Almost Sure and 𝐿^𝑝 Convergence of Split-Step Backward Euler Method for Stochastic Delay Differential Equation

Abstract

The convergence of the split-step backward Euler (SSBE) method applied to stochastic differential equation with variable delay is proven in 𝐿^𝑝-sense. Almost sure convergence is derived from the 𝐿^𝑝 convergence by Chebyshev’s inequality and the Borel-Cantelli lemma; meanwhile, the convergence rate is obtained.

1. Introduction

In probability theory, there are several types of convergence of sequences of random variables such as convergence in 𝑝th mean (𝐿^𝑝 sense), almost sure, in probability, and in distribution. As we know, the almost sure (a.s.) convergence and the convergence in 𝐿^𝑝 sense each imply the convergence in probability, and the convergence in probability implies the convergence in distribution. Among them, the almost sure convergence, also known as convergence with probability one, is the convergence concept most closely related to that of nonrandom sequences. The mean-square convergence analysis of numerical schemes for solving stochastic delay differential equation (SDDE) has gained considerable research attention, and we refer here to the papers of Baker and Buckwar [1], Buckwar [2], Hu et al. [3], Liu et al. [4], and Mao and Sabanis [5] just to mention a few of them. In particular, a type of split-step method for stochastic differential equation (SDE) was first introduced by Higham et al. [6] and, subsequently, the method was extended to solve a linear SDDE with constant delay (see [7]) and to solve an SDDE with variable delay (see [8]). However, the almost sure and 𝐿^𝑝 convergence of a numerical method for an SDDE are rarely investigated in the literature. Until recently, Gyöngy and Sabanis [9] proved the almost sure convergence of Euler approximations for a class of SDDE under local Lipschitz and monotonicity conditions.

In this paper we study the following nonlinear SDDE: d𝑥(𝑡)=𝑓(𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)))d𝑡+𝑔(𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)))d𝑊(𝑡),𝑡≥0 (1) with initial data 𝑥(𝑡)=𝜓(𝑡) for 𝑡∈[−𝜏,0]. Here the time delay 𝜏(𝑡) is a real-valued function satisfying 𝜏(𝑡)≥𝜏_min>0 and −𝜏∶=inf{𝑡−𝜏(𝑡)∶𝑡≥0}. Unlike the delay in [9], however, the function 𝜏(𝑡) will not be limited to an increasing function of  𝑡. The drift function 𝑓 and diffusion function 𝑔 are all continuous, and 𝑓,𝑔∶ℝ×ℝ→ℝ. The SDDE (1) is driven by a scalar Wiener process 𝑊(𝑡).

We can write (1) in the following integral form: 𝑥(𝑡)=𝑥(0)+^𝑡₀𝑓(𝑥(𝑠),𝑥(𝑠−𝜏(𝑠)))d𝑠+^𝑡₀𝑔(𝑥(𝑠),𝑥(𝑠−𝜏(𝑠)))d𝑊(𝑠). (2)

Here we focus mainly on the SSBE method proposed by Wang and Gan [8]. Although the convergence rate in mean-square (𝐿²) sense has been obtained, the convergence analysis in 𝐿^𝑝 sense is more difficult; for example, Jensen's inequality cannot be used directly. The aim of this paper is to obtain the almost sure convergence rate together with 𝐿^𝑝 convergence rate of the SSBE method for SDDE (1). In the next section, we recall the SSBE method and give some assumptions. The main convergence results are given in Section 3.

2. The SSBE Method

First, let us state the numerical method. In the following, we consider a uniform mesh ℐ_𝑁={𝑡₀,𝑡₁,…,𝑡_𝑁}, where the positive integer 𝑁 is given, the time step size ℎ_𝑁=𝑇/𝑁, and 𝑡_𝑛=𝑛ℎ_𝑁 for 0≤𝑛≤𝑁. The numerical approximation of 𝑥(𝑡) at time 𝑡_𝑛 is denoted by 𝑌_𝑛. The SSBE method ([8]) for SDDE (1) can be written as 𝑌^∗_𝑛=𝑌_𝑛+ℎ_𝑁𝑓𝑌^∗_𝑛,𝑍_𝑛,𝑌_𝑛+1=𝑌^∗_𝑛+𝑔𝑌^∗_𝑛,𝑍_𝑛Δ𝑤_𝑛, (3) where Δ𝑤_𝑛∶=𝑤(𝑡_𝑛+1)−𝑤(𝑡_𝑛) and 𝑍_𝑛=𝜓𝑡_𝑛−𝜏𝑡_𝑛,𝑡_𝑛−𝜏𝑡_𝑛<0,𝜇𝑌_{𝑛−𝑞𝑛+1}+1−𝜇𝑌_{𝑛−𝑞𝑛},0≤𝑡_𝑛−𝜏𝑡_𝑛∈𝑡_{𝑛−𝑞𝑛},𝑡_{𝑛−𝑞𝑛+1}, (4) for 0≤𝜇<1 and positive integer 𝑞_𝑛≥1.

To prove the almost sure convergence of the SSBE method (3), we need to define its continuous extension. Let us introduce two step processes as follows: 𝑦^∗(𝑠)∶=^∞_𝑘=0𝟏_{{𝑡𝑘≤𝑠<𝑡𝑘+1}}𝑌^∗_𝑘,𝑧(𝑠)∶=^∞_𝑘=0𝟏_{{𝑡𝑘≤𝑠<𝑡𝑘+1}}𝑍_𝑘, (5) where 𝟏_𝑆 is the indicator function of set 𝑆.

The continuous SSBE approximate solution is then defined by 𝑦(𝑡)𝜓(𝑡)[],∶=𝜓(0)+,𝑡∈−𝜏,0_𝑡0𝑓𝑦^∗(𝑠),𝑧(𝑠)d𝑠+^𝑡₀𝑔𝑦^∗(𝑠),𝑧(𝑠)d𝑊(𝑠),𝑡∈[0,𝑇]. (6) It is not difficult to see that 𝑦(𝑡_𝑘)=𝑌_𝑘 for every 𝑘≥0.

At various points in this paper, we will assume subsets of the following set of conditions [8].(A1)The SDDE (1) has a unique solution 𝑥(𝑡) on [−𝜏,𝑇]. The functions 𝑓(𝑥,𝑦) and 𝑔(𝑥,𝑦) are both locally Lipschitz continuous in 𝑥 and 𝑦; that is, there exists a constant 𝐿_𝐷 such that 𝑓𝑥,𝑦−𝑓𝑥,𝑦∨𝑔𝑥,𝑦−𝑔𝑥,𝑦≤𝐿_𝐷|𝑥−𝑥|+𝑦−𝑦, (7) for all 𝑡≥0 and those 𝑥,𝑦,𝑥,𝑦∈ℝ with |𝑥|∨|𝑦|∨|𝑥|∨|𝑦|≤𝐷. Here, ∨ is the maximal operator.(A2)The function values 𝑓(0,0) and 𝑔(0,0) are bounded. The exact solution 𝑥(𝑡) and its continuous-time approximation solution 𝑦(𝑡) have 𝑝th moment bounds; that is, there exist constants 𝐿_𝐴>0 and integer 𝑝>2 such that Esup_{0≤𝑡≤𝑇}|𝑥(𝑡)|^𝑝∨Esup_{0≤𝑡≤𝑇}𝑦(𝑡)^𝑝≤𝐿_𝐴. (8)(A3)(The Hölder continuity of the initial data) There exist constants 𝐾₁>0 and 𝐾₂>0 such that for all −𝜏≤𝑠<𝑡≤0 and positive integer 𝑝, E𝜓(𝑡)−𝜓(𝑠)^2𝑝≤𝐾₁|𝑡−𝑠|^𝑝, (9) and 𝜏(𝑡) is a continuous function satisfying |𝜏(𝑡)−𝜏(𝑠)|≤𝐾₂|𝑡−𝑠|. (10)

3. The Convergence Analysis

In this section, we first give some lemmas for deriving the main theorem. We define three stopping times [8] as follows: 𝜌_𝐷{|𝑥(𝑡},𝜃=inf𝑡≥0∶)|≥𝐷_𝐷=inf𝑡≥0∶𝑦(𝑡)≥𝐷or𝑦^∗(𝑡)≥𝐷, (11) and 𝜎_𝐷=𝜌_𝐷∧𝜃_𝐷, where ∧ is the minimal operator. Further, the infimum of the empty set is set as ∞.

In what follows, constant 𝐾 is generic, which depends on 𝑓,𝑔, the initial data 𝜓, the interval of integration [0,𝑇], and 𝐷, but it is independent of the discretization parameters 𝑛 and ℎ_𝑁.

Lemma 1. Let 𝑦(𝑡) be the solution of (6). Under assumptions (𝐴1) and (𝐴2), one has E𝟏_{{𝑡≤𝜃𝐷}}𝑦^∗(𝑡)−𝑦𝑡_𝑘^𝑝≤𝐾ℎ^𝑝_𝑁,E𝟏_{{𝑡≤𝜃𝐷}}𝑦(𝑡)−𝑦𝑡_𝑘^𝑝≤𝐾ℎ_𝑁^𝑝/2 (12)(13)
for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1] with 𝑘=0,1,…,𝑁−1, and integer 𝑝>2.

Proof. We note that for |𝑥|∨|𝑦|≤𝐷, 𝑓𝑥,𝑦²≤2𝑓𝑥,𝑦−𝑓(0,0)²+2𝑓(0,0)²≤𝐾1+|𝑥|²+𝑦². (14)
Combining (3), (5), (6), and (14), we obtain E𝟏_{{𝑡≤𝜃𝐷}}𝑦^∗(𝑡)−𝑦𝑡_𝑘^𝑝≤Eℎ_𝑁𝑓𝑌^∗_𝑘,𝑍_𝑘^𝑝≤𝐾ℎ^𝑝_𝑁 (15) under assumptions (A1) and (A2).
Now we prove (13). From (6), we have 𝑦(𝑡)=𝑦𝑡_𝑘+^𝑡_𝑡𝑘𝑓𝑦^∗(𝑠),𝑧(𝑠)d𝑠+^𝑡_𝑡𝑘𝑔𝑦^∗(𝑠),𝑧(𝑠)d𝑊(𝑠), (16) for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1]. To estimate E|𝑦(𝑡)−𝑦(𝑡_𝑘)|^𝑝, we will first apply the elementary inequality, which states that, for every 𝑟>0, it follows that |𝑎+𝑏|^𝑟≤2^𝑟−1∨1|𝑎|^𝑟+|𝑏|^𝑟. (17) Then, for 𝑝>2 and 𝑡_𝑘<𝜃_𝐷, E𝟏_{{𝑡≤𝜃𝐷}}𝑦(𝑡)−𝑦𝑡_𝑘^𝑝≤2^𝑝−1E^{𝑡∧𝜃𝐷}_𝑡𝑘𝑓𝑦^∗(𝑠),𝑧(𝑠)d𝑠^𝑝+^{𝑡∧𝜃𝐷}_𝑡𝑘𝑔𝑦^∗(𝑠),𝑧(𝑠)d𝑊(𝑠)^𝑝. (18)
Applying Hölder’s inequality to the first integral of (18), as well as Burkholder-Davis-Gundy inequality to the Itô integral of (18), we obtain E𝟏_{{𝑡≤𝜃𝐷}}𝑦(𝑡)−𝑦𝑡_𝑘^𝑝≤2^𝑝−1𝑡−𝑡_𝑘^𝑝−1𝑡_𝑡𝑘E𝑓𝑦^∗𝑠∧𝜃_𝐷,𝑧𝑠∧𝜃_𝐷^𝑝d𝑠_𝐽1+𝐾𝑡−𝑡_𝑘^𝑝/2−1×^𝑡_𝑡𝑘E𝑔𝑦^∗𝑠∧𝜃_𝐷,𝑧𝑠∧𝜃_𝐷^𝑝d𝑠_𝐽2. (19)
We then have 𝐽₁=^𝑡_𝑡𝑘E𝑓𝑦^∗𝑠∧𝜃_𝐷,𝑧𝑠∧𝜃_𝐷^2𝑝/2d𝑠≤^𝑡_𝑡𝑘E𝐾_𝐷1+𝑦^∗𝑠∧𝜃_𝐷²+𝑧𝑠∧𝜃_𝐷^2𝑝/2d𝑠≤2^𝑝/2−1𝐾_𝐷^𝑝/2×^𝑡_𝑡𝑘E1+2^𝑝/2−1𝑦^∗𝑠∧𝜃_𝐷^𝑝+𝑧𝑠∧𝜃_𝐷^𝑝d𝑠≤2^𝑝/2−1𝐾_𝐷^𝑝/2×1+2^𝑝/2−1E𝑦^∗𝑠∧𝜃_𝐷^𝑝+2^𝑝/2−1E𝑧𝑠∧𝜃_𝐷^𝑝×𝑡−𝑡_𝑘=𝐾𝑡−𝑡_𝑘 (20) under the assumption (A2).
Similarly, replacing 𝑓 by 𝑔 and repeating the previous procedure, we obtain 𝐽₂≤𝐾(𝑡−𝑡_𝑘). Therefore, by substituting this result and (20) into (19), the proof is complete.

Lemma 2. Let 𝑦(𝑡) be the solution of (6). Under assumptions (𝐴1), (𝐴2), and (𝐴3), one has E𝟏_{{𝑡≤𝜃𝐷}}𝑦(𝑡−𝜏(𝑡))−𝑧(𝑡)^𝑝≤𝐾ℎ_𝑁^𝑝/2 (21) for 𝑡∈[𝑡_𝑛,𝑡_𝑛+1] with 𝑛=0,1,…,𝑁−1, and integer 𝑝>2.

Proof. To show the desired result, let us consider the following four possible cases:(1)𝑡_𝑛−𝜏(𝑡_𝑛)≥0,𝑡−𝜏(𝑡)≥0,(2)𝑡_𝑛−𝜏(𝑡_𝑛)≥0,𝑡−𝜏(𝑡)<0,(3)𝑡_𝑛−𝜏(𝑡_𝑛)<0,𝑡−𝜏(𝑡)≥0,(4)𝑡_𝑛−𝜏(𝑡_𝑛)<0,𝑡−𝜏(𝑡)<0.In case (1), without loss of generality, we assume 𝑡−𝜏(𝑡)∈[𝑡_𝑖,𝑡_𝑖+1), 𝑡_𝑛−𝜏(𝑡_𝑛)=(1−𝜇)𝑡_𝑗+𝜇𝑡_𝑗+1∈[𝑡_𝑗,𝑡_𝑗+1), 𝑖>𝑗≥0. Thus, from (4), (5), and (6), we have 𝑦((𝑡(𝑡)=(𝑡)𝑡−𝜏))−𝑧𝑡−𝜏−𝑡_𝑖𝑓𝑌^∗_𝑖,𝑍_𝑖+𝑔𝑌^∗_𝑖,𝑍_𝑖Δ𝑤_𝑖−1−𝜇^𝑖−1_𝑘=𝑗ℎ_𝑁𝑓𝑌^∗_𝑘+1,𝑍_𝑘+1+𝑔𝑌^∗_𝑘,𝑍_𝑘Δ𝑤_𝑘−𝜇^𝑖−1_𝑘=𝑗+1𝟏_{{𝑖>𝑗+1}}ℎ_𝑁𝑓𝑌^∗_𝑘+1,𝑍_𝑘+1+𝑔𝑌^∗_𝑘,𝑍_𝑘Δ𝑤_𝑘. (22)
We note that EΔ𝑤_𝑙^𝑝=Γ𝑝+1/2√𝜋2ℎ_𝑁^𝑝/2 (23) with 𝑙=0,1,2,…,𝑁−1. Therefore, under assumptions (A1) and (A2), applying (17) and (23), we can derive (21) from (22). Under assumptions (A1)–(A3), the proof of  the three other cases follows in a similar manner that of Lemma 2.5 in [8]. Then combining these results gives the required result.

Lemma 3. Let 𝑥(𝑡) be the solution of  SDDE (1) and let 𝑦(𝑡) be the solution of (6). Under assumptions (𝐴1), (𝐴2), and (𝐴3), one obtains sup_{[]𝑡∈0,𝑇}E𝑦𝑡∧𝜎_𝐷−𝑥𝑡∧𝜎_𝐷^𝑝≤𝐾ℎ_𝑁^𝑝/2 (24) for integer 𝑝>2.

Proof. Suppose that 𝑡∈[𝑡_𝑘,𝑡_𝑘+1] and 𝑡_𝑘≤𝜎_𝐷. From (2) and (6), we have 𝑦(𝑡)−𝑥(𝑡)=𝑦𝑡_𝑘−𝑥𝑡_𝑘+^𝑡_𝑡𝑘𝑓𝑦^∗(𝑠)(𝑠(𝑥(𝑠)((𝑠,𝑧)−𝑓,𝑥𝑠−𝜏)))d𝑠+^𝑡_𝑡𝑘𝑔𝑦^∗(𝑠)(𝑠)(𝑥(𝑠)((𝑠,𝑧−𝑔,𝑥𝑠−𝜏)))d𝑊_𝑠. (25) Let 𝜙(𝑡) denote the difference 𝑦(𝑡)−𝑥(𝑡), by using Itô's formula to the function 𝑢(𝜙)=|𝜙|^𝑝 [10] and then, by taking expectation, we getE𝑦(𝑡∧𝜎_𝐷)−𝑥𝑡∧𝜎_𝐷^𝑝≤E𝑦𝑡_𝑘−𝑥𝑡_𝑘^𝑝+𝑝E^{𝑡∧𝜎𝐷}_𝑡𝑘𝑓𝑦^∗(𝑠),𝑧(𝑠)−𝑓(𝑥(𝑠),𝑥(𝑠−𝜏(𝑠)))𝜙(𝑠)^𝑝−1d𝑠_𝐼1(𝑡)+𝑝𝑝−12E^{𝑡∧𝜎𝐷}_𝑡𝑘𝑔𝑦^∗(𝑠),𝑧(𝑠)−𝑔(𝑥(𝑠),𝑥(𝑠−𝜏(𝑠)))²𝜙(𝑠)^𝑝−2d𝑠_𝐼2(𝑡)+𝑝E^{𝑡∧𝜎𝐷}_𝑡𝑘𝑔𝑦^∗(𝑠),𝑧(𝑠)−𝑔(𝑥(𝑠),𝑥(𝑠−𝜏(𝑠)))𝜙(𝑠)^𝑝−1d𝑊(𝑠)._𝐼3(𝑡) (26)Let 𝑉(𝑡) denote the left side of (26); that is, 𝑉(𝑡)∶=E𝑦𝑡∧𝜎_𝐷−𝑥𝑡∧𝜎_𝐷^𝑝. (27)
As we know, if 𝜚 is a stopping time and if 𝑀_𝑡 is a martingale, then 𝑀_𝑡∧𝜚 is a martingale too. Therefore, the expectation 𝐼₃(𝑡) vanishes because of martingale property for Itô integral; then, the inequality (26) implies that 𝑉(𝑡)≤𝑉𝑡_𝑘+𝑝𝐼₁(𝑡)+𝑝𝑝−12𝐼₂(𝑡) (28) for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1] with 𝑘=0,1,…,𝑁−1.
By applying the local Lipschitz condition and the elementary Young inequality, we obtain 𝐼₁(𝑡)≤E^{𝑡∧𝜎𝐷}_𝑡𝑘𝐿_𝐷𝑦^∗(𝑠)(𝑠+|𝑧(𝑠)((𝑠×𝜙(𝑠−𝑥)−𝑥𝑠−𝜏))|)^𝑝−1d𝑠≤𝐿_𝐷E^{𝑡∧𝜎𝐷}_𝑡𝑘𝑦^∗(𝑠)(𝑠+𝑦(𝑠)(𝑠+𝑧(𝑠)((𝑠+𝑦((𝑠((𝑠×𝜙(𝑠−𝑦)−𝑥)−𝑦𝑠−𝜏))𝑠−𝜏))−𝑥𝑠−𝜏)))^𝑝−1d𝑠≤𝐿_𝐷E^{𝑡∧𝜎𝐷}_𝑡𝑘𝑦^∗(𝑠)−𝑦𝑡_𝑘+𝑦𝑡_𝑘(𝑠+𝑦(𝑠)(𝑠+𝑧(𝑠)((𝑠+𝑦((𝑠((𝑠×𝜙(𝑠−𝑦)−𝑥)−𝑦𝑠−𝜏))𝑠−𝜏))−𝑥𝑠−𝜏)))^𝑝−1d𝑠≤𝐿_𝐷1+4𝑝𝑝−1^𝑡_𝑡𝑘𝑉(𝑠)d1𝑠+𝑝^𝑡_𝑡𝑘𝑉(𝑠−𝜏(𝑠))d𝑠+1𝑝^𝑡_𝑡𝑘E𝑦^∗𝑠∧𝜎_𝐷−𝑦(𝑡_𝑘)^𝑝d𝑠+1𝑝^𝑡_𝑡𝑘E𝑦𝑠∧𝜎_𝐷−𝑦𝑡_𝑘^𝑝d𝑠+1𝑝^𝑡_𝑡𝑘E𝑦𝑠∧𝜎_𝐷−𝜏𝑠∧𝜎_𝐷−𝑧𝑠∧𝜎_𝐷^𝑝d𝑠, (29) which implies that 𝐼₁(𝑡)≤𝐿_𝐷1+4𝑝𝑝−1^𝑡_𝑡𝑘𝑉(𝑠)d𝑠+𝐿_𝐷𝑝^𝑡_𝑡𝑘𝑉(𝑠−𝜏(𝑠))d𝑠+𝐾ℎ_𝑁^𝑝/2+1 (30) by Lemmas 1 and 2.
A similar result can be derived for 𝐼₂(𝑡) so that 𝑉(𝑡)≤𝑉𝑡_𝑘+𝐾ℎ_𝑁^𝑝/2+1+𝛼^𝑡_𝑡𝑘𝑉(𝑠)d𝑠+𝛽^𝑡_𝑡𝑘𝑉(𝑠−𝜏(𝑠))d𝑠 (31) from (28), where 𝛼 and 𝛽 are generic constants independent of ℎ_𝑁.
Now we will proceed by using an induction argument over consecutive intervals of length 𝜏_min up to the end of the interval [0,𝑇].
Step 1. Given 𝑡∈[0,𝜏_min], it is easy to show that 𝑠−𝜏(𝑠)≤𝑡−𝜏_min≤0 for 𝑠≤𝑡, so ^𝑡_𝑡𝑘𝑉(𝑠−𝜏(𝑠))d𝑠=E^{𝑡∧𝜎𝐷}_𝑡𝑘𝜓(𝑠−𝜏(𝑠))−𝜓(𝑠−𝜏(𝑠))^𝑝d𝑠=0. (32) Inserting this into (31) gives 𝑉(𝑡)≤𝑉𝑡_𝑘+𝐾ℎ_𝑁^𝑝/2+1+^𝑡_𝑡𝑘𝛼𝑉(𝑠)d𝑠≤𝑉𝑡_𝑘+𝐾ℎ_𝑁^𝑝/2+1𝑒^{𝛼(𝑡−𝑡𝑘)} (33) by applying the Gronwall inequality [11].
Further, from (33), we have 𝑉𝑡_𝑘≤𝑉𝑡_𝑘−1𝑒^𝛼ℎ𝑁+𝐾ℎ_𝑁^𝑝/2+1𝑒^𝛼ℎ𝑁≤𝑉𝑡₀𝑒^{𝛼ℎ𝑁𝑘}+𝐾ℎ_𝑁^𝑘𝑝/2+1_𝑗=1𝑒^{𝛼ℎ𝑁𝑗}≤𝐾ℎ_𝑁^𝑝/2. (34) Combining (33) and (34) gives 𝑉(𝑡)≤𝐾ℎ_𝑁^𝑝/2 for 𝑡∈[0,𝜏_min].
Step 2. Suppose 𝑡∈[𝑚𝜏_min,(𝑚+1)𝜏_min] with 𝑚=1,2,…, which implies 𝑠−𝜏(𝑠)≤𝑡−𝜏_min≤𝑚𝜏_min for 𝑠≤𝑡; then we can obtain ^𝑡_𝑡𝑘𝑉(𝑠−𝜏(𝑠))d𝑠≤𝐾ℎ_𝑁^𝑝/2+1 (35) from the previous recursive step. Hence, the inequality (31) also gives 𝑉(𝑡)≤𝑉𝑡_𝑘+𝐾ℎ_𝑁^𝑝/2+1+^𝑡_𝑡𝑘𝛼𝑉(𝑠)d𝑠. (36)
Finally, we can obtain the desired result using an approach similar to Step 1.

The next lemma shows that the sequence of the approximate solutions 𝑦(𝑡) tends to the exact solution 𝑥(𝑡) as ℎ_𝑁↓0 in the sense of 𝐿^𝑝-norm under local condition.

Lemma 4. We assume that the conditions of Lemma 3 are fulfilled. Then, for 𝑝>2, one has Esup_{[]𝑡∈0,𝑇}𝑥𝑡∧𝜎_𝐷−𝑦𝑡∧𝜎_𝐷^𝑝≤𝐾ℎ_𝑁^𝑝/2,𝑎𝑠ℎ_𝑁↓0. (37)

Proof. Using (2), (6), (17), the Burkholder-Davis-Gundy inequality, and Hölder's inequality, we have Esup_{[]𝑡∈0,𝑇}𝑥𝑡∧𝜎_𝐷−𝑦𝑡∧𝜎_𝐷^𝑝≤2^𝑝−1Esup_{[]𝑡∈0,𝑇}^{𝑡∧𝜎𝐷}₀𝑓𝑦^∗(𝑠)(𝑠)(𝑥(𝑠)((𝑠,𝑧−𝑓,𝑥𝑠−𝜏)))d𝑠^𝑝+2^𝑝−1Esup_{[]𝑡∈0,𝑇}^{𝑡∧𝜎𝐷}₀𝑔𝑦^∗(𝑠)(𝑠(𝑥(𝑠),𝑥((𝑠,𝑧)−𝑔𝑠−𝜏)))]d𝑊(𝑠)^𝑝≤2^𝑝−1(𝑇∨1)^𝑝−1𝑇₀E𝑓𝑦^∗(𝑠)(𝑠)(𝑥(𝑠)((𝑠,𝑧−𝑓,𝑥𝑠−𝜏)))^𝑝d𝑠+2^𝑝−1(𝑇∨1)^{𝑝/2−1𝑇}₀E𝑔𝑦^∗(𝑠)(𝑠)(𝑥(𝑠)((𝑠,𝑧−𝑔,𝑥𝑠−𝜏)))^𝑝d𝑠. (38)
Combining Lemmas 1, 2, and 3, we get sup_{𝑠∈[0,𝑇]}E𝑦^∗𝑠∧𝜎_𝐷−𝑥𝑠∧𝜎_𝐷^𝑝≤𝐾ℎ_𝑁^𝑝/2,sup_{[]𝑠∈0,𝑇}E𝑧𝑠∧𝜎_𝐷−𝑥𝑠∧𝜎_𝐷−𝜏𝑠∧𝜎_𝐷^𝑝≤𝐾ℎ_𝑁^𝑝/2 (39) by first using Minkowski’s inequality.
Applying the local Lipschitz condition (7) to (38), the relation (37) then follows from (39).

The following two theorems are the main results of this paper. They give, respectively, the 𝐿^𝑝 error and almost sure error of the SSBE method (3).

Theorem 5. One assumes that the conditions of Lemma 3 are fulfilled. Then, for 𝑝>2 and sufficiently large 𝐷, the approximation (3) for SDDE (1) is convergent in 𝐿^𝑝 sense and Esup_{[]𝑡∈0,𝑇}𝑥(𝑡)−𝑦(𝑡)^𝑝≤𝐾ℎ_𝑁^𝑝/2,𝑎𝑠ℎ_𝑁↓0. (40)

The result can be proved via a similar approach to that in [5, Theorem 2.1] and [6, Theorem 2.2]; see the appendix.

Theorem 6. One assumes that the conditions of Lemma 3 are fulfilled. Then, sup_{[]𝑡∈0,𝑇}𝑥(𝑡)−𝑦(𝑡)=𝒪ℎ^𝛾_𝑁(𝑎.𝑠.) (41) for every positive number 𝛾<1/2 and sufficiently large 𝐷.

Proof. From Theorem 5, Chebyshev's inequality [11] gives ^∞_𝑁=1𝑃sup_{𝑡∈[0,𝑇]}𝑥(𝑡)−𝑦(𝑡)≥ℎ^𝛾_𝑁≤^∞_𝑁=1ℎ_𝑁^−𝑝𝛾Esup_{𝑡∈[0,𝑇]}𝑥(𝑡)−𝑦(𝑡)^𝑝≤𝐾^∞_𝑁=1𝑇𝑁^{𝑝/2−𝑝𝛾} (42) for any 𝛾>0 and 𝑝≥3.
The series ∑^∞_𝑁=1(𝑇/𝑁)^{𝑝/2−𝑝𝛾} is convergent for 𝑝/2−𝑝𝛾>1; that is, 𝛾<1/2−1/𝑝<1/2. Further, by applying the Borel-Cantelli lemma [11], the inequality (42) implies that 𝑃lim_𝑁→∞sup_{[]𝑡∈0,𝑇}𝑥(𝑡)−𝑦(𝑡)<ℎ^𝛾_𝑁=1. (43) Thus, under assumptions (A1)–(A3), the approximate solution 𝑦(𝑡) converges (a.s.) to the exact solution 𝑥(𝑡) uniformly on [0,𝑇] as ℎ_𝑁→0.

Appendix

Proof of Theorem 5. Obviously, we have Esup_{[]𝑡∈0,𝑇}𝑥(𝑡)−𝑦(𝑡)^𝑝=Esup_{[]𝑡∈0,𝑇}𝑥(𝑡)−𝑦(𝑡)^𝑝𝟏_{{𝜌𝐷>𝑇and𝜃𝐷>𝑇}}+Esup_{𝑡∈[0,𝑇]}𝑥(𝑡)−𝑦(𝑡)^𝑝𝟏_{{𝜌𝐷≤𝑇or𝜃𝐷≤𝑇}}. (A.1) By the improved Young inequality (see, e.g., [5, 6]) 𝐴𝐴𝐵≤𝜂^𝜇𝜇+1𝜂^(𝜈/𝜇)𝐵^𝜈𝜈∀𝐴,𝐵,𝜂,𝜇>0, (A.2)
when 𝜇⁻¹+𝜈⁻¹=1, we obtain Esup_{[]𝑡∈0,𝑇}𝑥(𝑡)−𝑦(𝑡)^𝑝𝟏_{{𝜌𝐷≤𝑇or𝜃𝐷≤𝑇}}≤E𝜂𝜇sup_{[]𝑡∈0,𝑇}𝑥(𝑡)−𝑦(𝑡)^𝜇𝑝+E1𝜈𝜂^(𝜈/𝜇)𝟏_{{𝜌𝐷≤𝑇or𝜃𝐷≤𝑇}}^𝜈≤𝜂𝜇Esup_{[]𝑡∈0,𝑇}𝑥(𝑡)−𝑦(𝑡)^𝜇𝑝+1𝜈𝜂^(𝜈/𝜇)𝑃𝜌_𝐷≤𝑇or𝜃_𝐷≤𝑇. (A.3) Note Esup_{[]𝑡∈0,𝑇}𝑥(𝑡)−𝑦(𝑡)^𝜇𝑝≤2^𝜇𝑝−1Esup_{[]𝑡∈0,𝑇}|𝑥(𝑡)|^𝜇𝑝+sup_{[]𝑡∈0,𝑇}𝑦(𝑡)^𝜇𝑝≤2^𝜇𝑝𝐿_𝐴. (A.4)
Furthermore, 𝑃𝜌_𝐷≤𝑇≤E𝟏_{{𝜌𝐷≤𝑇}}𝑥𝜌_𝐷^𝑝𝐷^𝑝≤𝐿_𝐴𝐷^𝑝, (A.5) and, similarly, 𝑃𝜃_𝐷≤𝐿≤𝑇^𝐴𝐷^𝑝. (A.6) Therefore, 𝑃𝜌_𝐷≤𝑇or𝜃_𝐷≤𝑇≤𝑃𝜌_𝐷≤𝑇+𝑃𝜃_𝐷≤𝑇≤2𝐿_𝐴𝐷^𝑝. (A.7) On the other side, Lemma 4 implies that Esup_{[]𝑡∈0,𝑇}𝑥(𝑡)−𝑦(𝑡)^𝑝𝟏_{{𝜌𝐷>𝑇and𝜃𝐷>𝑇}}≤Esup_{𝑡∈[0,𝑇]}𝑥𝑡∧𝜎_𝐷−𝑦𝑡∧𝜎_𝐷^𝑝≤𝐾ℎ_𝑁^𝑝/2. (A.8) Combining (A.1) and the above inequalities gives Esup_{𝑡∈[0,𝑇]}𝑥(𝑡)−𝑦(𝑡)^𝑝≤𝐾ℎ_𝑁^𝑝/2+𝜂2^𝜇𝑝𝐿_𝐴𝜇+2𝐿_𝐴𝜈𝜂^(𝜈/𝜇)𝐷^𝑝. (A.9)
Setting 𝜂=ℎ_𝑁^𝑝/2 and choosing 𝐷=ℎ_𝑁^{−(1/2)(1+(𝜈/𝜇))}, we obtain (40) for sufficiently large 𝐷.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the anonymous reviewers and editor for their helpful and constructive comments that greatly contributed to improving the paper. They would also like to thank Dr. Xiang Lv for his valuable discussion. This work was partially supported by E-Institutes of  Shanghai Municipal Education Commission (no. E03004), the National Natural Science Foundation of China (no. 10901106), and the Innovation Program of Shanghai Municipal Education Commission (no. 14YZ078).