Approximations of Numerical Method for Neutral Stochastic Functional Differential Equations with Markovian Switching

Abstract

Stochastic systems with Markovian switching have been used in a variety of application areas, including biology, epidemiology, mechanics, economics, and finance. In this paper, we study the Euler-Maruyama (EM) method for neutral stochastic functional differential equations with Markovian switching. The main aim is to show that the numerical solutions will converge to the true solutions. Moreover, we obtain the convergence order of the approximate solutions.

1. Introduction

Stochastic systems with Markovian switching have been successfully used in a variety of application areas, including biology, epidemiology, mechanics, economics, and finance [1]. As well as deterministic neutral functional differential equations and stochastic functional differential equations (SFDEs), most neutral stochastic functional differential equations with Markovian switching (NSFDEsMS) cannot be solved explicitly, so numerical methods become one of the powerful techniques. A number of papers have studied the numerical analysis of the deterministic neutral functional differential equations, for example, [2, 3] and references therein. The numerical solutions of SFDEs and stochastic systems with Markovian switching have also been studied extensively by many authors. Here we mention some of them, for example, [4–16]. moreover, Many well-known theorems in SFDEs are successfully extended to NSFDEs, for example, [17–24] discussed the stability analysis of the true solutions.

However, to the best of our knowledge, little is yet known about the numerical solutions for NSFDEsMS. In this paper we will extend the method developed by [5, 16] to NSFDEsMS and study strong convergence for the Euler-Maruyama approximations under the local Lipschitz condition, the linear growth condition, and contractive mapping. The three conditions are standard for the existence and uniqueness of the true solutions. Although the method of analysis borrows from [5], the existence of the neutral term and Markovian switching essentially complicates the problem. We develop several new techniques to cope with the difficulties which have risen from the two terms. Moreover, we also generalize the results in [19].

In Section 2, we describe some preliminaries and define the EM method for NSFDEsMS and state our main result that the approximate solutions strongly converge to the true solutions. The proof of the result is rather technical so we present several lemmas in Section 3 and then complete the proof in Section 4. In Section 5, under the global Lipschitz condition, we reveal the order of the convergence of the approximate solutions. Finally, a conclusion is made in Section 6.

2. Preliminaries and EM Scheme

Throughout this paper let (Ω,ℱ,ℙ) be a complete probability space with a filtration {ℱ_𝑡}_𝑡≥0 satisfying the usual conditions, that is, it is right continuous and increasing while ℱ₀ contains all P-null sets. Let 𝑤(𝑡)=(𝑤₁(𝑡),…,𝑤_𝑚(𝑡))^𝑇 be an 𝑚-dimensional Brownian motion defined on the probability space. Let |⋅| be the Euclidean norm in ℝ^𝑛. Let ℝ₊=[0,∞), and let 𝜏>0. Denoted by 𝐶([−𝜏,0],ℝ^𝑛) the family of continuous functions from [−𝜏,0] to ℝ^𝑛 with the norm ‖𝜑‖=sup_{−𝜏≤𝜃≤0}|𝜑(𝜃)|. Let 𝑝>0 and 𝐿^𝑝_ℱ0([−𝜏,0];ℝ^𝑛) be the family of ℱ₀-measurable 𝐶([−𝜏,0];ℝ^𝑛)-valued random variables 𝜉 such that 𝔼‖𝜉‖^𝑝<∞. If 𝑥(𝑡) is an ℝ^𝑛-valued stochastic process on 𝑡∈[−𝜏,∞), we let 𝑥_𝑡={𝑥(𝑡+𝜃)∶−𝜏≤𝜃≤0} for 𝑡≥0.

Let 𝑟(𝑡), 𝑡≥0, be a right-continuous Markov chain on the probability space taking values in a finite state space 𝑆={1,2,…,𝑁} with the generator Γ=(𝛾_𝑖𝑗)_𝑁×𝑁 given by 𝑃𝑟()(𝑡)=𝛾𝑡+Δ=𝑗∣𝑟=𝑖^𝑖𝑗Δ+𝑜(Δ)if𝑖≠𝑗,1+𝛾_𝑖𝑗Δ+𝑜(Δ)if𝑖=𝑗, (2.1) where Δ>0. Here 𝛾_𝑖𝑗≥0 is the transition rate from 𝑖 to 𝑗 if 𝑖≠𝑗 while 𝛾_𝑖𝑖=−∑_𝑖≠𝑗𝛾_𝑖𝑗. We assumethat the Markov chain 𝑟(⋅) is independent of the Brownian motion 𝑤(⋅). It is well known that almost every sample path of 𝑟(⋅) is a right-continuous step function with finite number of simple jumps in any finite subinterval of ℜ₊=[0,∞).

In this paper, we consider the 𝑛-dimensional NSFDEsMS 𝑑𝑥(𝑡)−𝑢𝑥_𝑡,𝑟(𝑡)=𝑓𝑥_𝑡,𝑟(𝑡)𝑑𝑡+𝑔𝑥_𝑡,𝑟(𝑡)𝑑𝑤(𝑡),𝑡≥0, (2.2) with initial data 𝑥₀=𝜉∈𝐿^𝑝_ℱ0([−𝜏,0];ℝ^𝑛) and 𝑟(0)=𝑖₀∈𝑆, where 𝑓∶𝐶([−𝜏,0];ℝ^𝑛)→ℝ^𝑛, 𝑔∶𝐶([−𝜏,0];ℝ^𝑛)→ℝ^𝑛×𝑚 and 𝑢∶𝐶([−𝜏,0];ℝ^𝑛)→ℝ^𝑛. As a standing hypothesis we assume that both 𝑓 and 𝑔 are sufficiently smooth so that (2.2) has a unique solution. We refer the reader to Mao [10, 12] for the conditions on the existence and uniqueness of the solution 𝑥(𝑡). The initial data 𝜉 and 𝑖₀ could be random, but the Markov property ensures that it is sufficient to consider only the case when both 𝑥₀ and 𝑖₀ are constants.

To analyze the Euler-Maruyama (EM) method, we need the following lemma (see [6, 7, 10–12, 16]).

Lemma 2.1. Given Δ>0, let 𝑟^Δ_𝑘=𝑟(𝑘Δ) for 𝑘≥0. Then {𝑟^Δ_𝑘,𝑘=0,1,2,…} is a discrete Markov chain with the one-step transition probability matrix 𝑃(Δ)=𝑃_𝑖𝑗(Δ)_𝑁×𝑁=𝑒^ΔΓ. (2.3)

For the completeness, we give the simulation of the Markov chain as follows. Given a stepsize Δ>0, we compute the one-step transition probability matrix 𝑃(Δ)=𝑃_𝑖𝑗(Δ)_𝑁×𝑁=𝑒^ΔΓ. (2.4) Let 𝑟^Δ₀=𝑖₀ and generate a random number 𝜉₁ which is uniformly distributed in [0,1]. Define 𝑟^Δ₁=𝑖₁,if𝑖₁∈𝑆−{𝑁}suchthat∑^𝑖1−1_𝑗=1𝑃_𝑖0,𝑗(Δ)≤𝜉₁<∑^𝑖1_𝑗=1𝑃_𝑖0,𝑗(Δ),𝑁,if∑^𝑁−1_𝑗=1𝑃_𝑖0,𝑗(Δ)≤𝜉₁, (2.5) where we set ∑⁰_𝑖=1𝑃_𝑖0,𝑗(Δ)=0 as usual. Generate independently a new random number 𝜉₂ which is again uniformly distributed in [0,1], and then define 𝑟^Δ₂=𝑖₂,if𝑖₂∈𝑆−{𝑁}suchthat∑^𝑖2−1_𝑗=1𝑃_𝑟^Δ_1,𝑗(Δ)≤𝜉₂<∑^𝑖2_𝑗=1𝑃_𝑟^Δ_1,𝑗(Δ),𝑁,if∑^𝑁−1_𝑗=1𝑃_𝑟^Δ_1,𝑗(Δ)≤𝜉₂. (2.6)

Repeating this procedure a trajectory of {𝑟^Δ_𝑘,𝑘=0,1,2,…} can be generated. This procedure can be carried out independently to obtain more trajectories.

Now we can define the Euler-Maruyama (EM) approximate solution for (2.2) on the finite time interval [0,𝑇]. Without loss of any generality, we may assume that 𝑇/𝜏 is a rational number; otherwise we may replace 𝑇 by a larger number. Let the step size Δ∈(0,1) be a fraction of 𝜏 and 𝑇, namely, Δ=𝜏/𝑁=𝑇/𝑀 for some integers 𝑁>𝜏 and 𝑀>𝑇. The explicit discrete EM approximate solution 𝑦(𝑘Δ), 𝑘≥−𝑁 is defined as follows: 𝑦(𝑘Δ)=𝜉(𝑘Δ),−𝑁≤𝑘≤0,𝑦((𝑘+1)Δ)=𝑦(𝑘Δ)+𝑢𝑦_𝑘Δ,𝑟^Δ_𝑘−𝑢𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘−1+𝑓𝑦_𝑘Δ,𝑟^Δ_𝑘Δ+𝑔𝑦_𝑘Δ,𝑟^Δ_𝑘Δ𝑤_𝑘,0≤𝑘≤𝑀−1, (2.7) where Δ𝑤_𝑘=𝑤((𝑘+1)Δ)−𝑤(𝑘Δ) and 𝑦_𝑘Δ={𝑦_𝑘Δ(𝜃)∶−𝜏≤𝜃≤0} is a 𝐶([−𝜏,0];ℝ^𝑛)-valued random variable defined by 𝑦_𝑘Δ(𝜃)=𝑦((𝑘+𝑖)Δ)+𝜃−𝑖ΔΔ𝑦((𝑘+𝑖+1)Δ)−𝑦)Δ=()((𝑘+𝑖)Δ−𝜃−𝑖ΔΔ𝑦((𝑘+𝑖)Δ)+𝜃−𝑖ΔΔ𝑦((𝑘+𝑖+1)Δ), (2.8) for 𝑖Δ≤𝜃≤(𝑖+1)Δ, 𝑖=−𝑁,−𝑁+1,…,−1, where in order for 𝑦_−Δ to be well defined, we set 𝑦(−(𝑁+1)Δ)=𝜉(−𝑁Δ).

That is, 𝑦_𝑘Δ(⋅) is the linear interpolation of 𝑦((𝑘−𝑁)Δ), 𝑦((𝑘−𝑁+1)Δ),…,𝑦(𝑘Δ). We hence have 𝑦_𝑘Δ(𝜃=())Δ−𝜃−𝑖ΔΔ𝑦((𝑘+𝑖)Δ)+𝜃−𝑖ΔΔ𝑦)Δ≤((𝑘+𝑖+1)𝑦((𝑘+𝑖)Δ)∨𝑦((𝑘+𝑖+1)Δ). (2.9) We therefore obtain 𝑦_𝑘Δ(𝜃)=max_{−𝑁≤𝑖≤0}𝑦((𝑘+𝑖)Δ),forany𝑘=−1,0,1,…,𝑀−1. (2.10) It is obvious that ‖𝑦_−Δ‖≤‖𝑦₀‖.

In our analysis it will be more convenient to use continuous-time approximations. We hence introduce the 𝐶([−𝜏,0];ℝ^𝑛)-valued step process 𝑦_𝑡=^𝑀−2_𝑘=0𝑦_𝑘Δ1_{[𝑘Δ,(𝑘+1)Δ)}(𝑡)+𝑦_(𝑀−1)Δ1_{[(𝑀−1)Δ,𝑀Δ]}(𝑡),𝑟(𝑡)=^𝑀−1_𝑘=0𝑟^Δ_𝑘1_{[𝑘Δ,(𝑘+1)Δ)}(𝑡), (2.11) and we define the continuous EM approximate solution as follows: let 𝑦(𝑡)=𝜉(𝑡) for −𝜏≤𝑡≤0, while for 𝑡∈[𝑘Δ,(𝑘+1)Δ], 𝑘=0,1,…,𝑀−1, 𝑦(𝑡)=𝜉(0)+𝑢𝑦_(𝑘−1)Δ+𝑡−𝑘ΔΔ𝑦_𝑘Δ−𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘−𝑢𝑦_−Δ,𝑟^Δ₀+^𝑡₀𝑓𝑦_𝑠,𝑟(𝑠)𝑑𝑠+^𝑡₀𝑔𝑦_𝑠,𝑟(𝑠)𝑑𝑤(𝑠). (2.12) Clearly, (2.12) can also be written as 𝑦(𝑡)=𝑦(𝑘Δ)+𝑢𝑦_(𝑘−1)Δ+𝑡−𝑘ΔΔ𝑦_𝑘Δ−𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘−𝑢𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘−1+^𝑡_𝑘Δ𝑓𝑦_𝑠,𝑟(𝑠)𝑑𝑠+^𝑡_𝑘Δ𝑔𝑦_𝑠,𝑟(𝑠)𝑑𝑤(𝑠). (2.13) In particular, this shows that 𝑦(𝑘Δ)=𝑦(𝑘Δ), that is, the discrete and continuous EM approximate solutions coincide at the grid points. We know that 𝑦(𝑡) is not computable because it requires knowledge of the entire Brownian path, not just its Δ-increments. However, 𝑦(𝑘Δ)=𝑦(𝑘Δ), so the error bound for 𝑦(𝑡) will automatically imply the error bound for 𝑦(𝑘Δ). It is then obvious that 𝑦_𝑘Δ≤𝑦_𝑘Δ,∀𝑘=0,1,2,…,𝑀−1. (2.14) Moreover, for any 𝑡∈[0,𝑇], sup_{0≤𝑡≤𝑇}𝑦_𝑡=sup_{0≤𝑘≤𝑀−1}𝑦_𝑘Δ≤sup_{0≤𝑘≤𝑀−1}𝑦_𝑘Δ=sup_{0≤𝑘≤𝑀−1}sup_{−𝜏≤𝜃≤0}𝑦(𝑘Δ+𝜃)≤sup_{0≤𝑡≤𝑇}sup_{−𝜏≤𝜃≤0}𝑦(𝑡+𝜃)≤sup_{−𝜏≤𝑠≤𝑇}𝑦(𝑠), (2.15) and letting [𝑡/Δ] be the integer part of 𝑡/Δ, then 𝑦_𝑡=𝑦_[𝑡/Δ]Δ≤𝑦_[𝑡/Δ]Δ≤sup_{−𝜏≤𝑠≤𝑡}𝑦(𝑠). (2.16) These properties will be used frequently in what follows, without further explanation.

For the existence and uniqueness of the solution of (2.2) and the boundedness of the solution’s moments, we impose the following hypotheses (e.g., see [11]).

Assumption 2.2. For each integer 𝑗≥1 and 𝑖∈𝑆, there exists a positive constant 𝐶_𝑗 such that 𝑓𝜑,𝑖−𝑓𝜓,𝑖²∨𝑔𝜑,𝑖−𝑔𝜓,𝑖²≤𝐶_𝑗𝜑−𝜓² (2.17) for 𝜑, 𝜓∈𝐶([−𝜏,0];ℝ^𝑛) with ‖𝜑‖∨‖𝜓‖≤𝑗.

Assumption 2.3. There is a constant 𝐾>0 such that 𝑓𝜑,𝑖²∨𝑔𝜑,𝑖²≤𝐾1+𝜑², (2.18) for 𝜑∈𝐶([−𝜏,0];ℝ^𝑛) and 𝑖∈𝑆.

Assumption 2.4. There exists a constant 𝜅∈(0,1) such that for all 𝜑, 𝜓∈𝐶([−𝜏,0];ℝ^𝑛) and 𝑖∈𝑆, 𝑢𝜑,𝑖−𝑢𝜓,𝑖≤𝜅𝜑−𝜓, (2.19) for 𝜑, 𝜓∈𝐶([−𝜏,0];ℝ^𝑛) and 𝑢(0,𝑖)=0.

We also impose the following condition on the initial data.

Assumption 2.5. 𝜉∈𝐿^𝑝_ℱ0([−𝜏,0];ℝ^𝑛) for some 𝑝≥2, and there exists a nondecreasing function 𝛼(⋅) such that 𝔼sup_{−𝜏≤𝑠≤𝑡≤0}𝜉(𝑡)−𝜉(𝑠)²≤𝛼(𝑡−𝑠), (2.20) with the property 𝛼(𝑠)→0 as 𝑠→0.

From Mao and Yuan [11], we may therefore state the following theorem.

Theorem 2.6. Let 𝑝≥2. If Assumptions 2.3–2.5 are satisfied, then 𝔼sup_{−𝜏≤𝑡≤𝑇}|𝑥(𝑡)|^𝑝≤𝐻_{𝜅,𝑝,𝑇,𝐾,𝜉}, (2.21) for any 𝑇>0, where 𝐻_{𝜅,𝑝,𝑇,𝐾,𝜉} is a constant dependent on 𝜅, 𝑝, 𝑇, 𝐾, 𝜉.

The primary aim of this paper is to establish the following strong mean square convergence theorem for the EM approximations.

Theorem 2.7. If Assumptions 2.2–2.5 hold, lim_Δ→0𝔼sup_{0≤𝑡≤𝑇}𝑥(𝑡)−𝑦(𝑡)²=0. (2.22)

The proof of this theorem is very technical, so we present some lemmas in the next section, and then complete the proof in the sequent section.

3. Lemmas

Lemma 3.1. If Assumptions 2.3–2.5 hold, for any 𝑝≥2, there exists a constant 𝐻(𝑝) such that 𝔼sup_{−𝜏≤𝑡≤𝑇}𝑦(𝑡)^𝑝≤𝐻𝑝, (3.1) where 𝐻(𝑝) is independent of Δ.

Proof. For 𝑡∈[𝑘Δ,(𝑘+1)Δ], 𝑘=0,1,2,…,𝑀−1, set 𝑦(𝑡)∶=𝑦(𝑡)−𝑢(𝑦_(𝑘−1)Δ+(𝑡−𝑘Δ)(𝑦_𝑘Δ−𝑦_(𝑘−1)Δ)/Δ,𝑟^Δ_𝑘) and ℎ(𝑡)∶=𝔼sup_{−𝜏≤𝑠≤𝑡}𝑦(𝑠)^𝑝,ℎ(𝑡)∶=𝔼sup_{0≤𝑠≤𝑡}𝑦(𝑠)^𝑝. (3.2) Recall the inequality that for 𝑝≥1 and any 𝜀>0, |𝑥+𝑦|^𝑝≤(1+𝜀)^𝑝−1(|𝑥|^𝑝+𝜀^1−𝑝|𝑦|^𝑝). Then we have, from Assumption 2.4, 𝑦(𝑡)^𝑝≤(1+𝜀)^𝑝−1𝑦(𝑡)^𝑝+𝜀^1−𝑝𝑢𝑦_(𝑘−1)Δ+(𝑡−𝑘Δ)𝑦_𝑘Δ−𝑦_(𝑘−1)ΔΔ,𝑟^Δ_𝑘^𝑝≤(1+𝜀)^𝑝−1𝑦(𝑡)^𝑝+𝜀^1−𝑝𝜅^𝑝𝑦_(𝑘−1)Δ+𝑡−𝑘ΔΔ𝑦_𝑘Δ−𝑦_(𝑘−1)Δ^𝑝. (3.3) By ‖𝑦_−Δ‖≤‖𝑦₀‖, noting 𝑘=0,1,2,…,𝑀−1, 𝑦_(𝑘−1)Δ+𝑡−𝑘ΔΔ𝑦_𝑘Δ−𝑦_(𝑘−1)Δ^𝑝≤(𝑘+1)Δ−𝑡Δ𝑦_(𝑘−1)Δ+𝑡−𝑘ΔΔ𝑦_𝑘Δ^𝑝≤(𝑘+1)Δ−𝑡Δsup_{−𝜏≤𝑠≤𝑡}𝑦(𝑠)+𝑡−𝑘ΔΔsup_{−𝜏≤𝑠≤𝑡}𝑦(𝑠)^𝑝≤sup_{−𝜏≤𝑠≤𝑡}𝑦(𝑠)^𝑝. (3.4) Consequently, choose 𝜀=𝜅/(1−𝜅), then 𝑦(𝑡)^𝑝≤(1−𝜅)^1−𝑝𝑦(𝑡)^𝑝+𝜅sup_{−𝜏≤𝑠≤𝑡}𝑦(𝑠)^𝑝. (3.5) Hence, ℎ(𝑡)≤𝔼𝜉^𝑝+𝔼sup_{0≤𝑠≤𝑡}𝑦(𝑠)^𝑝≤𝔼𝜉^𝑝+𝜅ℎ(𝑡)+(1−𝜅)^1−𝑝ℎ(𝑡), (3.6) which implies ℎ(𝑡)≤𝔼𝜉^𝑝+ℎ(𝑡)1−𝜅(1−𝜅)^𝑝. (3.7) Since 𝑦(𝑡)=𝑦(0)+^𝑡₀𝑓𝑦_𝑠,𝑟(𝑠)𝑑𝑠+^𝑡₀𝑔𝑦_𝑠,𝑟(𝑠)𝑑𝑤(𝑠), (3.8) with 𝑦(0)=𝑦(0)−𝑢(𝑦_−Δ), by the Hölder inequality, we have 𝑦(𝑡)^𝑝≤3^𝑝−1𝑦(0)^𝑝+𝑡^𝑝−1𝑡₀𝑓𝑦_𝑠,𝑟(𝑠)^𝑝𝑑𝑠+^𝑡₀𝑔𝑦_𝑠,𝑟(𝑠)𝑑𝑤(𝑠)^𝑝. (3.9) Hence, for any 𝑡₁∈[0,𝑇], ℎ𝑡₁≤3^𝑝−1𝔼𝑦(0)^𝑝+𝑇^𝑝−1𝔼^𝑡1₀𝑓𝑦_𝑠,𝑟(𝑠)^𝑝𝑑𝑠+𝔼sup_{0≤𝑡≤𝑡1}^𝑡₀𝑔𝑦_𝑠,𝑟(𝑠)𝑑𝑤(𝑠)^𝑝. (3.10) By Assumption 2.4 and the fact ‖𝑦_−Δ‖≤‖𝑦₀‖, we compute that 𝔼𝑦(0)^𝑝=𝔼𝑦(0)−𝑢𝑦_−Δ,𝑟^Δ₀^𝑝≤𝔼𝑦(0)+𝜅𝑦_−Δ^𝑝≤𝔼𝑦(0)+𝜅𝑦₀^𝑝≤𝔼𝜉(0)+𝜅𝜉^𝑝≤(1+𝜅)^𝑝𝔼𝜉^𝑝. (3.11) Assumption 2.3 and the Hölder inequality give 𝔼^𝑡1₀𝑓𝑦_𝑠,𝑟(𝑠)^𝑝𝑑𝑠≤𝔼^𝑡1₀𝐾^𝑝/21+𝑦_𝑠^2𝑝/2𝑑𝑠≤𝐾^𝑝/22^(𝑝−2)/2𝔼^𝑡1₀1+𝑦_𝑠^𝑝𝑑𝑠≤𝐾^𝑝/22^(𝑝−2)/2𝑇+^𝑡1₀𝔼sup_{−𝜏≤𝑡≤𝑠}𝑦(𝑡)^𝑝𝑑𝑠. (3.12) Applying the Burkholder-Davis-Gundy inequality, the Hölder inequality and Assumption 2.3 yield 𝔼sup_{0≤𝑡≤𝑡1}^𝑡₀𝑔𝑦_𝑠,𝑟(𝑠)𝑑𝑤(𝑠)^𝑝≤𝐶_𝑝𝔼^𝑡1₀𝑔𝑦_𝑠,𝑟(𝑠)²𝑑𝑠^𝑝/2≤𝐶_𝑝𝑇^(𝑝−2)/2𝔼^𝑡1₀𝑔𝑦_𝑠,𝑟(𝑠)^𝑝𝑑𝑠≤𝐶_𝑝𝑇^(𝑝−2)/2𝔼^𝑡1₀𝐾^𝑝/21+𝑦_𝑠^2𝑝/2𝑑𝑠≤𝐶_𝑝𝑇^(𝑝−2)/2𝐾^𝑝/22^(𝑝−2)/2𝔼^𝑡1₀1+𝑦_𝑠^𝑝𝑑𝑠≤𝐶_𝑝𝑇^(𝑝−2)/2𝐾^𝑝/22^(𝑝−2)/2𝑇+^𝑡1₀𝔼sup_{−𝜏≤𝑡≤𝑠}𝑦(𝑡)^𝑝𝑑𝑠, (3.13) where 𝐶_𝑝 is a constant dependent only on 𝑝. Substituting (3.11), (3.12), and (3.13) into (3.10) gives ℎ𝑡₁≤3^𝑝−1(1+𝜅)^𝑝𝔼𝜉^𝑝+𝐾^𝑝/22^(𝑝−2)/2𝑇^𝑝+𝐶_𝑝(2𝑇)^(𝑝−2)/2𝐾^𝑝/2𝑇+3^𝑝−1𝐾^𝑝/22^(𝑝−2)/2𝑇^𝑝−1+𝐶_𝑝(2𝑇)^(𝑝−2)/2𝐾^𝑝/2𝑡1₀𝔼sup_{−𝜏≤𝑡≤𝑠}𝑦(𝑡)^𝑝𝑑𝑠=∶𝐶₁+𝐶₂^𝑡1₀ℎ(𝑠)𝑑𝑠. (3.14) Hence from (3.7), we have ℎ𝑡₁≤𝔼𝜉^𝑝+11−𝜅(1−𝜅)^𝑝𝐶₁+𝐶₂^𝑡1₀ℎ(𝑠)≤𝔼𝜉𝑑𝑠_𝑝+𝐶1−𝜅(1−¹𝜅)^𝑝+𝐶₂(1−𝜅)^𝑝𝑡1₀ℎ(𝑠)𝑑𝑠. (3.15) By the Gronwall inequality we find that ℎ(𝑇)≤𝔼𝜉^𝑝+𝐶1−𝜅(1−¹𝜅)^𝑝𝑒^{𝐶2𝑇/(1−𝜅)𝑝}. (3.16) From the expressions of 𝐶₁ and 𝐶₂, we know that they are positive constants dependent only on 𝜉, 𝜅, 𝐾, 𝑝, and 𝑇, but independent of Δ. The proof is complete.

Lemma 3.2. If Assumptions 2.3–2.5 hold, then for any integer 𝑙>1, 𝔼sup_{0≤𝑘≤𝑀−1}𝑦_𝑘Δ−𝑦_(𝑘−1)Δ²≤𝑐^′₁+𝑐₁𝛼(Δ)+𝑐₁(𝑙)Δ^{(𝑙−1)/𝑙}=∶𝛾(Δ), (3.17) where 𝑐₁=1/(1−2𝜅), 𝑐^′₁=(8𝜅/(1−2𝜅))𝐻(2), and  𝑐₁(𝑙) is a constant dependent on 𝑙 but independent of Δ.

Proof. For 𝜃∈[𝑖Δ,(𝑖+1)Δ], where 𝑖=−𝑁,−(𝑁+1),…,−1, from (2.7), 𝑦_𝑘Δ−𝑦_(𝑘−1)Δ≤(𝑖+1)Δ−𝜃Δ𝑦((𝑘+𝑖)Δ)−𝑦)Δ+((𝑘−1+𝑖)𝜃−𝑖ΔΔ𝑦((𝑘+𝑖+1)Δ)−𝑦)Δ≤((𝑘+𝑖)𝑦((𝑘+𝑖)Δ)−𝑦((𝑘−1+𝑖)Δ)∨𝑦((𝑘+𝑖+1)Δ)−𝑦((𝑘+𝑖)Δ), (3.18) so 𝑦_𝑘Δ−𝑦_(𝑘−1)Δ≤sup_{−𝑁≤𝑖≤0}𝑦((𝑘+𝑖)Δ)−𝑦((𝑘−1+𝑖)Δ). (3.19) We therefore have 𝔼sup_{0≤𝑘≤𝑀−1}𝑦_𝑘Δ−𝑦_(𝑘−1)Δ²≤𝔼sup_{0≤𝑘≤𝑀−1}sup_{−𝑁≤𝑖≤0}𝑦((𝑘+𝑖)Δ)−𝑦((𝑘−1+𝑖)Δ)²≤𝔼sup_{−𝑁≤𝑘≤𝑀−1}𝑦(𝑘Δ)−𝑦((𝑘−1)Δ)². (3.20) When −𝑁≤𝑘≤0, by Assumption 2.5 and 𝑦(−(𝑁+1)Δ)=𝜉(−𝑁Δ), 𝔼sup_{−𝑁≤𝑘≤0}𝑦(𝑘Δ)−𝑦((𝑘−1)Δ)²≤𝔼sup_{−𝑁≤𝑘≤0}𝜉(𝑘Δ)−𝜉((𝑘−1)Δ)²≤𝛼(Δ). (3.21) When 1≤𝑘≤𝑀−1, from (2.13), we have 𝑦(𝑘Δ)−𝑦((𝑘−1)Δ)=𝑢𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘−1−𝑢𝑦_(𝑘−2)Δ,𝑟^Δ_𝑘−2+𝑓𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘−1Δ+𝑔𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘−1Δ𝑤_𝑘−1. (3.22) Recall the elementary inequality, for any 𝑥, 𝑦>0 and 𝜀∈(0,1), (𝑥+𝑦)²≤𝑥²/𝜀+𝑦²/(1−𝜀). Then 𝑦(𝑘Δ)−𝑦((𝑘−1)Δ)²≤1𝜀𝑢𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘−1−𝑢𝑦_(𝑘−2)Δ,𝑟^Δ_𝑘−1+𝑢𝑦_(𝑘−2)Δ,𝑟^Δ_𝑘−1−𝑢𝑦_(𝑘−2)Δ,𝑟^Δ_𝑘−2²+1𝑓1−𝜀𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘−1Δ+𝑔𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘−1Δ𝑤_𝑘−1²≤2𝜅²𝜀𝑦_(𝑘−1)Δ−𝑦_(𝑘−2)Δ²+8𝜅²𝜀𝑦_(𝑘−2)Δ²+2𝑓1−𝜀𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘−1²Δ²+2𝑔1−𝜀𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘−1Δ𝑤_𝑘−1². (3.23) Consequently 𝔼sup_{1≤𝑘≤𝑀−1}𝑦(𝑘Δ)−𝑦((𝑘−1)Δ)²≤2𝜅²𝜀𝔼sup_{1≤𝑘≤𝑀−1}𝑦_(𝑘−1)Δ−𝑦_(𝑘−2)Δ²+8𝜅²𝜀𝔼sup_{1≤𝑘≤𝑀−1}𝑦_(𝑘−2)Δ²+2Δ²𝔼1−𝜀sup_{1≤𝑘≤𝑀−1}𝑓𝑦_(𝑘−1)Δ,𝑟(𝑡)²+2𝔼1−𝜀sup_{1≤𝑘≤𝑀−1}𝑔𝑦_(𝑘−1)Δ,𝑟(𝑡)Δ𝑤_𝑘−1². (3.24) We deal with these four terms, separately. By (3.21), 𝔼sup_{1≤𝑘≤𝑀−1}𝑦_(𝑘−1)Δ−𝑦_(𝑘−2)Δ²≤𝔼sup_{1≤𝑘≤𝑀−1}sup_{−𝑁≤𝑖≤0}𝑦((𝑘+𝑖−1)Δ)−𝑦((𝑘−2+𝑖)Δ)²≤𝔼sup_{−𝑁≤𝑘≤𝑀−1}𝑦(𝑘Δ)−𝑦((𝑘−1)Δ)²≤𝔼sup_{−𝑁≤𝑘≤0}𝑦(𝑘Δ)−𝑦((𝑘−1)Δ)²+𝔼sup_{1≤𝑘≤𝑀−1}𝑦(𝑘Δ)−𝑦((𝑘−1)Δ)²≤𝛼(Δ)+𝔼sup_{1≤𝑘≤𝑀−1}𝑦(𝑘Δ)−𝑦((𝑘−1)Δ)². (3.25) Noting that 𝔼[sup_{−𝜏≤𝑡≤𝑇}|𝑦(𝑡)|²]≤𝔼[sup_{−𝜏≤𝑡≤𝑇}|𝑦(𝑡)|²]≤𝐻(2) (where 𝐻(𝑝) has been defined in Lemma 3.1), by Assumption 2.3 and (2.15), 𝔼sup_{1≤𝑘≤𝑀−1}𝑓𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘−1²≤𝔼sup_{1≤𝑘≤𝑀−1}𝐾1+𝑦_(𝑘−1)Δ²≤𝐾+𝐾𝔼sup_{1≤𝑘≤𝑀−1}sup_{−𝑁≤𝑖≤0}𝑦((𝑘−1+𝑖)Δ)²≤𝐾+𝐾𝔼sup_{−𝑁≤𝑘≤𝑀−1}𝑦(𝑘Δ)²≤𝐾+𝐾𝔼sup_{−𝜏≤𝑡≤𝑇}𝑦(𝑡)²≤𝐾(1+𝐻(2)). (3.26) By the Hölder inequality, for any integer 𝑙>1, 𝔼sup_{1≤𝑘≤𝑀−1}𝑔𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘−1Δ𝑤_𝑘−1²≤𝔼sup_{1≤𝑘≤𝑀−1}𝑔𝑦_(𝑘−1)Δ,𝑟(𝑡)²sup_{1≤𝑘≤𝑀−1}Δ𝑤_𝑘−1²≤𝔼sup_{1≤𝑘≤𝑀−1}𝑔𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘−1^{2𝑙/(𝑙−1)(𝑙−1)/𝑙}𝔼sup_{1≤𝑘≤𝑀−1}Δ𝑤_𝑘−1^2𝑙1/𝑙≤𝔼sup_{0≤𝑘≤𝑀−1}𝐾1+𝑦_𝑘Δ^{2𝑙/(𝑙−1)(𝑙−1)/𝑙}𝔼^𝑀−1_𝑘=0Δ𝑤_𝑘^2𝑙1/𝑙≤𝐾^{𝑙/(1−𝑙)}𝔼1+sup_{0≤𝑘≤𝑀−1}𝑦_𝑘Δ^{2𝑙/(𝑙−1)(𝑙−1)/𝑙𝑀−1}_𝑘=0𝔼Δ𝑤_𝑘^2𝑙1/𝑙≤2^1/(𝑙−1)𝐾^{𝑙/(1−𝑙)}1+𝔼sup_{0≤𝑘≤𝑀−1}𝑦_𝑘Δ^{2𝑙/(𝑙−1)(𝑙−1)/𝑙𝑀−1}_𝑘=0(2𝑙−1)!!Δ^𝑙1/𝑙≤2^1/(𝑙−1)𝐾^{𝑙/(1−𝑙)}1+𝐻2𝑙𝑙−1^{(𝑙−1)/𝑙}(2𝑙−1)!!𝑇Δ^{𝑙−11/𝑙}≤𝐷(𝑙)Δ^{(𝑙−1)/𝑙}, (3.27) where 𝐷(𝑙) is a constant dependent on 𝑙.
Substituting (3.25), (3.26), and (3.27) into (3.24), choosing 𝜀=𝜅, and noting Δ∈(0,1), we have 𝔼sup_{1≤𝑘≤𝑀−1}𝑦(𝑘Δ)−𝑦((𝑘−1)Δ)²≤2𝜅𝛼(Δ)+1−2𝜅8𝜅𝐻(2)+((2(𝑙)1−2𝜅2𝐾1+𝐻))+2𝐷(1−2𝜅)²Δ^{(𝑙−1)/𝑙}. (3.28) Combining (3.21) with (3.28), from (3.20), we have 𝔼sup_{0≤𝑘≤𝑀−1}𝑦_𝑘Δ−𝑦_(𝑘−1)Δ²≤𝔼sup_{−𝑁≤𝑘≤𝑀−1}𝑦(𝑘Δ)−𝑦((𝑘−1)Δ)²≤𝔼sup_{−𝑁≤𝑘≤0}𝑦(𝑘Δ)−𝑦((𝑘−1)Δ)²+𝔼sup_{1≤𝑘≤𝑀−1}𝑦(𝑘Δ)−𝑦((𝑘−1)Δ)²≤1𝛼(Δ)+1−2𝜅8𝜅𝐻(2)+((2(𝑙)1−2𝜅2𝐾1+𝐻))+2𝐷(1−2𝜅)²Δ^{(𝑙−1)/𝑙}, (3.29) as required.

Lemma 3.3. If Assumptions 2.3–2.5 hold, 𝔼sup_{0≤𝑠≤𝑇}𝑦_𝑠−𝑦_𝑠²≤𝑐^′₂+𝑐₂𝛼(2Δ)+𝑐₂(𝑙)Δ^{(𝑙−1)/𝑙}=∶𝛽(Δ), (3.30) where 𝑐₂, 𝑐^′₂ are a constant independent of  𝑙  and Δ, 𝑐₂(𝑙) is a constant dependent on  𝑙  but independent of Δ.

Proof. Fix any 𝑠∈[0,𝑇] and 𝜃∈[−𝜏,0]. Let 𝑘_𝑠∈{0,1,2,…,𝑀−1}, 𝑘_𝜃∈{−𝑁,−𝑁+1,…,−1}, and 𝑘_𝑠𝜃∈{−𝑁,−𝑁+1,…,𝑀−1} be the integers for which 𝑠∈[𝑘_𝑠Δ,(𝑘_𝑠+1)Δ], 𝜃∈[𝑘_𝜃Δ,(𝑘_𝜃+1)Δ], and 𝑠+𝜃∈[𝑘_𝑠𝜃Δ,(𝑘_𝑠𝜃+1)Δ], respectively. Clearly, 0≤𝑠+𝜃−𝑘_𝑠+𝑘_𝜃𝑘≤2Δ,_𝑠𝜃−𝑘_𝑠+𝑘_𝜃∈{0,1,2}. (3.31) From (2.7), 𝑦_𝑠=𝑦_𝑘𝑠Δ(𝜃)=𝑦𝑘_𝑠+𝑘_𝜃Δ+𝜃−𝑘_𝜃ΔΔ𝑦𝑘_𝑠+𝑘_𝜃+1Δ−𝑦𝑘_𝑠+𝑘_𝜃Δ, (3.32) which yields 𝑦_𝑠−𝑦_𝑠=𝑦(𝑠+𝜃)−𝑦_𝑘𝑠Δ(𝜃≤𝑦()−)𝑠+𝜃𝑦𝑘_𝑠+𝑘_𝜃Δ+𝜃−𝑘_𝜃ΔΔ𝑦𝑘_𝑠+𝑘_𝜃+1Δ−𝑦𝑘_𝑠+𝑘_𝜃Δ≤𝑦(𝑠+𝜃)−𝑦𝑘_𝑠+𝑘_𝜃Δ+𝑦𝑘_𝑠+𝑘_𝜃+1Δ−𝑦𝑘_𝑠+𝑘_𝜃Δ, (3.33) so by (3.20) and Lemma 3.2, noting 𝑦(𝑀Δ)=𝑦((𝑀−1)Δ) from (2.15), 𝔼sup_{0≤𝑠≤𝑇}𝑦_𝑠−𝑦_𝑠²≤2𝔼sup_{0≤𝑠≤𝑇}sup_{−𝜏≤𝜃≤0}𝑦(𝑠+𝜃)−𝑦𝑘_𝑠+𝑘_𝜃Δ²+2𝔼sup_{0≤𝑘𝑠≤𝑀−1}sup_{−𝑁≤𝑘𝜃≤0}𝑦𝑘_𝑠+𝑘_𝜃+1Δ−𝑦𝑘_𝑠+𝑘_𝜃Δ²≤2𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇}𝑦(𝑠+𝜃)−𝑦𝑘_𝑠+𝑘_𝜃Δ²+2𝛾(Δ). (3.34) Therefore, it is a key to compute 𝔼(sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇}|𝑦(𝑠+𝜃)−𝑦((𝑘_𝑠+𝑘_𝜃)Δ)|²). We discuss the following four possible cases.
Case 1 (𝑘_𝑠+𝑘_𝜃≥0). We again divide this case into three possible subcases according to 𝑘_𝑠𝜃−(𝑘_𝑠+𝑘_𝜃)∈{0,1,2}.
Subcase  1 (𝑘_𝑠𝜃−(𝑘_𝑠+𝑘_𝜃)=0). From (2.13), 𝑦(𝑠+𝜃)−𝑦𝑘_𝑠+𝑘_𝜃Δ=𝑢𝑦_{(𝑘𝑠𝜃−1)Δ}+𝑠+𝜃−𝑘_𝑠𝜃ΔΔ𝑦_{𝑘𝑠𝜃Δ}−𝑦_{(𝑘𝑠𝜃−1)Δ},𝑟^Δ_𝑘𝑠𝜃−𝑢𝑦_{(𝑘𝑠𝜃−1)Δ},𝑟^Δ_{𝑘𝑠𝜃−1}+_𝑘^𝑠+𝜃_𝑠𝜃Δ𝑓𝑦_𝑣,𝑟(𝑣)𝑑𝑣+_𝑘^𝑠+𝜃_𝑠𝜃Δ𝑔𝑦_𝑣,𝑟(𝑣)𝑑𝑤(𝑣), (3.35) which yields 𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠+𝑘𝜃≥0}𝑦(𝑠+𝜃)−𝑦𝑘_𝑠+𝑘_𝜃Δ²≤3𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠𝜃≥0}𝑢𝑦_{(𝑘𝑠𝜃−1)Δ}+𝑠+𝜃−𝑘_𝑠𝜃ΔΔ𝑦_{𝑘𝑠𝜃Δ}−𝑦_{(𝑘𝑠𝜃−1)Δ},𝑟^Δ_𝑘𝑠𝜃−𝑢𝑦_{(𝑘𝑠𝜃−1)Δ},𝑟^Δ_{𝑘𝑠𝜃−1}²+3𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠𝜃≥0𝑘}^𝑠+𝜃_𝑠𝜃Δ𝑓𝑦_𝑟,𝑟(𝑟)𝑑𝑟²+3𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠𝜃≥0𝑘}^𝑠+𝜃_𝑠𝜃Δ𝑔𝑦_𝑟,𝑟(𝑟)𝑑𝑤(𝑟)². (3.36) From Assumption 2.4, (3.24), (3.26), and Lemma 3.2, we have 𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠𝜃≥0}𝑢𝑦_{(𝑘𝑠𝜃−1)Δ}+𝑠+𝜃−𝑘_𝑠𝜃ΔΔ𝑦_{𝑘𝑠𝜃Δ}−𝑦_{(𝑘𝑠𝜃−1)Δ},𝑟^Δ_𝑘𝑠𝜃−𝑢𝑦_{(𝑘𝑠𝜃−1)Δ},𝑟^Δ_{𝑘𝑠𝜃−1}²≤2𝜅²𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠𝜃≥0}𝑠+𝜃−𝑘_𝑠𝜃ΔΔ𝑦_{𝑘𝑠𝜃Δ}−𝑦_{(𝑘𝑠𝜃−1)Δ}²+8𝜅²𝐻(2)≤2𝜅²𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠𝜃≥0}𝑦_{𝑘𝑠𝜃Δ}−𝑦_{(𝑘𝑠𝜃−1)Δ}²+8𝜅²𝐻(2)≤2𝜅²𝔼sup_{0≤𝑘𝑠𝜃≤𝑀−1}𝑦_{𝑘𝑠𝜃Δ}−𝑦_{(𝑘𝑠𝜃−1)Δ}²+8𝜅²𝐻(2)≤2𝜅²𝛾(Δ)+8𝜅²𝐻(2). (3.37)
By the Hölder inequality, Assumption 2.3, and Lemma 3.1, 𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠𝜃≥0𝑘}^𝑠+𝜃_𝑠𝜃Δ𝑓𝑦_𝑟,𝑟(𝑟)𝑑𝑟²≤Δ𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠𝜃≥0𝑘}^𝑠+𝜃_𝑠𝜃Δ𝑓𝑦_𝑟,𝑟(𝑟)²𝑑𝑟≤𝐾Δ𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠𝜃≥0𝑘}^𝑠+𝜃_𝑠𝜃Δ1+𝑦_𝑟²𝑑𝑟≤𝐾Δ𝔼^𝑇₀1+sup_{0≤𝑟≤𝑇}𝑦_𝑟²𝑑𝑟≤𝐾Δ^𝑇₀1+𝔼sup_{−𝜏≤𝑡≤𝑇}𝑦(𝑡)|²𝑑𝑟≤𝐾Δ^𝑇₀[(2[(21+𝐻)]𝑑𝑟≤𝐾𝑇1+𝐻)]Δ. (3.38) Setting 𝑣=𝑠+𝜃 and 𝑘_𝑣=𝑘_𝑠𝜃 and applying the Hölder inequality yield 𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠𝜃≥0𝑘}^𝑠+𝜃_𝑠𝜃Δ𝑔𝑦_𝑟,𝑟(𝑟)𝑑𝑤(𝑟)²=𝔼sup_{0≤𝑣≤𝑇,0≤𝑘𝑣≤𝑀−1}𝑔𝑦_𝑘𝑣Δ,𝑟(𝑣)𝑤(𝑣)−𝑤𝑘_𝑣Δ²≤𝔼sup_{0≤𝑣≤𝑇,0≤𝑘𝑣≤𝑀−1}𝑔𝑦_𝑘𝑣Δ,𝑟(𝑣)^{2𝑙/(𝑙−1)(𝑙−1)/𝑙}×𝔼sup_{0≤𝑣≤𝑇,0≤𝑘𝑣≤𝑀−1}𝑤(𝑣)−𝑤𝑘_𝑣Δ^2𝑙1/𝑙. (3.39) The Doob martingale inequality gives 𝔼sup_{0≤𝑣≤𝑇,0≤𝑘𝑣≤𝑀−1}𝑤(𝑣)−𝑤𝑘_𝑣Δ^2𝑙=𝔼sup_{0≤𝑘𝑣≤𝑀−1}sup_{𝑘𝑣𝑘Δ≤𝑣≤𝑣Δ+1}𝑤(𝑣)−𝑤𝑘_𝑣Δ^2𝑙≤𝔼^𝑀−1_𝑘𝑣=0sup_{𝑘𝑣𝑘Δ≤𝑣≤𝑣Δ+1}𝑤(𝑣)−𝑤𝑘_𝑣Δ^2𝑙=^𝑀−1_𝑘𝑣=0𝔼sup_{𝑘𝑣𝑘Δ≤𝑣≤𝑣Δ+1}𝑤(𝑣)−𝑤𝑘_𝑣Δ^2𝑙≤2𝑙2𝑙−1^{2𝑙𝑀−1}_𝑘𝑣=0𝔼𝑤𝑘_𝑣+1Δ−𝑤𝑘_𝑣Δ^2𝑙. (3.40) By (3.27), we therefore have 𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠𝜃≥0𝑘}^𝑠+𝜃_𝑠𝜃Δ𝑔𝑦_𝑟,𝑟(𝑟)𝑑𝑤(𝑟)²≤2𝑙2𝑙−1²𝔼sup_{0≤𝑘𝑣≤𝑀−1}𝑔𝑦_𝑘𝑣Δ,𝑟(𝑣)^{2𝑙/(𝑙−1)(𝑙−1)/𝑙}×^𝑀−1_𝑘𝑣=0𝔼𝑤𝑘_𝑣+1Δ−𝑤𝑘_𝑣Δ^2𝑙1/𝑙≤2𝑙2𝑙−1²𝐷(𝑙)Δ^{(𝑙−1)/𝑙}. (3.41) Substituting (3.37), (3.38), and (3.41) into (3.36) and noting Δ∈(0,1) give 𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠+𝑘𝜃≥0}𝑦(𝑠+𝜃)−𝑦𝑘_𝑠+𝑘_𝜃Δ²≤6𝜅²𝛾(Δ)+24𝜅²𝐻(2)+3𝐾𝑇(1+𝐻(2))+2𝑙2𝑙−1²𝐷(𝑙)Δ^{(𝑙−1)/𝑙}=∶6𝜅²𝛾(Δ)+24𝜅²𝐻(2)+𝑐₁(𝑙)Δ^{(𝑙−1)/𝑙}. (3.42)Subcase  2 (𝑘_𝑠𝜃−(𝑘_𝑠+𝑘_𝜃)=1). From (2.13), 𝑦(𝑠+𝜃)−𝑦𝑘_𝑠+𝑘_𝜃Δ=𝑦𝑘_𝑠𝜃Δ−𝑦𝑘_𝑠+𝑘_𝜃Δ+𝑦(𝑠+𝜃)−𝑦𝑘_𝑠𝜃Δ≤𝑢𝑦_{(𝑘𝑠+𝑘𝜃)Δ},𝑟^Δ_{𝑘𝑠+𝑘𝜃}−𝑢𝑦_{(𝑘𝑠+𝑘𝜃−1)Δ},𝑟^Δ_{𝑘𝑠+𝑘𝜃−1}+𝑓𝑦_{(𝑘𝑠+𝑘𝜃)Δ},𝑟^Δ_{𝑘𝑠+𝑘𝜃}Δ+𝑔𝑦_{(𝑘𝑠+𝑘𝜃)Δ},𝑟^Δ_{𝑘𝑠+𝑘𝜃}Δ𝑤_{𝑘𝑠+𝑘𝜃}+𝑦(𝑠+𝜃)−𝑦𝑘_𝑠𝜃Δ, (3.43) so we have 𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠+𝑘𝜃≥0}𝑦(𝑠+𝜃)−𝑦𝑘_𝑠+𝑘_𝜃Δ²≤4𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠+𝑘𝜃≥0}𝑢𝑦_{(𝑘𝑠+𝑘𝜃)Δ},𝑟^Δ_{𝑘𝑠+𝑘𝜃}−𝑢𝑦_{(𝑘𝑠+𝑘𝜃−1)Δ},𝑟^Δ_{𝑘𝑠+𝑘𝜃−1}²+𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠+𝑘𝜃≥0}𝑓𝑦_{(𝑘𝑠+𝑘𝜃)Δ},𝑟^Δ_{𝑘𝑠+𝑘𝜃}Δ²+𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠+𝑘𝜃≥0}𝑔𝑦_{(𝑘𝑠+𝑘𝜃)Δ},𝑟^Δ_{𝑘𝑠+𝑘𝜃}Δ𝑤_{𝑘𝑠+𝑘𝜃}²+𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠+𝑘𝜃≥0}𝑦(𝑠+𝜃)−𝑦𝑘_𝑠𝜃Δ². (3.44) Since 𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠+𝑘𝜃≥0}𝑢𝑦_{(𝑘𝑠+𝑘𝜃)Δ},𝑟^Δ_{𝑘𝑠+𝑘𝜃}−𝑢𝑦_{(𝑘𝑠+𝑘𝜃−1)Δ},𝑟^Δ_{𝑘𝑠+𝑘𝜃−1}²≤2𝜅²𝛾(Δ)+8𝜅²𝐻(2), (3.45) from (3.26), (3.27), and the Subcase 1, noting Δ∈(0,1), we have 𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠+𝑘𝜃≥0}𝑦(𝑠+𝜃)−𝑦𝑘_𝑠+𝑘_𝜃Δ²≤42𝜅²𝛾(Δ)+8𝜅²𝐻(2)+𝐾[1+𝐻(2)]Δ²+𝐷(𝑙)Δ^{(𝑙−1)/𝑙}+32𝜅²𝛾(Δ)+8𝜅²𝐻(2)(Δ)+𝑐₁(𝑙)Δ^{(𝑙−1)/𝑙}=∶32𝜅²𝛾(Δ)+128𝜅²𝐻(2)+𝑐₂(𝑙)Δ^{(𝑙−1)/𝑙}. (3.46)Subcase  3 (𝑘_𝑠𝜃−(𝑘_𝑠+𝑘_𝜃)=2). From (2.13), we have 𝑦(𝑠+𝜃)−𝑦𝑘_𝑠+𝑘_𝜃Δ=𝑦(𝑠+𝜃)−𝑦𝑘_𝑠𝜃−1Δ+𝑦𝑘_𝑠𝜃−1Δ−𝑦𝑘_𝑠+𝑘_𝜃Δ, (3.47) so from the Subcase 2, we have 𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠+𝑘𝜃≥0}𝑦(𝑠+𝜃)−𝑦𝑘_𝑠+𝑘_𝜃Δ²≤2𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠+𝑘𝜃≥0}𝑦(𝑠+𝜃)−𝑦𝑘_𝑠𝜃−1Δ²+2𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠+𝑘𝜃≥0}𝑦𝑘_𝑠𝜃−1Δ−𝑦𝑘_𝑠+𝑘_𝜃Δ²≤232𝜅²𝛾(Δ)+128𝜅²𝐻(2)+𝑐₂(𝑙)Δ^{(𝑙−1)/𝑙}+22𝜅²𝛾(Δ)+8𝜅²𝐻(2)+𝐾[1+𝐻(2)]Δ²+𝐷(𝑙)Δ^{(𝑙−1)/𝑙}=∶68𝜅²𝛾(Δ)+272𝜅²𝐻(2)+𝑐₃(𝑙)Δ^{(𝑙−1)/𝑙}. (3.48) From these three subcases, we have 𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇,𝑘𝑠+𝑘𝜃≥0}𝑦(𝑠+𝜃)−𝑦𝑘_𝑠+𝑘_𝜃Δ²≤106𝜅²𝛾(Δ)+424𝜅²𝐻(2)+𝑐₁(𝑙)+𝑐₂(𝑙)+𝑐₃(𝑙)Δ^{(𝑙−1)/𝑙}. (3.49)
Case  2  (𝑘_𝑠+𝑘_𝜃=−1 and 0≤𝑠+𝜃≤Δ). In this case, applying Assumption 2.5 and case 1, we have 𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇}𝑦(𝑠+𝜃)−𝑦𝑘_𝑠+𝑘_𝜃Δ²≤2𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇}𝑦(𝑠+𝜃)−𝑦(0)²+2𝐸𝑦(0)−𝑦(−Δ)²≤212𝜅²𝛾(Δ)+848𝜅²𝐻(2)+2𝑐₁(𝑙)+𝑐₂(𝑙)+𝑐₃(𝑙)Δ^{(𝑙−1)/𝑙}+2𝛼(Δ). (3.50)
Case 3 (𝑘_𝑠+𝑘_𝜃=−1 and −Δ≤𝑠+𝜃≤0). In this case, using Assumption 2.5, 𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇}𝑦(𝑠+𝜃)−𝑦𝑘_𝑠+𝑘_𝜃Δ²≤𝛼(Δ). (3.51)
Case  4 (𝑘_𝑠+𝑘_𝜃≤−2). In this case, 𝑠+𝜃≤0, so using Assumption 2.5, 𝔼sup_{−𝜏≤𝜃≤0,0≤𝑠≤𝑇}𝑦(𝑠+𝜃)−𝑦𝑘_𝑠+𝑘_𝜃Δ²≤𝛼(2Δ). (3.52) Substituting these four cases into (3.34) and noting the expression of 𝛾(Δ), there exist 𝑐₂, 𝑐^′₂, and 𝑐₂(𝑙) such that 𝔼sup_{0≤𝑠≤𝑇}𝑦_𝑠−𝑦_𝑠²≤𝑐₂𝛼(2Δ)+𝑐^′₂+𝑐₂(𝑙)Δ^{(𝑙−1)/𝑙}, (3.53) This proof is complete.

Remark 3.4. It should be pointed out that much simpler proofs of Lemmas 3.2 and 3.3 can be obtained by choosing 𝑙=2 if we only want to prove Theorem 2.7. However, here we choose 𝑙>1 to control the stochastic terms 𝛽(Δ) and 𝛾(Δ) by Δ^{(𝑙−1)/𝑙} in Section 3, which will be used to show the order of the strong convergence.

Lemma 3.5. If Assumption 2.4 holds, 𝔼sup_{0≤𝑡≤𝑇}𝑢𝑦_𝑡,𝑟(𝑡)−𝑢𝑦_𝑡,𝑟(𝑡)²≤8𝜅²𝐻(2)𝐿Δ∶=𝜁Δ (3.54) where 𝐿 is a positive constant independent of Δ.

Proof. Let 𝑛=[𝑇/Δ], the integer part of 𝑇/Δ, and 1_𝐺 be the indication function of the set 𝐺. Then, by Assumption 2.4 and Lemma 3.1, we obtain 𝔼sup_{0≤𝑡≤𝑇}𝑢𝑦_𝑡,𝑟(𝑡)−𝑢𝑦_𝑡,𝑟(𝑡)²≤max_{0≤𝑘≤𝑛}𝔼sup_{𝑡𝑠∈𝑘,𝑡𝑘+1}𝑢𝑦_𝑠,𝑟(𝑠)−𝑢𝑦_𝑠,𝑟(𝑠)²≤2max_{0≤𝑘≤𝑛}𝔼sup_{𝑡𝑠∈𝑘,𝑡𝑘+1}𝑢𝑦_𝑠,𝑟(𝑠)−𝑢𝑦_𝑠,𝑟(𝑠)²1_{{𝑟(𝑠)≠𝑟(𝑡𝑘)}}≤4max_{0≤𝑘≤𝑛}𝔼sup_{𝑡𝑠∈𝑘,𝑡𝑘+1}|𝑢(𝑦_𝑠,𝑟(𝑠))|²+𝑢𝑦_𝑠,𝑟(𝑠)²1_{{𝑟(𝑠)≠𝑟(𝑡𝑘)}}≤8𝜅²max_{0≤𝑘≤𝑛}𝔼sup_{0≤𝑡≤𝑇}𝑦_𝑡²𝔼1_{{𝑟(𝑠)≠𝑟(𝑡𝑘)}}≤8𝜅²𝐻(2)𝔼1_{{𝑟(𝑠)≠𝑟(𝑡𝑘)}}=8𝜅²𝐻(2)𝔼𝔼1_{{𝑟(𝑠)≠𝑟(𝑡𝑘)}}∣𝑡_𝑘, (3.55) where in the last step we use the fact that 𝑦_𝑡𝑘 and 1_{{𝑟(𝑠)≠𝑟(𝑡𝑘)}} are conditionally independent with respect to the 𝜎− algebra generated by 𝑟(𝑡_𝑘). But, by the Markov property, 𝔼1_{{𝑟(𝑠)≠𝑟(𝑡𝑘)}}∣𝑟𝑡_𝑘=_𝑖∈𝑆1_{{𝑟(𝑡𝑘)=𝑖}}𝑃𝑟(𝑠)≠𝑖∣𝑟𝑡_𝑘=𝑖=_𝑖∈𝑆1_{{𝑟(𝑡𝑘)=𝑖}𝑗≠𝑖}𝛾_𝑖𝑗𝑠−𝑡_𝑘+𝑜𝑠−𝑡_𝑘≤_𝑖∈𝑆1_{{𝑟(𝑡𝑘)=𝑖}}max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑗Δ+𝑜(Δ)≤𝐿Δ, (3.56) where 𝐿 is a positive constant independent of Δ. Then 𝔼sup_{0≤𝑡≤𝑇}𝑢𝑦_𝑡,𝑟(𝑡)−𝑢𝑦_𝑡,𝑟(𝑡)²≤8𝜅²𝐻(2)𝐿Δ. (3.57) This proof is complete.

Lemma 3.6. If Assumption 2.3 holds, there is a constant 𝐶, which is independent of Δ such that 𝔼^𝑇₀𝑓𝑦_𝑠,𝑟(𝑠)−𝑓𝑦_𝑠,𝑟(𝑠)²𝔼𝑑𝑠≤𝐶Δ,_𝑇0𝑔𝑦_𝑠,𝑟(𝑠)−𝑔𝑦_𝑠,𝑟(𝑠)²𝑑𝑠≤𝐶Δ. (3.58)(3.59)

Proof. Let 𝑛=[𝑇/Δ], the integer part of 𝑇/Δ. Then 𝔼^𝑇₀𝑓𝑦_𝑠,𝑟(𝑠)−𝑓𝑦_𝑠,𝑟(𝑠)²𝑑𝑠=^𝑛_𝑘=0𝔼^𝑡𝑘+1_𝑡𝑘𝑓𝑦_𝑡𝑘,𝑟(𝑠)−𝑓𝑦_𝑡𝑘,𝑟𝑡_𝑘²𝑑𝑠, (3.60) with 𝑡_𝑛+1 being 𝑇. Let 1_𝐺 be the indication function of the set 𝐺. Moreover, in what follows, 𝐶 is a generic positive constant independent of Δ, whose values may vary from line to line. With these notations we derive, using Assumption 2.3, that 𝔼^𝑡𝑘+1_𝑡𝑘𝑓𝑦_𝑡𝑘,𝑟(𝑠)−𝑓𝑦_𝑡𝑘,𝑟𝑡_𝑘²𝑑𝑠≤2𝔼^𝑡𝑘+1_𝑡𝑘|𝑓(𝑦_𝑡𝑘,𝑟(𝑠))|²+𝑓𝑦_𝑡𝑘,𝑟𝑡_𝑘²1_{{𝑟(𝑠)≠𝑟(𝑡𝑘)}}𝑑𝑠≤𝐶𝔼^𝑡𝑘+1_𝑡𝑘1+𝑦_𝑡𝑘²1_{{𝑟(𝑠)≠𝑟(𝑡𝑘)}}𝑑𝑠≤𝐶^𝑡𝑘+1_𝑡𝑘𝔼𝔼1+𝑦_𝑡𝑘²1_{{𝑟(𝑠)≠𝑟(𝑡𝑘)}}∣𝑟𝑡_𝑘𝑑𝑠≤𝐶^𝑡𝑘+1_𝑡𝑘𝔼𝔼1+𝑦_𝑡𝑘²∣𝑟𝑡_𝑘𝔼1_{{𝑟(𝑠)≠𝑟(𝑡𝑘)}}∣𝑟𝑡_𝑘𝑑𝑠, (3.61) where in the last step we use the fact that 𝑦_𝑡𝑘 and 1_{{𝑟(𝑠)≠𝑟(𝑡𝑘)}} are conditionally independent with respect to the 𝜎− algebra generated by 𝑟(𝑡_𝑘). But, by the Markov property, 𝔼1_{{𝑟(𝑠)≠𝑟(𝑡𝑘)}}∣𝑟𝑡_𝑘=_𝑖∈𝑆1_{{𝑟(𝑡𝑘)=𝑖}}𝑃𝑟(𝑠)≠𝑖∣𝑟𝑡_𝑘=𝑖=_𝑖∈𝑆1_{{𝑟(𝑡𝑘)=𝑖}𝑗≠𝑖}𝛾_𝑖𝑗𝑠−𝑡_𝑘+𝑜𝑠−𝑡_𝑘≤_𝑖∈𝑆1_{{𝑟(𝑡𝑘)=𝑖}}max_{1≤𝑖≤𝑁}−𝛾_𝑖𝑗Δ+𝑜(Δ)≤𝐶Δ. (3.62) So, by Lemma 3.1, 𝔼^𝑡𝑘+1_𝑡𝑘𝑓𝑦_𝑡𝑘,𝑟(𝑠)−𝑓𝑦_𝑡𝑘,𝑟𝑡_𝑘²𝑑𝑠≤𝐶Δ^𝑡𝑘+1_𝑡𝑘1+𝔼𝑦_𝑡𝑘²𝑑𝑠≤𝐶Δ². (3.63) Therefore 𝔼^𝑇₀𝑓𝑦_𝑠,𝑟(𝑠)−𝑓𝑦_𝑠,𝑟(𝑠)²𝑑𝑠≤𝐶Δ. (3.64) Similarly, we can show (3.59).

4. Convergence of the EM Methods

In this section we will use the lemmas above to prove Theorem 2.7. From Lemma 3.1 and Theorem 2.6, there exists a positive constant 𝐻 such that 𝔼sup_{−𝜏≤𝑡≤𝑇}|𝑥(𝑡)|^𝑝∨𝔼sup_{−𝜏≤𝑡≤𝑇}𝑦(𝑡)^𝑝≤𝐻. (4.1) Let 𝑗 be a sufficient large integer. Define the stopping times 𝑢_𝑗∶=inf𝑡≥0∶𝑥_𝑡≥𝑗,𝑣_𝑗∶=inf𝑡≥0∶𝑦_𝑡≥𝑗,𝜌_𝑗∶=𝑢_𝑗∧𝑣_𝑗, (4.2) where we set inf∅=∞ as usual. Let 𝑒(𝑡)∶=𝑥(𝑡)−𝑦(𝑡). (4.3) Obviously, 𝔼sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²=𝔼sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²1_{{𝑢𝑗>𝑇,𝑣𝑗>𝑇}}+𝔼sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²1_{{𝑢𝑗≤𝑇or𝑣𝑗≤𝑇}}. (4.4) Recall the following elementary inequality: 𝑎^𝛾𝑏^1−𝛾≤𝛾𝑎+1−𝛾𝑏,∀𝑎,𝑏>0,𝛾∈[0,1]. (4.5) We thus have, for any 𝛿>0, 𝔼sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²1_{{𝑢𝑗≤𝑇or𝑣𝑗≤𝑇}}≤𝔼𝛿sup_{{}0≤𝑡≤𝑇}|𝑒(𝑡)|^𝑝2/𝑝𝛿^{−2/(𝑝−2)}1_{{𝑢𝑗≤𝑇or𝑣𝑗≤𝑇}}^{(𝑝−2)/𝑝}≤2𝛿𝑝𝔼sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|^𝑝+𝑝−2𝑝𝛿^2/(𝑝−2)ℙ𝑢_𝑗≤𝑇or𝑣_𝑗≤𝑇. (4.6) Hence 𝔼sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²≤𝔼sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²1_{{𝜌𝑗>𝑇}}+2𝛿𝑝𝔼sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|^𝑝+𝑝−2𝑝𝛿^2/(𝑝−2)ℙ𝑢_𝑗≤𝑇or𝑣_𝑗≤𝑇. (4.7) Now, ℙ𝑢_𝑗≤𝑇≤𝔼1_{{𝑢𝑗≤𝑇}}𝑥_𝑡^𝑝𝑗^𝑝≤1𝑗^𝑝𝔼sup_{−𝜏≤𝑡≤𝑇}|𝑥(𝑡)|^𝑝≤𝐻𝑗^𝑝. (4.8) Similarly, ℙ𝑣_𝑗≤𝐻≤𝑇𝑗^𝑝. (4.9) Thus ℙ𝑣_𝑗≤𝑇or𝑢_𝑗≤𝑇≤ℙ𝑣_𝑗≤𝑇+ℙ𝑢_𝑗≤2𝐻≤𝑇𝑗^𝑝. (4.10) We also have 𝔼sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|^𝑝≤2^𝑝−1𝔼sup_{0≤𝑡≤𝑇}𝑥_𝑡^𝑝+𝑦_𝑡^𝑝≤2^𝑝𝐻. (4.11) Moreover, 𝔼sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²1_{{𝜌𝑗>𝑇}}=𝔼sup_{0≤𝑡≤𝑇}𝑒𝑡∧𝜌_𝑗²1_{{𝜌𝑗>𝑇}}≤𝔼sup_{0≤𝑡≤𝑇}𝑒𝑡∧𝜌_𝑗². (4.12) Using these bounds in (4.7) yields 𝔼sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²≤𝔼sup_{0≤𝑡≤𝑇}𝑒𝑡∧𝜌_𝑗²+2^𝑝+1𝛿𝐻𝑝+2𝑝−2𝐻𝑝𝛿^2/(𝑝−2)𝑗^𝑝. (4.13) Setting 𝑣∶=𝑡∧𝜌_𝑗 and for any 𝜀∈(0,1), by the Hölder inequality, when 𝑣∈[𝑘Δ,(𝑘+1)Δ], for 𝑘=0,1,2,…,𝑀−1, |𝑒(𝑣)|²=𝑥(𝑣)−𝑦(𝑣)²≤𝑢𝑥_𝑣,𝑟(𝑣)−𝑢𝑦_(𝑘−1)Δ+(𝑣−𝑘Δ)Δ𝑦_𝑘Δ−𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘+^𝑣₀𝑓𝑥_𝑠,𝑟(𝑠)−𝑓𝑦_𝑠,𝑟(𝑠)𝑑𝑠+^𝑣₀𝑔𝑥_𝑠,𝑟(𝑠)−𝑔𝑦_𝑠,𝑟(𝑠)𝑑𝑤(𝑠)²≤1𝜀𝑢𝑥_𝑣,𝑟(𝑣)−𝑢𝑦_(𝑘−1)Δ+(𝑣−𝑘Δ)Δ𝑦_𝑘Δ−𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘²+2𝑇1−𝜀^𝑣₀𝑓𝑥_𝑠,𝑟(𝑠)−𝑓𝑦_𝑠,𝑟(𝑠)²𝑑𝑠+^𝑣₀𝑔𝑥_𝑠,𝑟(𝑠)−𝑔𝑦_𝑠,𝑟(𝑠)𝑑𝑤(𝑠)². (4.14) Assumption 2.4 yields 𝑢𝑥_𝑣,𝑟(𝑣)−𝑢𝑦_(𝑘−1)Δ+𝑣−𝑘ΔΔ𝑦_𝑘Δ−𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘²≤2𝜅²𝑥_𝑣−𝑦_(𝑘−1)Δ−𝑣−𝑘ΔΔ𝑦_𝑘Δ−𝑦_(𝑘−1)Δ²+2𝑢𝑦_(𝑘−1)Δ+𝑣−𝑘ΔΔ𝑦_𝑘Δ−𝑦_(𝑘−1)Δ,𝑟(𝑣)−𝑢𝑦_(𝑘−1)Δ+𝑣−𝑘ΔΔ𝑦_𝑘Δ−𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘²≤2𝜅²𝑥_𝑣−𝑦_𝑣+𝑦_𝑣−𝑦_𝑣+𝑦_𝑘Δ−𝑦_(𝑘−1)Δ−𝑣−𝑘ΔΔ𝑦_𝑘Δ−𝑦_(𝑘−1)Δ²+2𝑢𝑦_(𝑘−1)Δ+𝑣−𝑘ΔΔ𝑦_𝑘Δ−𝑦_(𝑘−1)Δ,𝑟(𝑣)−𝑢𝑦_(𝑘−1)Δ+𝑣−𝑘ΔΔ𝑦_𝑘Δ−𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘²≤2𝜅²𝜀𝑥_𝑣−𝑦_𝑣²+4𝜅²𝑦1−𝜀^𝑣−𝑦_𝑣²+𝑦_𝑘Δ−𝑦_(𝑘−1)Δ²+2𝑢𝑦_(𝑘−1)Δ+𝑣−𝑘ΔΔ𝑦_𝑘Δ−𝑦_(𝑘−1)Δ,𝑟(𝑣)−𝑢(𝑦_(𝑘−1)Δ+𝑣−𝑘ΔΔ(𝑦_𝑘Δ−𝑦_(𝑘−1)Δ),𝑟^Δ_𝑘)². (4.15) Then, we have |𝑒(𝑣)|²≤2𝜅²𝜀²𝑥_𝑣−𝑦_𝑣²+4𝜅²𝜀()𝑦1−𝜀^𝑣−𝑦_𝑣²+𝑦_𝑘Δ−𝑦_(𝑘−1)Δ²+2𝑢𝑦_(𝑘−1)Δ+𝑣−𝑘ΔΔ𝑦_𝑘Δ−𝑦_(𝑘−1)Δ,𝑟(𝑣)−𝑢𝑦_(𝑘−1)Δ+𝑣−𝑘ΔΔ𝑦_𝑘Δ−𝑦_(𝑘−1)Δ,𝑟^Δ_𝑘²+2𝑇1−𝜀^𝑣₀𝑓𝑥_𝑠,𝑟(𝑠)−𝑓𝑦_𝑠,𝑟(𝑠)²𝑑𝑠+^𝑣₀𝑔𝑥_𝑠,𝑟(𝑠)−𝑔𝑦_𝑠,𝑟(𝑠)𝑑𝑤(𝑠)². (4.16) Hence, for any 𝑡₁∈[0,𝑇], by Lemmas 3.2–3.5, 𝔼sup_{0≤𝑡≤𝑡1}𝑒𝑡∧𝜌_𝑗²≤2𝜅²𝜀²𝔼sup_{0≤𝑡≤𝑡1}𝑥_{𝑡∧𝜌𝑗}−𝑦_{𝑡∧𝜌𝑗}²+4𝜅²𝜀()𝔼1−𝜀sup_{0≤𝑡≤𝑡1}𝑦_{𝑡∧𝜌𝑗}−𝑦_{𝑡∧𝜌𝑗}²+𝔼sup_{0≤𝑘≤𝑀−1}𝑦_{𝑘Δ∧𝜌𝑗}−𝑦_{(𝑘−1)Δ∧𝜌𝑗}²+2𝔼sup_{0≤𝑘≤𝑀−1}𝑢𝑦_(𝑘−1)Δ+𝑘Δ∧𝜌_𝑗−𝑘ΔΔ𝑦_𝑘Δ−𝑦_(𝑘−1)Δ,𝑟𝑘Δ∧𝜌_𝑗−𝑢𝑦_(𝑘−1)Δ+𝑘Δ∧𝜌_𝑗−𝑘ΔΔ𝑦_𝑘Δ−𝑦_(𝑘−1)Δ,𝑟𝑘Δ∧𝜌_𝑗²+2𝑇𝔼1−𝜀^{𝑡1∧𝜌𝑗}₀𝑓𝑥_𝑠,𝑟(𝑠)−𝑓𝑦_𝑠,𝑟(𝑠)²+2𝑑𝑠𝔼1−𝜀sup_{0≤𝑡≤𝑡1}^{𝑡∧𝜌𝑗}₀𝑔𝑥_𝑠,𝑟(𝑠)−𝑔𝑦_𝑠,𝑟(𝑠)𝑑𝑤(𝑠)²≤2𝜅²𝜀²𝔼sup_{0≤𝑡≤𝑡1}𝑥_{𝑡∧𝜌𝑗}−𝑦_{𝑡∧𝜌𝑗}²+4𝜅²𝜀()𝛾(Δ)(Δ)+1−𝜀+𝛽+2𝜁Δ2𝑇𝔼1−𝜀^{𝑡1∧𝜌𝑗}₀𝑓𝑥_𝑠,𝑟(𝑠)−𝑓𝑦_𝑠,𝑟(𝑠)²+2𝑑𝑠𝔼1−𝜀sup_{0≤𝑡≤𝑡1}^{𝑡∧𝜌𝑗}₀𝑔𝑥_𝑠,𝑟(𝑠)−𝑔𝑦_𝑠,𝑟(𝑠)𝑑𝑤(𝑠)². (4.17) Since 𝑥(𝑡)=𝑦(𝑡)=𝜉(𝑡) when 𝑡∈[−𝜏,0], we have 𝔼sup_{0≤𝑡≤𝑡1}𝑥_{𝑡∧𝜌𝑗}−𝑦_{𝑡∧𝜌𝑗}²≤𝔼sup_{−𝜏≤𝜃≤0}sup_{0≤𝑡≤𝑡1}𝑥𝑡∧𝜌_𝑗+𝜃−𝑦𝑡∧𝜌_𝑗+𝜃²≤𝔼sup_{−𝜏≤𝑡≤𝑡1}𝑥𝑡∧𝜌_𝑗−𝑦𝑡∧𝜌_𝑗²=𝔼sup_{0≤𝑡≤𝑡1}𝑥𝑡∧𝜌_𝑗−𝑦𝑡∧𝜌_𝑗². (4.18) By Assumption 2.2, Lemma 3.3, and Lemma 3.6, we may compute 𝔼^{𝑡1∧𝜌𝑗}₀𝑓𝑥_𝑠,𝑟(𝑠)−𝑓𝑦_𝑠,𝑟(𝑠)²𝑑𝑠≤𝔼^{𝑡1∧𝜌𝑗}₀𝑓𝑥_𝑠,𝑟(𝑠)−𝑓𝑦_𝑠,𝑟(𝑠)+𝑓𝑦_𝑠,𝑟(𝑠)−𝑓𝑦_𝑠,𝑟(𝑠)²𝑑𝑠≤2𝐶_𝑗𝔼^{𝑡1∧𝜌𝑗}₀𝑥_𝑠−𝑦_𝑠+𝑦_𝑠−𝑦_𝑠²𝑑𝑠+2𝐶Δ≤4𝐶_𝑗𝔼^{𝑡1∧𝜌𝑗}₀𝑥_𝑠−𝑦_𝑠²𝑑𝑠+4𝐶_𝑗𝔼^{𝑡1∧𝜌𝑗}₀𝑦_𝑠−𝑦_𝑠²𝑑𝑠+2𝐶Δ≤4𝐶_𝑗𝔼^{𝑡1∧𝜌𝑗}₀sup_{−𝜏≤𝜃≤0}𝑥(𝑠+𝜃)−𝑦(𝑠+𝜃)²𝑑𝑠+4𝐶_𝑗𝑇𝛽(Δ)+2𝐶Δ≤4𝐶_𝑗𝔼^𝑡1₀sup_{−𝜏≤𝜃≤0}𝑥𝑠∧𝜌_𝑗+𝜃−𝑦𝑠∧𝜌_𝑗+𝜃²𝑑𝑠+4𝐶_𝑗𝑇𝛽(Δ)+2𝐶Δ≤4𝐶_𝑗𝔼^𝑡1₀sup_{−𝜏≤𝑟≤𝑠}𝑥𝑟∧𝜌_𝑗−𝑦𝑟∧𝜌_𝑗²𝑑𝑠+4𝐶_𝑗𝑇𝛽(Δ)+2𝐶Δ=4𝐶_𝑗^𝑡1₀𝔼sup_{0≤𝑟≤𝑠}𝑥𝑟∧𝜌_𝑗−𝑦𝑟∧𝜌_𝑗²𝑑𝑠+4𝐶_𝑗𝑇𝛽(Δ)+2𝐶Δ. (4.19) By the Doob martingale inequality, Lemma 3.3, Lemma 3.6, and Assumption 2.2, we compute 𝔼sup_{0≤𝑡≤𝑡1}^{𝑡∧𝜌𝑗}₀𝑔𝑥_𝑠,𝑟(𝑠)−𝑔𝑦_𝑠,𝑟(𝑠)𝑑𝑤(𝑠)²=𝔼sup_{0≤𝑡≤𝑡1}^{𝑡∧𝜌𝑗}₀𝑔𝑥_𝑠,𝑟(𝑠)−𝑔𝑦_𝑠,𝑟(𝑠)+𝑔𝑦_𝑠,𝑟(𝑠)−𝑔𝑦_𝑠,𝑟(𝑠)𝑑𝑤(𝑠)²≤8𝐶_𝑗𝔼^{𝑡1∧𝜌𝑗}₀𝑥_𝑠−𝑦_𝑠²𝑑𝑠+8𝐶Δ≤16𝐶_𝑗^𝑡1₀𝔼sup_{0≤𝑟≤𝑠}𝑥𝑟∧𝜌_𝑗−𝑦𝑟∧𝜌_𝑗²𝑑𝑠+16𝐶_𝑗𝑇𝛽(Δ)+8𝐶Δ. (4.20) Therefore, (4.17) can be written as 1−2𝜅²𝜀²𝔼sup_{0≤𝑡≤𝑡1}𝑒𝑡∧𝜌_𝑗²≤4𝜅²𝜀()𝛽(Δ)(Δ+1−𝜀+𝛾)8𝐶_𝑗𝑇(𝑇+4)𝛽(Δ)+()1−𝜀4𝐶Δ𝑇+81−𝜀+2𝜁Δ+8𝐶_𝑗(𝑇+4)1−𝜀^𝑡1₀𝔼sup_{0≤𝑣≤𝑠}𝑒𝑣∧𝜌_𝑗²𝑑𝑠. (4.21) Choosing 𝜀=(1+√2𝜅)/2 and noting 𝜅∈(0,1), we have 𝔼sup_{0≤𝑡≤𝑡1}𝑒𝑡∧𝜌_𝑗²≤16𝜅²1+√2𝜅1−√2𝜅²1+3√𝛽(Δ)(Δ+2𝜅+𝛾)16𝐶_𝑗𝑇1+√2𝜅²(𝑇+4)1−√2𝜅²1+3√𝛽(Δ)+()√2𝜅8𝐶Δ𝑇+81+2𝜅²1−√2𝜅²1+3√+2√2𝜅1+2𝜅²1−√2𝜅²1+3√+2𝜅𝜁Δ16𝐶_𝑗1+√2𝜅²(𝑇+4)1−√2𝜅²1+3√2𝜅^𝑡10𝔼sup_{0≤𝑠≤𝑣}𝑒𝑠∧𝜌_𝑗²≤𝑑𝑣161−√2𝜅²𝛽(Δ)+𝛾(Δ)+16𝐶_𝑗𝑇(𝑇+4)1−√2𝜅²𝛽(Δ)+8𝐶Δ(𝑇+8)1−√2𝜅²+21−√2𝜅𝜁Δ+16𝐶_𝑗(𝑇+4)1−√2𝜅^2𝑡10𝔼sup_{0≤𝑠≤𝑣}𝑒𝑠∧𝜌_𝑗²𝑑𝑣. (4.22) By the Gronwall inequality, we have 𝔼sup_{0≤𝑡≤𝑡1}𝑒𝑡∧𝜌_𝑗²≤16(1−√2𝜅)²𝛽(Δ)+𝛾(Δ)+16𝐶_𝑗𝑇(𝑇+4)1−√2𝜅²𝛽(Δ)+8𝐶Δ(𝑇+8)1−√2𝜅²+2𝜁Δ1−√2𝜅×𝑒^{√(16/(1−2𝜅)2)𝐶𝑗𝑇(𝑇+4)}. (4.23) By (4.13), 𝔼sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²≤161−√2𝜅²𝛽(Δ)+𝛾(Δ)+16𝐶_𝑗𝑇(𝑇+4)1−√2𝜅²𝛽(Δ)+8𝐶Δ(𝑇+8)1−√2𝜅²+2𝜁Δ1−√2𝜅×𝑒^{√(16/(1−2𝜅)2)𝐶𝑗𝑇(𝑇+4)}+2^𝑝+1𝛿𝐻𝑝+2𝑝−2𝐻𝑝𝛿^2/(𝑝−2)𝑗^𝑝. (4.24) Given any 𝜖>0, we can now choose 𝛿 sufficient small such that 2^𝑝+1𝛿𝐻/𝑝≤𝜖/3, then choose 𝑗 sufficient large such that 2𝑝−2𝐻𝑝𝛿^2/(𝑝−2)𝑗^𝑝<𝜖3, (4.25) and finally choose Δ so small such that 161−√2𝜅²𝛽(Δ)+𝛾(Δ)+16𝐶_𝑗𝑇(𝑇+4)1−√2𝜅²𝛽(Δ)+8𝐶Δ(𝑇+8)1−√2𝜅²+2𝜁Δ1−√2𝜅×𝑒^{√(16/(1−2𝜅)2)𝐶𝑗𝑇(𝑇+4)}<𝜖3 (4.26) and thus 𝔼(sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²)≤𝜖 as required.

Remark 4.1. Obviously, according to Theorem 2.7, for neutral stochastic delay differential equations with Markovian switching [19], we can easily obtain that the numerical solutions converge to the true solutions in mean square under Assumptions 2.2–2.4.

5. Convergence Order of the EM Method

To reveal the convergence order of the EM method, we need the following assumptions.

Assumption 5.1. There exists a constant 𝑄 such that for all 𝜑, 𝜓∈𝐶([−𝜏,0];ℝ^𝑛),  𝑖∈𝑆, and 𝑡∈[0,𝑇], 𝑓𝜑,𝑖−𝑓𝜓,𝑖²∨𝑔𝜑,𝑖−𝑔𝜓,𝑖²≤𝑄𝜑−𝜓². (5.1)

It is easy to see from the global Lipschitz condition that, for any 𝜑∈𝐶([−𝜏,0];ℝ^𝑛), 𝑓𝜑,𝑖²∨𝑔𝜓,𝑖²≤2𝑓(0,𝑖)²∨𝑔(0,𝑖)²+𝑄𝜑². (5.2) In other words, the global Lipschitz condition implies linear growth condition with the growth coefficient 𝐾=2𝑓(0,𝑖)²∨𝑔(0,𝑖)²∨𝑄. (5.3)

Assumption 5.2. 𝜉∈𝐿^𝑝_ℱ0([−𝜏,0];ℝ^𝑛) for some 𝑝≥2, and there exists a positive constant 𝜆 such that 𝔼sup_{−𝜏≤𝑠≤𝑡≤0}𝜉(𝑡)−𝜉(𝑠)²≤𝜆(𝑡−𝑠). (5.4)

We can state another theorem, which reveals the order of the convergence.

Theorem 5.3. If Assumptions 5.1, 5.2, and 2.4 hold, for any positive constant 𝑙>1, 𝔼sup_{0≤𝑡≤𝑇}𝑥(𝑡)−𝑦(𝑡)²≤𝑂Δ^1−1/𝑙. (5.5)

Proof. Since 𝛼(Δ) may be replaced by 𝜆Δ, from Lemmas 3.2 and 3.3, there exist constants 𝑐₁̃(𝑙) and 𝑐₂̃(𝑙) such that 𝛽(Δ)≤𝑐₁̃(𝑙)Δ^{(𝑙−1)/𝑙} and 𝛾(Δ)≤𝑐₂̃(𝑙)Δ^{(𝑙−1)/𝑙}. Here we do not need to define the stopping times 𝑢_𝑗 and 𝑣_𝑗, and we may repeat the proof in Section 4 and directly compute 𝔼sup_{0≤𝑡≤𝑇}|𝑒(𝑡)|²≤161−√2𝜅²𝑐₁̃(𝑙)+𝑐₂̃(𝑙)+16𝑄𝑇(𝑇+4)1−√2𝜅²𝑐₁̃(𝑙)+8𝐶(𝑇+8)1−√2𝜅²+2𝜁1−√2𝜅×𝑒^{√(16/(1−2𝜅)2)𝑄𝑇(𝑇+4)}Δ^{(𝑙−1)/𝑙}≤𝑂Δ^{(𝑙−1)/𝑙}. (5.6) The proof is complete.

6. Conclusion

The EM method for neutral stochastic functional differential equations with Markovian switching is studied. The results show that the numerical solution converges to the true solution under the local Lipschitz condition. In addition, the results also show that the order of convergence of the numerical method is close to 1, although the order of the strong convergence in mean square for the EM scheme applied to both SDEs and SFDEs is one [6, 7, 11] under the global Lipschitz condition. Hence, we can control the numerical solution’s error; this method may value some path-dependent options more quickly and simply [25].

Acknowledgments

the work was supported by the Fundamental Research Funds for the Central Universities under Grant 2012089, China Postdoctoral Science Foundation funded project under Grant 2012M511615, the Research Fund for Wuhan Polytechnic University, and the State Key Program of National Natural Science of China (Grant no. 61134012).