝑠^𝑃≤2^𝑃𝑑^𝑃𝐾_𝑑^𝑃/2ℎ^𝑃. (22) By using the Burkholder-Davis-Gundy inequality and Assumptions 2 and 3, we have 𝐸sup_{()ℎ𝑛ℎ≤𝑡≤𝑛+1}^𝑡_𝑛ℎ𝑔𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)𝑑𝑊(𝑠)^𝑃≤𝐶_𝑃𝐸^(𝑛+1)ℎ_𝑛ℎ𝑔𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)²𝑑𝑠^𝑃/2≤𝐶_𝑃ℎ^(𝑃/2)−1𝐸^(𝑛+1)ℎ_𝑛ℎ𝑔𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)^𝑃𝑑𝑠≤𝐶_𝑝2^𝑃𝑑^𝑃𝐾_𝑑^𝑃/2ℎ^𝑃/2,𝐸sup_{()ℎ𝑛ℎ≤𝑡≤𝑛+1}^𝑡_{𝑛ℎ𝑅𝑛}ℎ𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠),𝑢𝑁(𝑑𝑠,𝑑𝑢)^𝑃≤𝐶_𝑃𝐸^(𝑛+1)ℎ_{𝑛ℎ𝑅𝑛}ℎ𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠),𝑢²𝜋(𝑑𝑢)𝑑𝑠^𝑃/2≤𝐶_𝑃ℎ^(𝑃/2)−1𝐸×^(𝑛+1)ℎ_{𝑛ℎ𝑅𝑛}ℎ𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠),𝑢²𝜋(𝑑𝑢)^𝑃/2𝑑𝑠≤𝐶_𝑃ℎ^(𝑃/2)−1𝐸×^(𝑛+1)ℎ_{𝑛ℎ𝑅𝑛}2ℎ𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)(,𝑢−ℎ0,0,𝑟(𝑠),𝑢)|²+|ℎ(0,0,𝑟(𝑠),𝑢)|²𝜋(𝑑𝑢)^𝑃/2𝑑𝑠≤𝐶_𝑃ℎ^(𝑃/2)−1𝐸^(𝑛+1)ℎ_𝑛ℎ2𝐾_𝑑𝑍₁(𝑠)²+𝑍₂(𝑠)^2𝑃/2𝑑𝑠≤𝐶_𝑝2^𝑃𝑑^𝑃𝐾_𝑑^𝑃/2ℎ^𝑃/2. (23)(24) Combing (22)–(24) together, we have 𝐸𝑌(𝑡)−𝑍₁(𝑡)^𝑃≤3^𝑃−12^𝑃𝑑^𝑃𝐾_𝑑^𝑃/2ℎ^𝑃+2𝐶_𝑝2^𝑃𝑑^𝑃𝐾_𝑑^𝑃/2ℎ^𝑃/2,𝐸^{𝑡∧𝜏𝑑}₀𝑌(𝑠)−𝑍₁(𝑠)^𝑃𝑑𝑠≤𝐶₁(𝑑)ℎ^𝑃/2𝑇, (25) where 𝐶₁(𝑑)=3^𝑃−1[2^𝑃𝑑^𝑃𝐾_𝑑^𝑃/2+2𝐶_𝑝2^𝑃𝑑^𝑃𝐾_𝑑^𝑃/2]. The proof is complete.

Lemma 6. Under Assumptions 2 and 3, for any 𝑡∈[0,𝑇] and 𝑃≥2, then 𝐸^𝑡₀𝑌𝑞𝑠∧𝜏_𝑑−𝑍₂𝑠∧𝜏_d^𝑃𝑑𝑠≤𝐶₂(𝑑)ℎ^𝑃/2𝑇, (26) where 𝐶₂(𝑑) is a positive constant independent of the stepsize ℎ.

The proof of this lemma is similar to that of Lemma 5.

Lemma 7. Under Assumptions 2 and 3, for any 𝑡∈[0,𝑇] and 𝑃≥2, then 𝐸^𝑡₀𝑓𝑍₁𝑠∧𝜏_𝑑,𝑍₂𝑠∧𝜏_𝑑,𝑟𝑠∧𝜏_𝑑−𝑓𝑍₁𝑠∧𝜏_𝑑,𝑍₂𝑠∧𝜏_𝑑,𝑟𝑠∧𝜏_𝑑^𝑃𝑑𝑠∨𝐸^𝑡₀𝑔𝑍₁𝑠∧𝜏_𝑑,𝑍₂𝑠∧𝜏_𝑑,𝑟𝑠∧𝜏_𝑑−𝑔𝑍₁𝑠∧𝜏_𝑑,𝑍₂𝑠∧𝜏_𝑑,𝑟𝑠∧𝜏_𝑑^𝑃𝑑𝑠∨𝐸^𝑡_0𝑅𝑛ℎ𝑍₁𝑠∧𝜏_𝑑,𝑍₂𝑠∧𝜏_𝑑,𝑟𝑠∧𝜏_𝑑𝑍,𝑢−ℎ₁𝑠∧𝜏_𝑑,𝑍₂𝑠∧𝜏_𝑑,𝑟𝑠∧𝜏_𝑑,𝑢^𝑃×𝜋(𝑑𝑢)𝑑𝑠≤𝐶₃(𝑑)ℎ^𝑃/2+∘ℎ^𝑃/2, (27) where 𝐶₃(𝑑) is a positive constant independent of the stepsize ℎ.

The proof of this lemma is similar to that of [16, 24].

Proof of Theorem 4. Combining (4) and (14), one has 𝑋(𝑡)(𝑡)=−𝑌_𝑡0𝑓𝑋(𝑠)(𝑠𝑍,𝑋𝑞𝑠,𝑟)−𝑓₁(𝑠),𝑍₂(𝑠),𝑟(𝑠+)𝑑𝑠_𝑡0𝑔𝑋(𝑠)(𝑠)𝑍,𝑋𝑞𝑠,𝑟−𝑔₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)(𝑠)+𝑑𝑊_{𝑡0𝑅𝑛}ℎ𝑋(𝑠)(𝑠)𝑍,𝑋𝑞𝑠,𝑟,𝑢−ℎ₁(𝑠),𝑍₂(𝑠),𝑟(𝑠),𝑢𝑁(𝑑𝑠,𝑑𝑢). (28) Then applying the generalized Itô’s formula, we can show that |𝑋(𝑡)−𝑌(𝑡)|²=2^𝑡₀|𝑋(𝑠)(𝑠×𝑓𝑋(𝑠)(𝑠)𝑍−𝑌)|,𝑋𝑞𝑠,𝑟−𝑓₁(𝑠),𝑍₂(𝑠),𝑟(𝑠+)𝑑𝑠_𝑡0𝑔𝑋(𝑠),𝑋𝑞𝑠,𝑟(𝑠)−𝑔𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)²+𝑑𝑠_{𝑡0𝑅𝑛}ℎ𝑋(𝑠)(𝑠)𝑍,𝑋𝑞𝑠,𝑟,𝑢−ℎ₁(𝑠),𝑍₂(𝑠),𝑟(𝑠),𝑢²𝜋(𝑑𝑢)𝑑𝑠+2^𝑡₀|𝑋(𝑠)(𝑠×𝑔𝑋(𝑠)(𝑠)𝑍−𝑌)|,𝑋𝑞𝑠,𝑟−𝑔₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)(𝑠)+𝑑𝑊_{𝑡0𝑅𝑛}2|𝑋(𝑠)(𝑠×ℎ𝑋(𝑠)(𝑠)𝑍−𝑌)|,𝑋𝑞𝑠,𝑟,𝑢−ℎ₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)𝑁()+,𝑢𝑑𝑠,𝑑𝑢_{𝑡0𝑅𝑛}ℎ𝑋(𝑠)(𝑠)𝑍,𝑋𝑞𝑠,𝑟,𝑢−ℎ₁(𝑠),𝑍₂(𝑠),𝑟(𝑠),𝑢²𝑁(𝑑𝑠,𝑑𝑢). (29) Hence, for any 𝑡∈[0,𝑇], we get 𝐸sup_{0≤𝑡≤𝑇∧𝜏𝑑}|𝑋(𝑡)−𝑌(𝑡)|²≤𝐸^{𝑇∧𝜏𝑑}₀|𝑋(𝑠)−𝑌(𝑠)|²𝑑𝑠+𝐸^𝜏𝑑₀^∧𝑇𝑓𝑋(𝑠)(𝑠)𝑍,𝑋𝑞𝑠,𝑟−𝑓₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)²𝑑𝑠+𝐸^𝜏𝑑₀^∧𝑇𝑔𝑋(𝑠)(𝑠)𝑍,𝑋𝑞𝑠,𝑟−𝑔₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)²𝑑𝑠+𝐸^𝜏𝑑₀^∧𝑇_𝑅𝑛ℎ𝑋(𝑠)(𝑠)𝑍,𝑋𝑞𝑠,𝑟,𝑢−ℎ₁(𝑠),𝑍₂(𝑠),𝑟(𝑠),𝑢²𝜋(𝑑𝑢)𝑑𝑠+2𝐸sup_{0≤𝑡≤𝜏𝑑∧𝑇}^𝑡₀𝑋𝑠∧𝜏_𝑑−𝑌𝑠∧𝜏_𝑑×𝑔𝑋𝑠∧𝜏_𝑑,𝑋𝑞𝑠∧𝜏_𝑑(𝑠𝑍,𝑟)−𝑔₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)𝑑𝑊(𝑠)+𝐸sup_{0≤𝑡≤𝜏𝑑∧𝑇}^𝑡_0𝑅𝑛2|𝑋(𝑠)(𝑠×ℎ𝑋(𝑠)(𝑠)𝑍−𝑌)|,𝑋𝑞𝑠,𝑟,𝑢−ℎ₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)×𝑁(),𝑢𝑑𝑠,𝑑𝑢+𝐸sup_{0≤𝑡≤𝜏𝑑∧𝑇}^𝑡_0𝑅𝑛ℎ𝑋(𝑠)(𝑠)𝑍,𝑋𝑞𝑠,𝑟,𝑢−ℎ₁(𝑠),𝑍₂(𝑠),𝑟(𝑠),𝑢²×𝑁(𝑑𝑠,𝑑𝑢). (30) By Assumption 3 and Lemmas 5–7, we have 𝐸^𝜏𝑑₀^∧𝑇𝑓𝑋(𝑠)(𝑠)𝑍,𝑋𝑞𝑠,𝑟−𝑓₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)²𝑑𝑠≤2𝐸^𝜏𝑑₀^∧𝑇𝑓𝑋(𝑠)(𝑠)𝑍,𝑋𝑞𝑠,𝑟−𝑓₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)²𝑑𝑠+2𝐸^𝜏𝑑₀^∧𝑇𝑓𝑍₁(𝑠),𝑍₂(𝑠)(𝑠)𝑍,𝑟−𝑓₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)²𝑑𝑠≤2𝐾_𝑑𝐸^𝜏𝑑₀^∧𝑇𝑋(𝑠)−𝑍₁(𝑠)²+𝑋𝑞𝑠−𝑍₂(𝑠)²𝑑𝑠+2𝐶₃(𝑑)ℎ≤2𝐾_𝑑𝐸×^𝜏𝑑₀^∧𝑇2|𝑋(𝑠)−𝑌(𝑠)|²+𝑌(𝑠)−𝑍₁(𝑠)²+2𝑋𝑞𝑠−𝑌𝑞𝑠²+2𝑌𝑞𝑠−𝑍₂(𝑠)²𝑑𝑠+2𝐶₃(𝑑)[ℎ+∘(ℎ)]≤4𝐾_𝑑𝐶₁(𝑑)+𝐶₂(𝑑)ℎ𝑇+2𝐶₃(𝑑)[ℎ+∘(ℎ)]+8𝐾_𝑑^𝜏𝑑₀^∧𝑇𝐸sup_{0≤𝑢≤𝑠}|𝑋(𝑢)−𝑌(𝑢)|²𝑑𝑠. (31) Similarly, by Assumption 3 and Lemmas 5–7, we obtain 𝐸^𝜏𝑑₀^∧𝑇𝑔𝑋(𝑠)(𝑠)𝑍,𝑋𝑞𝑠,𝑟−𝑔₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)²𝑑𝑠≤4𝐾_𝑑𝐶₁(𝑑)+𝐶₂(𝑑)ℎ𝑇+2𝐶₃(𝑑)[ℎ+∘(ℎ)]+8𝐾_𝑑^𝜏𝑑₀^∧𝑇𝐸sup_{0≤𝑢≤𝑠}|𝑋(𝑢)−𝑌(𝑢)|²𝐸𝑑𝑠,_{𝜏𝑑0∧𝑇𝑅𝑛}ℎ𝑋(𝑠),𝑋𝑞𝑠,𝑟(𝑠),𝑢−ℎ(𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠),𝑢)²𝜋(𝑑𝑢)𝑑𝑠≤4𝐾_𝑑𝐶₁(𝑑)+𝐶₂(𝑑)ℎ𝑇+2𝐶₃(𝑑)[ℎ+∘(ℎ)]+8𝐾_𝑑^𝜏𝑑₀^∧𝑇𝐸sup_{0≤𝑢≤𝑠}|𝑋(𝑢)−𝑌(𝑢)|²𝑑𝑠. (32)(33) On the other hand, by the Burkholder-Davis-Gundy inequality, Young’s inequality, and Lemmas 5–7, we have for any 𝜀₁>0𝐸sup_{0≤𝑡≤𝜏𝑑∧𝑇}^𝑡₀|𝑋(𝑠)(𝑠×𝑔𝑋(𝑠)(𝑠)𝑍−𝑌)|,𝑋𝑞𝑠,𝑟−𝑔₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)𝑑𝑊(𝑠)≤3𝐸sup_{0≤𝑡≤𝜏𝑑∧𝑇}|𝑋(𝑡)(𝑡×−𝑌)|_{𝜏𝑑0∧𝑇}𝑔𝑋(𝑡)(𝑡)𝑍,𝑋𝑞𝑡,𝑟−𝑔₁(𝑡),𝑍₂(𝑡),𝑟(𝑡)²𝑑𝑡^1/2≤3𝜀₁𝐸sup_{0≤𝑡≤𝜏𝑑∧𝑇}|𝑋(𝑡)−𝑌(𝑡)|^21/2×1𝜀₁𝐸^𝜏𝑑₀^∧𝑇𝑔𝑋(𝑡)(𝑡)𝑍,𝑋𝑞𝑡,𝑟−𝑔₁(𝑡),𝑍₂(𝑡),𝑟(𝑡)²𝑑𝑡^1/2≤6𝜀₁𝐸sup_{0≤𝑡≤𝜏𝑑∧𝑇}|𝑋(𝑡)−𝑌(𝑡)|²+6𝜀₁𝐸^𝜏𝑑₀^∧𝑇𝑔𝑋(𝑡)(𝑡)𝑍,𝑋𝑞𝑡,𝑟−𝑔₁(𝑡),𝑍₂(𝑡),𝑟(𝑡)²𝑑𝑡≤6𝜀₁𝐸sup_{0≤𝑡≤𝜏𝑑∧𝑇}|𝑋(𝑡)−𝑌(𝑡)|²+24𝜀₁𝐾_𝑑𝐶₁(𝑑)+𝐶₂(𝑑+)ℎ𝑇12𝜀₁𝐶₃(𝑑(ℎ+)[ℎ+∘)]48𝜀₁𝐾_𝑑^𝜏𝑑₀^∧𝑇𝐸sup_{0≤𝑢≤𝑠}|𝑋(𝑢)−𝑌(𝑢)|²𝑑𝑠. (34) We have for any 𝜀₂>0𝐸sup_{0≤𝑡≤𝜏𝑑∧𝑇}^𝑡_0𝑅𝑛2|𝑋(𝑠)(𝑠ℎ𝑋(𝑠)(𝑠)𝑍−𝑌)|,𝑋𝑞𝑠,𝑟,𝑢−ℎ₁(𝑠),𝑍₂(𝑠),𝑟(𝑠),𝑢𝑁(𝑑𝑠,𝑑𝑢)≤𝐶𝐸_{𝑡∈𝐷𝑝,𝑡≤𝜏∧𝑇}|𝑋(𝑡)−𝑌(𝑡)|²×ℎ𝑋(𝑡),𝑋𝑞𝑡,𝑟(𝑡),𝑝_𝑡−ℎ𝑍₁(𝑡),𝑍₂(𝑡),𝑟(𝑡),𝑝_𝑡^21/2≤𝐶𝐸sup_{0≤𝑡≤𝜏𝑑∧𝑇}|𝑋(𝑡)(𝑡𝐸×−𝑌)|_{𝑡∈𝐷𝑝,𝑡≤𝜏𝑑∧𝑇}ℎ𝑋(𝑡),𝑋𝑞𝑡,𝑟(𝑡),𝑝_𝑡−ℎ𝑍₁(𝑡),𝑍₂(𝑡),𝑟(𝑡),𝑝_𝑡^21/2≤𝐶𝐸𝜀₂sup_{0≤𝑡≤𝜏𝑑∧𝑇}|𝑋(𝑡)−𝑌(𝑡)|^21/21×𝐸𝜀_{2𝑡∈𝐷𝑝,𝑡≤𝜏𝑑∧𝑇}ℎ𝑋(𝑡),𝑋𝑞𝑡,𝑟(𝑡),𝑝_𝑡−ℎ𝑍₁(𝑡),𝑍₂(𝑡),𝑟(𝑡),𝑝_𝑡^21/2≤𝐶𝜀₂𝐸sup_{0≤𝑡≤𝜏𝑑∧𝑇}|𝑋(𝑡)−𝑌(𝑡)|²+𝐶𝜀₂𝐸_{𝑡∈𝐷𝑝,𝑡≤𝜏𝑑∧𝑇}ℎ𝑋(𝑡),𝑋𝑞𝑡,𝑟(𝑡),𝑝_𝑡−ℎ𝑍₁(𝑡),𝑍₂(𝑡),𝑟(𝑡),𝑝_𝑡²≤𝐶𝜀₂𝐸sup_{0≤𝑡≤𝜏𝑑∧𝑇}|𝑋(𝑡)−𝑌(𝑡)|²+𝐶𝜀₂𝐸^𝜏𝑑₀^∧𝑇_𝑅𝑛ℎ𝑋(𝑡)(𝑡)𝑍,𝑋𝑞𝑡,𝑟,𝑢−ℎ₁(𝑡),𝑍₂(𝑡),𝑟(𝑡),𝑢²𝜋(𝑑𝑢)𝑑𝑡≤𝐶𝜀₂𝐸sup_{0≤𝑡≤𝜏𝑑∧𝑇}|𝑋(𝑡)−𝑌(𝑡)|²+4𝐶𝜀₂𝐾_𝑑𝐶₁(𝑑)+𝐶₂(𝑑)ℎ𝑇+2𝐶𝜀₂𝐶₃(𝑑(ℎ+)[ℎ+∘)]8𝐶𝜀₂𝐾_𝑑^𝜏𝑑₀^∧𝑇𝐸sup_{0≤𝑢≤𝑠}|𝑋(𝑢)−𝑌(𝑢)|²𝑑𝑠, (35) where 𝐶 is some constant that may change from line to line. Similar, we have 𝐸sup_{0≤𝑡≤𝜏𝑑∧𝑇}^𝑡_0𝑅𝑛ℎ𝑋(𝑠)(𝑠)𝑍,𝑋𝑞𝑠,𝑟,𝑢−ℎ₁(𝑠),𝑍₂(𝑠),𝑟(𝑠),𝑢²𝑁(𝑑𝑠,𝑑𝑢)≤𝐶𝐸_{𝑡∈𝐷𝑝,𝑡≤𝜏𝑑∧𝑇}ℎ𝑋(𝑡),𝑋𝑞𝑡,𝑟(𝑡),𝑝_𝑡−ℎ𝑍₁(𝑡),𝑍₂(𝑡),𝑟(𝑡),𝑝_𝑡^41/2≤𝐶𝐸_{𝑠∈𝐷𝑝,𝑠≤𝜏𝑑∧𝑇}ℎ𝑋(𝑡),𝑋𝑞𝑡,𝑟(𝑡),𝑝_𝑡−ℎ𝑍₁(𝑡),𝑍₂(𝑡),𝑟(𝑡),𝑝_𝑡²≤𝐶𝐸^𝜏𝑑₀^∧𝑇_𝑅𝑛ℎ𝑋(𝑡)(𝑡)𝑍,𝑋𝑞𝑡,𝑟,𝑢−ℎ₁(𝑡),𝑍₂(𝑡),𝑟(𝑡),𝑢²𝜋(𝑑𝑢)𝑑𝑡≤4𝐶𝐾_𝑑𝐶₁(𝑑)+𝐶₂(𝑑)ℎ𝑇+2𝐶𝐶₃(𝑑)[ℎ+∘(ℎ)]+8𝐶𝐾_𝑑^𝜏𝑑₀^∧𝑇𝐸sup_{0≤𝑢≤𝑠}|𝑋(𝑢)−𝑌(𝑢)|²𝑑𝑠. (36) Substituting (31)–(36) into (30), we obtain that 𝐸sup_{0≤𝑡≤𝜏𝑑∧𝑇}|𝑋(𝑡)−𝑌(𝑡)|²≤6𝜀₁+𝐶𝜀₂𝐸sup_{0≤𝑡≤𝜏𝑑∧𝑇}|𝑋(𝑡)−𝑌(𝑡)|²+10+24𝜀₁+2𝐶1+𝜀₂𝜀₂𝐾_𝑑𝐶₁(𝑑)+𝐶₂(𝑑+)𝑇ℎ4+12𝜀₁𝐶₃(𝑑(ℎ+)[ℎ+∘)]1+20+48𝜀₁+4𝐶1+𝜀₂𝜀₂𝐾_𝑑×^𝑇₀𝐸sup_{0≤𝑢≤𝜏∧𝑠}|𝑋(𝑢)−𝑌(𝑢)|²𝑑𝑠. (37) By choosing 𝜀₁>0, 𝜀₂>0 sufficiently small and letting 6𝜀₁+𝐶𝜀₂=2/3, we have 𝐸sup_{0≤𝑡≤𝜏∧𝑇}|𝑋(𝑡)−𝑌(𝑡)|²≤310+24𝜀₁+2𝐶1+𝜀₂𝜀₂𝐾_𝑑𝐶₁(𝑑)+𝐶₂(𝑑)𝑇ℎ+34+12𝜀₁𝐶₃(𝑑)[ℎ+∘(ℎ)]+31+20+48𝜀₁+4𝐶1+𝜀₂𝜀₂𝐾_𝑑×^𝑇₀𝐸sup_{0≤𝑢≤𝜏∧𝑠}|𝑋(𝑢)−𝑌(𝑢)|²𝑑𝑠. (38) Therefore, we apply Gronwall's inequality to get 𝐸sup_{0≤𝑡≤𝜏∧𝑇}|𝑋(𝑡)−𝑌(𝑡)|²≤310+24𝜀₁+2𝐶1+𝜀₂𝜀₂𝐾_𝑑𝐶₁(𝑑)+𝐶₂(𝑑)𝑇ℎ+34+12𝜀₁𝐶₃(𝑑)[ℎ+∘(ℎ)]𝑒^{3[1+(20+(48/𝜀1)+4𝐶((1+𝜀2)/𝜀2))𝐾𝑑]}. (39) This completes the proof.

Remark 8. Under the local Lipschitz condition, Theorem 4 not only tells us the strong convergence of the approximate solutions to the true solutions but also tells us the rate of the convergence with order 1/2 by (39).

Remark 9. When 𝑟(𝑡)≡0 or ℎ≡0, (4) reduces to which was studied by Fan et al. [14], Xiao and Zhang [15], and Milošević and Jovanović [16]. Our results in the present paper generalized and improved the results in [14–16].

4. Convergence of Numerical Solutions in Probability

In this section, by applying Theorem 4, we will show the convergence in probability of the approximate solutions to the true solutions under local Lipschitz condition. Before we give the convergence theorem, we need some additional conditions based on Lyapunov-type functions.

Assumption 10. For 𝑥∈𝑅^𝑛 and 𝑖∈𝑆, there exist a positive function 𝑉∈𝐶²(𝑅^𝑛×𝑆;𝑅⁺), 𝐾>0, and two constants 𝜆₁>𝜆₂≥0 such that lim_|𝑥|→∞𝑉(𝑥,𝑖)=∞,𝐿𝑉𝑥,𝑦,𝑖≤𝐾1−𝜆₁𝑉(𝑥,𝑖)+𝜆₂𝑉𝑦,𝑖. (40)(41)

Assumption 11. There exists a positive constant 𝐿_𝑑 such that, for all 𝑥,𝑦∈𝑅^𝑛 and 𝑖∈𝑆 with |𝑥|∨|𝑦|≤𝑑, 𝑉(𝑥,𝑖)−𝑉𝑦,𝑖∨𝑉_𝑥(𝑥,𝑖)−𝑉_𝑥∨𝑉𝑦,𝑖_𝑥𝑥(𝑥,𝑖)−𝑉_𝑥𝑥𝑦,𝑖≤𝐿_𝑑𝑥−𝑦. (42)

Now, let us state our convergence theorem.

Theorem 12. Let the assumptions of Theorem 4 hold. Also assume that there exists a 𝐶² function 𝑉∶𝑅^𝑛×𝑆→𝑅₊ satisfying (40)–(42). Then the Euler approximate solutions converges to the true solutions of (4) in the sense of the probability.
That is, lim_ℎ→0sup_{0≤𝑡≤𝑇}|𝑌(𝑡)−𝑋(𝑡)|²=0,𝑖𝑛𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦. (43)

Proof. The proof is rather technical, and we divide it into three steps.
Step 1. We assume the existence of the nonnegative Lyapunov function 𝑉(𝑥,𝑖) satisfying (40). Applying the Itô’s formula, 𝑉(𝑋(𝑡),𝑟(𝑡)) yields (𝑋(𝑡)(𝑡𝑋(𝑡)(𝑡)𝑑𝑉,𝑟))=𝐿𝑉,𝑋𝑞𝑡,𝑟𝑑𝑡+𝑉_𝑥(𝑋(𝑡)(𝑡𝑔𝑋(𝑡)(𝑡)(𝑡)+,𝑟)),𝑋𝑞𝑡,𝑟𝑑𝑊_𝑅𝑛𝑉𝑋(𝑡)𝑋(𝑡)(𝑡)(𝑡)(𝑋(𝑡)(𝑡𝑁().+ℎ,𝑋𝑞𝑡,𝑟,𝑢,𝑟−𝑉,𝑟))]𝑑𝑡,𝑑𝑢 (44) Integrating from 0 to 𝑡∧𝜃_𝑑 and taking expectations gives 𝐸𝑉𝑋𝑡∧𝜃_𝑑,𝑟𝑡∧𝜃_𝑑=𝑉𝑋₀,𝑟₀+𝐸∫^{𝑡∧𝜃𝑑}₀𝐿𝑉𝑋(𝑠),𝑋𝑞𝑠,𝑟(𝑠)𝑑𝑠. (45) By (41), we have 𝐸𝑉𝑋𝑡∧𝜃_𝑑,𝑟𝑡∧𝜃_𝑑≤𝑉𝑋₀,𝑟₀+𝐾𝐸^{𝑡∧𝜃𝑑}₀1−𝜆₁𝑉(𝑋(𝑠),𝑟(𝑠))+𝜆₂𝑉𝑋(𝑠)𝑋𝑞𝑠,𝑟𝑑𝑠≤𝑉₀,𝑟₀𝜆+𝐾𝑇+𝐾₁+𝜆₂₀^𝑡∧𝜃sup_{0≤𝑢≤𝑠}𝐸𝑉(𝑋(𝑢),𝑟(𝑢))𝑑𝑠. (46) Thus, for any 𝑡₁∈[0,𝑇], it follows that sup_{0≤𝑡≤𝑡1}𝐸𝑉𝑋𝑡∧𝜃_𝑑,𝑟𝑡∧𝜃_𝑑≤𝑉𝑋₀,𝑟₀+𝜆+𝐾𝑇₁+𝜆₂𝐾^𝑡1₀sup_{0≤𝑡≤𝑠}𝐸𝑉𝑋𝑡∧𝜃_𝑑,𝑟𝑡∧𝜃_𝑑𝑑𝑠. (47) Using the Gronwall inequality, we obtain that sup_{0≤𝑡≤𝑇}𝐸𝑉𝑥𝑡∧𝜃_𝑑,𝑟𝑡∧𝜃_𝑑≤𝑉𝑋₀,𝑟₀+𝐾𝑇𝑒^{(𝜆1+𝜆2)𝐾𝑇}. (48) Let 𝒱_𝑑=inf{𝑉(𝑥,𝑖)∶|𝑥|≥𝑑}. By (40), we have lim_𝑑→∞𝒱_𝑑=∞. Noting that |𝑋(𝜃_𝑑)|=𝑑, as 𝜃_𝑑<𝑇, we derive from (48) that 𝑉𝑋₀,𝑟₀+𝐾𝑇𝑒^{(𝜆1+𝜆2)𝐾𝑇}≥sup_{0≤𝑡≤𝑇}𝐸𝑉𝑥𝑡∧𝜃_𝑑,𝑟𝑡∧𝜃_𝑑≥𝐸𝑉𝑋𝜃_𝑑,𝑟𝜃_𝑑𝐼_{{𝜃𝑑<𝑇}}(𝑤)≥𝒱_𝑑𝑃𝜃_𝑑<𝑇. (49) That is, 𝑃𝜃_𝑑≤𝑉𝑋<𝑇⁰,𝑟₀+𝐾𝑇𝑒^{(𝜆1+𝜆2)𝐾𝑇}𝒱_𝑑. (50) Recall that 𝒱_𝑑→∞ as 𝑑→∞. For a given 𝑇, 𝑋₀, and 𝑟(0), it follows that 𝑉𝑋₀,𝑟₀+𝐾𝑇𝑒^{(𝜆1+𝜆2)𝐾𝑇}𝒱_𝑑→0, (51) as 𝑑→∞. Let 𝜀3=𝑉𝑋₀,𝑟₀+𝐾𝑇𝑒^{(𝜆1+𝜆2)𝐾𝑇}𝒱_𝑑∈(0,1). (52) Thus we have 𝑃𝜃_𝑑≤𝜀<𝑇3. (53)
Step 2. We will give the estimate of 𝑃(𝜌_𝑑<𝑇). By (14), applying the Itô’s formula to 𝑉(𝑌(𝑡),𝑟(𝑡)) yields 𝑑𝑉(𝑌(𝑡),𝑟(𝑡))=𝑉_𝑥(𝑌(𝑡),𝑟(𝑡))𝑓𝑍₁(𝑡),𝑍₂(𝑡),𝑟(𝑡)+12trace𝑔^⊤𝑍₁(𝑡),𝑍₂(𝑡),𝑟(𝑡)×𝑉_𝑥𝑥(𝑌(𝑡),𝑟(𝑡))𝑔𝑍₁(𝑡),𝑍₂(𝑡),𝑟(𝑡)𝑑𝑡+𝑉_𝑥(𝑌(𝑡),𝑟(𝑡))𝑔𝑍₁(𝑡),𝑍₂(𝑡),𝑟(𝑡)(𝑡)+𝑑𝑊_𝑅𝑛𝑉𝑌(𝑡)+ℎ𝑍₁(𝑡),𝑍₂(𝑡),𝑟(𝑡)(𝑡)(𝑌(𝑡)(𝑡,𝑢,𝑟−𝑉,𝑟))−𝑉_𝑥(𝑌(𝑡)(𝑡𝑍,𝑟))×ℎ₁(𝑡),𝑍₂(𝑡),𝑟(𝑡)Π()+,𝑢𝑑𝑢𝑑𝑡_𝑅𝑛𝑉𝑌(𝑡)+ℎ𝑍₁(𝑡),𝑍₂(𝑡),𝑟(𝑡)(𝑡(𝑌(𝑡)(𝑡𝑁()+,𝑢,𝑟)−𝑉,𝑟))𝑑𝑡,𝑑𝑢_𝑁𝑗=1𝛾_{𝑟(𝑡)𝑗}𝑉𝑌(𝑡),𝑗𝑑𝑡. (54) By (7), we have (𝑌(𝑡)(𝑡𝑍𝑑𝑉,𝑟))=𝐿𝑉₁(𝑡),𝑍₂(𝑡),𝑟(𝑡)+𝑉𝑑𝑡_𝑥(𝑌(𝑡),𝑟(𝑡))−𝑉_𝑥𝑍₁(𝑡),𝑟(𝑡)×𝑓𝑍₁(𝑡),𝑍₂(𝑡),𝑟(𝑡+1)𝑑𝑡2trace𝑔^⊤𝑍₁(𝑡),𝑍₂(𝑡),𝑟(𝑡)×𝑉_𝑥𝑥(𝑌(𝑡),𝑟(𝑡))−𝑉_𝑥𝑥𝑍₁(t),𝑟(𝑡𝑍)×𝑔₁(𝑡),𝑍₂(𝑡),𝑟(𝑡)+𝑑𝑡_𝑅𝑛𝑉𝑌(𝑡)+ℎ𝑍₁(𝑡),𝑍₂(𝑡),𝑟(𝑡)(𝑡)𝑍,𝑢,𝑟−𝑉₁(𝑡)+ℎ𝑍₁(𝑡),𝑍₂(𝑡),𝑟(𝑡),𝑢,𝑟(𝑡)−𝑉(𝑌(𝑡),𝑟(𝑡))−𝑉𝑍₁(𝑡),𝑟(𝑡)−𝑉_𝑥(𝑌(𝑡),𝑟(𝑡))−𝑉_𝑥𝑍₁(𝑡),𝑟(𝑡)×ℎ𝑍₁(𝑡),𝑍₂(𝑡),𝑟(𝑡),𝑢Π(𝑑𝑢)𝑑𝑡+𝑉_𝑥(𝑌(𝑡),𝑟(𝑡))𝑔𝑍₁(𝑡),𝑍₂(𝑡),𝑟(𝑡)𝑑𝑊(𝑡)+_𝑅𝑛𝑉𝑌(𝑡)+ℎ𝑍₁(𝑡),𝑍₂(𝑡),𝑟(𝑡)(𝑡)(𝑌(𝑡)(𝑡𝑁()+,𝑢,𝑟−𝑉,𝑟))𝑑𝑡,𝑑𝑢_𝑁𝑗=1𝛾_{𝑟(𝑡)𝑗}𝑉𝑌(𝑡),𝑗𝑑𝑡−^𝑁_𝑗=1𝛾_{𝑟(𝑡)𝑗}𝑉𝑍₁(𝑡),𝑗𝑑𝑡. (55) Integrating from 0 to 𝜌_𝑑∧𝑡, taking expectations, and by (41), we have 𝐸𝑉𝑌𝑡∧𝜌_𝑑,𝑟𝑡∧𝜌_𝑑≤𝑉𝑋₀,𝑟₀+𝐾𝑇+𝜆₁𝐾𝐸^𝜌𝑑₀^∧𝑡𝑉(𝑌(𝑠),𝑟(𝑠))𝑑𝑠+𝜆₂𝐾𝐸^𝜌𝑑₀^∧𝑡𝑉𝑌𝑞𝑠,𝑟(𝑠)𝑑𝑠−𝜆₁𝐾𝐸^𝜌𝑑₀^∧𝑡𝑉𝑍₁(𝑠),𝑟(𝑠)+𝑉(𝑌(𝑠),𝑟(𝑠))𝑑𝑠+ℐ₁+ℐ₂+ℐ₃+ℐ₄+ℐ₅, (56) where ℐ₁=𝜆₂𝐾𝐸^𝜌𝑑₀^∧𝑡𝑉𝑍₂(𝑠),𝑟(𝑠)𝑌(𝑠)ℐ−𝑉𝑞𝑠,𝑟𝑑𝑠,₂=𝐸^𝜌𝑑₀^∧𝑡𝑉_𝑥(𝑌(𝑠),𝑟(𝑠))−𝑉_𝑥𝑍₁(𝑠),𝑟(𝑠)×𝑓𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)ℐ𝑑𝑠,₃=12𝐸^𝜌𝑑₀^∧𝑡𝑉_𝑥𝑥(𝑌(𝑠),𝑟(𝑠))−𝑉_𝑥𝑥𝑍₁(𝑠),𝑟(𝑠)×𝑔𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)²ℐ𝑑𝑠,₄=𝐸^𝜌𝑑₀^∧𝑡_𝑅𝑛𝑉𝑌(𝑠)+ℎ𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠))(𝑠𝑍,𝑢,𝑟))−𝑉₁(𝑠)+ℎ𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠),𝑢,𝑟(𝑠)+𝑉(𝑌(𝑠),𝑟(𝑠))−𝑉𝑍₁(𝑠),𝑟(𝑠)+𝑉_𝑥(𝑌(𝑠),𝑟(𝑠))−𝑉_𝑥𝑍₁(𝑠),𝑟(𝑠)×ℎ𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)𝜋()ℐ,𝑢𝑑𝑢𝑑𝑠,₅=𝐸^𝜌𝑑₀^𝑁∧𝑡_𝑗=1𝛾_{𝑟(𝑠)𝑗}𝑉𝑌(𝑠),𝑗𝑑𝑠−𝐸^𝜌𝑑₀^𝑁∧𝑡_𝑗=1𝛾_{𝑟(𝑠)𝑗}𝑉𝑍₁(𝑠),𝑗𝑑𝑠. (57) By (42) and Young’s inequality, we have ℐ₁≤𝜆₂𝐾𝐸^𝜌𝑑₀^∧𝑡𝑉𝑍₂(𝑠),𝑟(𝑠)−𝑉𝑌𝑞𝑠,𝑟(𝑠)𝑑𝑠+𝜆₂𝐾𝐸^𝜌𝑑₀^∧𝑡𝑉𝑌𝑞𝑠,𝑟(𝑠)−𝑉𝑌𝑞𝑠,𝑟(𝑠)𝑑𝑠≤𝜆₂𝐾𝐿_𝑑𝐸^𝜌𝑑₀^∧𝑡𝑍₂(𝑠)−𝑌𝑞𝑠𝑑𝑠+𝜆₂𝐾𝐸^𝑇₀𝑉𝑌𝑞𝑠,𝑟(𝑠)−𝑉𝑌𝑞𝑠,𝑟(𝑠)𝑑𝑠≤𝜆₂𝐾𝐿_𝑑^𝜌𝑑₀^∧𝑡𝐸𝑍₂(𝑠)−𝑌𝑞𝑠^21/2𝑑𝑠+𝜆₂𝒥. (58) Let 𝑁=[𝑇/ℎ], the integer part of 𝑇/ℎ, and let 𝐼_𝐺 be the indicator function of the set 𝐺. Then 𝒥≤𝐾^𝑁_𝑛=0𝐸^(𝑛+1)ℎ_𝑛ℎ𝑉𝑌_[𝑞𝑛],𝑟^ℎ_𝑛−𝑉𝑌_[𝑞𝑛],𝑟(𝑠)𝑑𝑠≤𝐾^𝑁_𝑛=0𝐸^(𝑛+1)ℎ_𝑛ℎ𝑉𝑌_[𝑞𝑛],𝑟^ℎ_𝑛+𝑓𝑌_[𝑞𝑛],𝑟(𝑠)𝐼_{{𝑟(𝑠)≠𝑟ℎ𝑛}}𝑑𝑠. (59) By setting 𝑉₁=sup_{|𝑥|≤𝑅,𝑖∈𝑆}𝑉(𝑥,𝑖) and using the Markov property, we have 𝒥≤2𝐾𝑉₁^𝑁_𝑛=0𝐸^(𝑛+1)ℎ_𝑛ℎ𝐼_{{𝑟(𝑠)≠𝑟ℎ𝑛}}𝑑𝑠=2𝐾𝑉₁^𝑁_𝑛=0^(𝑛+1)ℎ_𝑛ℎ𝐸𝐸𝐼_{{𝑟(𝑠)≠𝑟ℎ𝑛}}∣𝑟^ℎ_𝑛𝑑𝑠=2𝐾𝑉₁^𝑁_𝑛=0^(𝑛+1)ℎ_𝑛ℎ𝐸_𝑖∈𝑆𝐼_{{𝑟ℎ𝑛=𝑖}}𝑃𝑟(𝑠)≠𝑖∣𝑟^ℎ_𝑛=𝑖𝑑𝑠=2𝐾𝑉₁×^𝑁_𝑛=0^(𝑛+1)ℎ_𝑛ℎ𝐸_𝑖∈𝑆𝐼_{{𝑟ℎ𝑛=𝑖}𝑗≠𝑖}𝛾_𝑖𝑗(𝑠−𝑛ℎ)+∘(𝑠−𝑛ℎ)×𝑑𝑠≤2𝐾𝑉₁^𝑁_𝑛=0^(𝑛+1)ℎ_𝑛ℎ𝐸max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑖ℎ+∘(ℎ)_𝑖∈𝑆𝐼_{{𝑟ℎ𝑛=𝑖}}𝑑𝑠=2𝐾𝑉₁max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑖ℎ+∘(ℎ)𝑇. (60) Inserting (60) into (58), we have, by Lemma 5, ℐ₁≤𝜆₂𝐾𝐿_𝑑^𝜌𝑑₀^∧𝑡𝐸𝑍₂(𝑠)−𝑌𝑞𝑠^21/2𝑑𝑠+2𝜆₂𝐾𝑉₁max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑖ℎ+∘(ℎ)𝑇≤𝜆₂𝐾𝐿_𝑑𝐶₂(𝑑)𝑇ℎ^1/2+2𝜆₂𝐾𝑉₁𝑇max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑖[ℎ+∘(ℎ)]. (61) Similarly, by Assumptions 2, 3, and 11, Lemma 5, and Markov property, we have ℐ₂≤𝐸^𝜌𝑑₀^∧𝑡𝑉_𝑥(𝑌(𝑠),𝑟(𝑠))−𝑉_𝑥𝑍₁(𝑠)(𝑠)×𝑓𝑍,𝑟₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)𝑑𝑠+𝐸^𝜌𝑑₀^∧𝑡𝑉_𝑥𝑍₁(𝑠),𝑟(𝑠)−𝑉_𝑥𝑍₁(𝑠),𝑟(𝑠)×𝑓𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)𝑑𝑠≤2𝐾_𝑑𝑑𝐸^𝜌𝑑₀^∧𝑡𝑉_𝑥(𝑌(𝑠),𝑟(𝑠))−𝑉_𝑥𝑍₁(𝑠),𝑟(𝑠)𝑑𝑠+2𝐾_𝑑𝑑𝐸^𝜌𝑑₀^∧𝑡𝑉_𝑥𝑍₁(𝑠),𝑟(𝑠)−𝑉_𝑥𝑍₁(𝑠),𝑟(𝑠)𝑑𝑠≤2𝐾_𝑑𝑑𝐿_𝑑𝐸^𝜌𝑑₀^∧𝑡𝑌(𝑠)−𝑍₁(𝑠)𝑑𝑠+2𝑑𝐾_𝑑𝐸^𝑇₀𝑉_𝑥𝑍₁(𝑠),𝑟(𝑠)−𝑉_𝑥𝑍₁(𝑠),𝑟(𝑠)𝑑𝑠≤2𝑑𝐾_𝑑𝐿_𝑑𝐶₁(𝑑)𝑇ℎ^1/2+4𝐾_𝑑𝑑𝑉₂𝑇max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑖[ℎ+∘(ℎ)], (62) where 𝑉₂=sup_{|𝑥|≤𝑅,𝑖∈𝑆}𝑉_𝑥(𝑥,𝑖). For the term ℐ₃ in (56), by Assumptions 2, 3, and 11 and Lemma 5, we have ℐ₃≤12𝐸^𝜌𝑑₀^∧𝑡𝑉_𝑥𝑥(𝑌(𝑠),𝑟(𝑠))−𝑉_𝑥𝑥𝑍₁(𝑠)(𝑠)×𝑔𝑍,𝑟₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)²+1𝑑𝑠2𝐸^𝜌𝑑₀^∧𝑡𝑉_𝑥𝑥𝑍₁(𝑠),𝑟(𝑠)−𝑉_𝑥𝑥𝑍₁(𝑠),𝑟(𝑠)×𝑔𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)²𝑑𝑠≤2𝑑²𝐾_𝑑𝐸^𝜌𝑑₀^∧𝑡𝑉_𝑥𝑥(𝑌(𝑠),𝑟(𝑠))−𝑉_𝑥𝑥𝑍₁(𝑠),𝑟(𝑠)𝑑𝑠+2𝑑²𝐾_𝑑𝐸^𝜌𝑑₀^∧𝑡𝑉_𝑥𝑥𝑍₁(𝑠),𝑟(𝑠)−𝑉_𝑥𝑥𝑍₁(𝑠),𝑟(𝑠)𝑑𝑠≤2𝑑²𝐾_𝑑𝐿_𝑑𝐸^𝜌𝑑₀^∧𝑡𝑌(𝑠)−𝑍₁(𝑠)𝑑𝑠+2𝑑²𝐾_𝑑𝐸^𝑇₀𝑉_𝑥𝑥𝑍₁(𝑠),𝑟(𝑠)−𝑉_𝑥𝑥𝑍₁(𝑠),𝑟(𝑠)𝑑𝑠≤2𝑑²𝐾_𝑑𝐿_𝑑𝐶₁(𝑑)𝑇ℎ^1/2+4𝑑²𝐾_𝑑𝑉₃𝑇max_1≤𝑖≤N−𝛾_𝑖𝑖[ℎ+∘(ℎ)], (63) where 𝑉₃=sup_{|𝑥|≤𝑅,𝑖∈𝑆}𝑉_𝑥𝑥(𝑥,𝑖). For the term ℐ₄ in (56), by Assumptions 2, 3, and 11 and Lemma 5, we have ℐ₄≤𝐸^𝜌𝑑₀^∧𝑡_𝑅𝑛𝑉𝑌(𝑠)+ℎ𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)(𝑠)𝑍,𝑢,𝑟−𝑉₁(𝑠)+ℎ𝑍₁(𝑠),Z₂(𝑠),𝑟(𝑠),𝑟(𝑠𝜋(),𝑢))|𝑑𝑢𝑑𝑠+𝐸^𝜌𝑑₀^∧𝑡_𝑅𝑛𝑉𝑍₁(𝑠)+ℎ𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)(𝑠)𝑍,𝑢,𝑟−𝑉₁(𝑠)+ℎ𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠),𝑢,𝑟(𝑠)×𝜋(𝑑𝑢)𝑑𝑠+𝐸^𝜌𝑑₀^∧𝑡_𝑅𝑛𝑉(𝑌(𝑠),𝑟(𝑠))−𝑉𝑍₁(𝑠)(𝑠)(),𝑟×𝜋𝑑𝑢𝑑𝑠+𝐸^𝜌𝑑₀^∧𝑡_𝑅𝑛𝑉𝑍₁(𝑠),𝑟(𝑠)−𝑉𝑍₁(𝑠),𝑟(𝑠)×𝜋(𝑑𝑢)𝑑𝑠+𝐸^𝜌𝑑₀^∧𝑡_𝑅𝑛𝑉_𝑥(𝑌(𝑠),𝑟(𝑠))−𝑉_𝑥𝑍₁(𝑠)(𝑠)×ℎ𝑍,𝑟₁(𝑠),𝑍₂(𝑠),𝑟(𝑠),𝑢𝜋(𝑑𝑢)𝑑𝑠+𝐸^𝜌𝑑∧t_0𝑅𝑛𝑉_𝑥𝑍₁(𝑠),𝑟(𝑠)−𝑉_𝑥𝑍₁(𝑠),𝑟(𝑠)×ℎ𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠),𝑢𝜋(𝑑𝑢)𝑑𝑠≤2𝐿_𝑑𝐸^𝜌𝑑₀^∧𝑡_𝑅𝑛𝑌(𝑠)−𝑍₁(𝑠)𝜋(𝑑𝑢)𝑑𝑠+2𝑑𝐾_𝑑𝑅𝑛𝜋(𝑑𝑢)×𝐸^𝜌𝑑₀^∧𝑡𝑉_𝑥(𝑌(𝑠),𝑟(𝑠))−𝑉_𝑥𝑍₁(𝑠),𝑟(𝑠)𝑑𝑠+_𝑅𝑛𝜋()𝐸×𝑑𝑢_{𝜌𝑑0∧𝑡}𝑉𝑍₁(𝑠),𝑟(𝑠)−𝑉𝑍₁(𝑠),𝑟(𝑠)𝑑𝑠+2𝑑𝐾_𝑑𝑅𝑛𝜋(𝑑𝑢)×𝐸^𝜌𝑑₀^∧𝑡𝑉_𝑥𝑍₁(𝑠),𝑟(𝑠)−𝑉_𝑥𝑍₁(𝑠),𝑟(𝑠)𝑑𝑠+𝐸^𝜌𝑑₀^∧𝑡_𝑅𝑛𝑉𝑍₁(𝑠)+ℎ𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)(𝑠)𝑍,𝑢,𝑟−𝑉₁(𝑠)+ℎ𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠),𝑢,𝑟(𝑠)×𝜋(𝑑𝑢)𝑑𝑠≤2_𝑅𝑛𝜋(𝑑𝑢)+2𝑑𝐾_𝑑𝑅𝑛𝜋(×𝑑𝑢)𝐶₁(𝑑)𝐿_𝑑𝑇ℎ^1/2+2_𝑅𝑛𝜋(𝑑𝑢)𝑉₁+2𝑑𝐾_𝑑𝑅𝑛𝜋(𝑑𝑢)𝑉₂×𝑇max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑖[ℎ+∘(ℎ)]+ℒ, (64) where ℒ≤𝐸^𝜌𝑑₀^∧𝑡_𝑅𝑛𝑉𝑍₁(𝑠)+ℎ𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠)(𝑠)𝑍,𝑢,𝑟−𝑉₁(𝑠),𝑟(𝑠)𝜋(𝑑𝑢)𝑑𝑠+𝐸^𝜌𝑑₀^∧𝑡_𝑅𝑛𝑉𝑍₁(𝑠)(𝑠)𝑍,𝑟−𝑉₁(𝑠),𝑟(𝑠)𝜋(𝑑𝑢)𝑑𝑠+𝐸^𝜌𝑑₀^∧𝑡_𝑅𝑛𝑉𝑍₁(𝑠),𝑟(𝑠𝑍)−𝑉₁(𝑠)+ℎ𝑍₁(𝑠),𝑍₂(𝑠),𝑟(𝑠),𝑢,𝑟(𝑠)𝜋(𝑑𝑢)𝑑𝑠≤4𝑑𝐿_𝑑𝐾_𝑑𝑅𝑛𝜋(𝑑𝑢)𝑇+2𝑉₁𝑇_𝑅𝑛𝜋(𝑑𝑢)max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑖[ℎ+∘(ℎ)]. (65) Inserting (65) into (64), we have ℐ₄≤2_𝑅𝑛𝜋(𝑑𝑢)+2𝑑𝐾_𝑑𝑅𝑛𝜋(𝑑𝑢)𝐶₁(𝑑)𝐿_𝑑𝑇ℎ^1/2+4𝑑𝐿_𝑑𝐾_𝑑𝑅𝑛𝜋(𝑑𝑢)𝑇+4_𝑅𝑛𝜋(𝑑𝑢)𝑉₁+𝑑𝐾_𝑑𝑅𝑛𝜋(𝑑𝑢)𝑉₂×𝑇max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑖[ℎ+∘(ℎ)]. (66) For the term ℐ₅, by Assumption 11, equation (56), and Lemma 5, we have ℐ₅≤𝐸^𝜌𝑑₀^𝑁∧𝑡_𝑗=1𝛾_{𝑟(𝑠)𝑗}𝑉𝑌(𝑠),𝑗−𝑉𝑍₁(𝑠),𝑗𝑑𝑠+𝐸^𝜌𝑑₀^𝑁∧𝑡_𝑗=1𝛾_{𝑟(𝑠)𝑗}−𝛾_{𝑟(𝑠)𝑗}𝑉𝑍₁(𝑠),𝑗𝑑𝑠≤𝑁max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑖𝐿_𝑑𝐶₁(𝑑)𝑇ℎ^1/2+2𝑉₁𝑁𝑇max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑖. (67) Combing these inequalities and (53), we derive that, for 𝑡₁∈[0,𝑇], sup_{0≤𝑡≤𝑡1}𝐸𝑉𝑌𝑡∧𝜌_𝑑,𝑟𝑡∧𝜌_𝑑≤𝑉𝑋₀,𝑟₀+𝐾+4𝑑𝐿_𝑑𝐾_𝑑𝑅𝑛𝜋(𝑑𝑢)+2𝑉₁𝑁max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑖𝑇+𝜆₁+𝜆₂𝐾×^𝑡1₀sup_{0≤𝑡≤𝑠}𝐸𝑉𝑌𝑡∧𝜌_𝑑𝑑𝑠+𝑀₁ℎ^1/2+𝑀₂[ℎ+∘(ℎ)], (68) where 𝑀₁=𝐿_𝑑𝐶₁(𝑑)𝑇2𝑑𝐾_𝑑+2𝑑²𝐾_𝑑+2_𝑅𝑛𝜋(𝑑𝑢)+4𝑑𝐾_𝑑𝑅𝑛𝜋(𝑑𝑢)+𝑁max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑖+𝜆₂𝐾𝐿_𝑑𝐶₂(𝑑)𝑀𝑇,₂=2𝜆₂𝐾+4_𝑅𝑛𝜋(𝑑𝑢)𝑉₁+4𝑑𝐾_𝑑+4𝑑𝐾_𝑑𝑅𝑛𝜋(𝑑𝑢)𝑉₂+4𝑑²𝐾_𝑑𝑉₃𝑇max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑖. (69) For arbitrary 0≤𝑡₁≤𝑇, by the Gronwall inequality, we get sup_{0≤𝑡≤𝑇}𝐸𝑉𝑌𝑡∧𝜌_𝑑,𝑟𝑡∧𝜌_𝑑≤𝑉𝑋₀,𝑟₀+𝐾+4𝑑𝐿_𝑑𝐾_𝑑𝑅𝑛𝜋(𝑑𝑢)+2𝑉₁𝑁max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑖𝑇+𝑀₁ℎ^1/2+𝑀₂[ℎ+∘(ℎ)]𝑒^{(𝜆1+𝜆2)𝐾𝑇}. (70) Noting that |𝑌(𝜌_𝑑)|=𝑑, as 𝜌_𝑑<𝑇, we derive from (70) that 𝑉𝑋₀,𝑟₀+𝐾+4𝑑𝐿_𝑑𝐾_𝑑𝑅𝑛𝜋(𝑑𝑢)+2𝑉₁𝑁max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑖𝑇+𝑀₁ℎ^1/2+𝑀₂[ℎ+∘(ℎ)]𝑒^{(𝜆1+𝜆2)𝐾𝑇}≥𝐸𝑉𝑌𝑡∧𝜌_𝑑,𝑟𝑡∧𝜌_𝑑≥𝐸𝑉𝑌𝜌_𝑑,𝑟𝜌_𝑑𝐼_{{𝜌𝑑<𝑇}}(𝑤)≥𝒱_𝑑𝑃𝜌_𝑑<𝑇. (71) So we have 𝑃𝜌_𝑑≤𝑉𝑋<𝑇₀,𝑟₀+𝐾+4𝑑𝐿_𝑑𝐾_𝑑∫_𝑅𝑛𝜋(𝑑𝑢)𝑇𝑒^{(𝜆1+𝜆2)𝐾𝑇}𝒱_𝑑+2𝑉₁𝑁max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑖𝑇𝑒^{(𝜆1+𝜆2)𝐾𝑇}𝒱_𝑑+𝑀₁ℎ^1/2+𝑀₂[ℎ+∘(ℎ)]𝑒^{(𝜆1+𝜆2)𝐾𝑇}𝒱_𝑑. (72)Step  3. Let 𝜖,𝛿∈(0,1) be arbitrarily small. By setting Ω=𝑤∶sup_{0≤𝑡≤𝑇}|𝑌(𝑡)−𝑋(𝑡)|²≥𝛿 (73) and using Theorem 4, we have 𝐶_𝑑[ℎ+∘(ℎ)]≥𝐸sup_{0≤𝑡≤𝑇}|𝑌(𝑡)−𝑋(𝑡)|²𝐼_{𝜏𝑑≥𝑇}(𝑤)≥𝐸sup_{0≤𝑡≤𝑇}|𝑌(𝑡)−𝑋(𝑡)|²𝐼_{𝜏𝑑≥𝑇}(𝑤)𝐼_Ω(𝑤𝐼)≥𝛿𝐸_{𝜏𝑑≥𝑇}(𝑤)𝐼_Ω(𝑤)=𝛿𝑃𝜏_𝑑≥𝑇∩Ω≥𝛿𝑃Ω−𝑃𝜏_𝑑<𝑇. (74) Combing (53) and (72) together, one gets 𝑃𝜏_𝑑𝜌<𝑇≤𝑃_𝑑<𝑇+𝑃𝜃_𝑑≤𝜀<𝑇3+𝑉𝑋₀,𝑟₀+𝐾+4𝑑𝐿_𝑑𝐾_𝑑∫_𝑅𝑛𝜋(𝑑𝑢)𝑇𝑒^{(𝜆1+𝜆2)𝐾𝑇}𝒱_𝑑+2𝑉₁𝑁max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑖𝑇𝑒^{(𝜆1+𝜆2)𝐾𝑇}𝒱_𝑑+𝑀₁ℎ^1/2+𝑀₂[ℎ+∘(ℎ)]𝑒^{(𝜆1+𝜆2)𝐾𝑇}𝒱_𝑑. (75) Hence on using (74), we conclude that 𝑃Ω=𝑃sup_{0≤𝑡≤𝑇}|𝑌(𝑡)−𝑋(𝑡)|²≤𝑉𝑋≥𝛿₀,𝑟₀+𝐾+4𝑑𝐿_𝑑𝐾_𝑑∫_𝑅𝑛𝜋(𝑑𝑢)𝑇𝑒^{(𝜆1+𝜆2)𝐾𝑇}𝒱_𝑑+2𝑉₁𝑁max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑖𝑇𝑒^{(𝜆1+𝜆2)𝐾𝑇}𝒱_𝑑+𝑀₁ℎ^1/2+𝑀₂[ℎ+∘(ℎ)]𝑒^{(𝜆1+𝜆2)𝐾𝑇}𝒱_𝑑+𝐶_𝑑𝛿[(ℎ+𝜀ℎ+∘)]3. (76) Recalling that 𝒱_𝑑→∞ as 𝑑→∞, we can choose 𝑑 sufficiently large for 𝑉𝑋₀,𝑟₀+𝐾+4𝑑𝐿_𝑑𝐾_𝑑∫_𝑅𝑛𝜋(𝑑𝑢)𝑇𝑒^{(𝜆1+𝜆2)𝐾𝑇}𝒱_𝑑+2𝑉₁𝑁max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑖𝑇𝑒^{(𝜆1+𝜆2)𝐾𝑇}𝒱_𝑑<𝜀3 (77) and then choose ℎ sufficiently small for 𝑀₁ℎ^1/2+𝑀₂[ℎ+∘(ℎ)]𝑒^{(𝜆1+𝜆2)𝐾𝑇}𝒱_𝑑+𝐶_𝑑𝛿[(ℎ<𝜀ℎ+∘)]3 (78) to obtain 𝑃sup_{0≤𝑡≤𝑇}|𝑌(𝑡)−𝑋(𝑡)|²≥𝛿<𝜀. (79) The proof of Theorem 12 is now complete.

5. An Example

In this section, we construct one example to demonstrate the effectiveness of this theory. Let 𝑟(𝑡) be a right-continuous Markov chain taking values in 𝑆={1,2} with the generator Γ=𝛾_𝑖𝑗2×2=−222−2 (80) Let 𝑁(𝑑𝑡,𝑑𝑢) be a compensated Poisson random measures and is given by 𝜋(𝑑𝑢)𝑑𝑡=𝜆𝑓(𝑢)𝑑𝑢𝑑𝑡, where 𝜆=2 and 𝑓(𝑢)=1√𝑒2𝜋𝑢^{−(𝑙𝑛𝑢)2/2},0≤𝑢<∞ (81) is the density function of a lognormal random variable. Of course 𝑁(𝑑𝑡,𝑑𝑢) and 𝑟(𝑡) are assumed to be independent. Consider a linear stochastic pantograph delay equations with Markovian switching and pure jumps (𝑡)(𝑟(𝑡𝑋(𝑡)+𝑑𝑋=𝑎))𝑑𝑡_|𝑢|≤1𝑢𝑏(𝑟(𝑡))𝑋(0.5𝑡)𝑁(𝑑𝑡,𝑑𝑢). (82) Here 𝑎(1)=−3, 𝑎(2)=−1, 𝑏(1)=1/√10, and 𝑏(2)=1/√6. Then (82) can be regarded as the result of the two equations (𝑡)(𝑡)1𝑑𝑋=−3𝑋𝑑𝑡+√10^|𝑢|≤1()𝑁(),(𝑡)(𝑡)1𝑢𝑋0.5𝑡𝑑𝑡,𝑑𝑢𝑑𝑋=−𝑋𝑑𝑡+√6^|𝑢|≤1𝑢𝑋(0.5𝑡)𝑁(𝑑𝑡,𝑑𝑢), (83) switching among each other according to the movement of the Markov chain 𝑟(𝑡). Obviously, (82) satisfies Assumptions 2 and 3. Given the stepsize ℎ, we can have the Euler method 𝑌_𝑛+1=𝑌_𝑛+𝑎𝑟^ℎ_𝑛𝑌_𝑛ℎ+^𝑡𝑛+1_{𝑡𝑛|𝑢|≤1}𝑢𝑏𝑟^ℎ_𝑛𝑌_[0.5𝑛]𝑁(𝑑𝑡,𝑑𝑢), (84) with 𝑌₀=𝑋(0). Let 𝑍₁(𝑡)=𝑌_𝑛, 𝑍₂(𝑡)=𝑌_[0.5𝑛], and 𝑟(𝑡)=𝑟^ℎ_𝑛, for 𝑡∈[𝑡_𝑛,𝑡_𝑛+1). Then we define the continuous Euler approximate solution 𝑌(𝑡)=𝑌(0)+^𝑡₀𝑎(𝑟(𝑠))𝑍₁(𝑠)+𝑑𝑠_{𝑡0|𝑢|≤1}𝑢𝑏(𝑟(𝑠))𝑍₂(𝑠)𝑁(𝑑𝑠,𝑑𝑢). (85) Since the conditions of Theorem 4 are satisfied, then the approximate solution (85) will converge to the true solution of (82) in the sense of the 𝐿²-norm. To examine the convergence in the sense of the probability, we construct a function 𝑉()=𝛽𝑥,𝑖¹𝑥²𝛽,if𝑖=1,₂𝑥²,if𝑖=2. (86) Then =𝐿𝑉𝑥,𝑦,𝑖2𝛽₂−8𝛽₁𝑥²+1∫10^|𝑢|≤1𝑢²𝜋(𝑑𝑢)𝛽₁𝑦²,if𝑖=1,2𝛽₁−4𝛽₂𝑥²+16∫_|𝑢|≤1𝑢²𝜋(𝑑𝑢)𝛽₂𝑦²,if𝑖=2. (87) From the properties of the lognormal distribution, we have 𝐿𝑉𝑥,𝑦,1≤−8−2𝛽₂𝛽₁𝛽₁𝑥²+15𝑒²𝛽₁𝑦²,𝐿𝑉𝑥,𝑦,2≤−4−2𝛽₁𝛽₂𝛽₂𝑥²+13𝑒²𝛽₂𝑦². (88)(89) If 6/(12−𝑒²)<(𝛽₂/𝛽₁)<4−(1/10)𝑒², then it follows that Assumptions 10 and 11 are satisfied. Consequently, the approximate solution (85) will converge to the true solution of (82) for any 𝑡∈[0,𝑇] in the sense of Theorem 12.

Acknowledgment

This work is sponsored by Qing Lan project of Jiangsu Province (2012) and supported by the Grant of Jiangsu Second Normal University (Jsie2011zd04) and the Jiangsu Government Overseas Study Scholarship.