Tamed Euler–Maruyama Method of Time-Changed McKean–Vlasov Neutral Stochastic Differential Equations with Super-Linear Growth

Abstract

This paper examines temporal symmetry breaking and structural duality in a class of time-changed McKean–Vlasov neutral stochastic differential equations. The system features super-linear drift coefficients satisfying a one-sided local Lipschitz condition and incorporates a fundamental duality: one drift component evolves under a random time change $E_t$, while the other progresses in regular time t. Within the symmetric framework of mean-field interacting particle systems, where particles exhibit permutation invariance, we establish strong convergence of the tamed Euler–Maruyama method over finite time intervals. By replacing the one-sided local condition with a globally symmetric Lipschitz assumption, we derive an explicit convergence rate for the numerical scheme. Two numerical examples validate the theoretical results.

1. Introduction

This work concerns the following time-changed MV-NSDE of Itô form:

$$\mathrm{d}[X_t-G\left(X_t\right)]=f\left(E_t\right)dt+h(E_t,X_t,{\mu }_t^X)\mathrm{d}{E}_t+g(E_t,X_t,{\mu }_t^X)\mathrm{d}{B}_{E_t},t\in [0,\mathrm{T}].$$

(1)

Here, the coefficients G, f, h and g satisfy specific regularity conditions (detailed in Assumption 1), while ${\mu }_t^X:={\mathcal{L}}_{X_t}$ denotes the distribution of X at time t. The process $B_t$ is a standard Brownian motion, and $E_t$ is defined as the inverse of a subordinator. The composition ${\left\{B_{E_t}\right\}}_{t\in [0,\mathrm{T}]}$, termed a time-changed Brownian motion, models subdiffusive behavior (see [1,2]). Formal mathematical definitions are provided in Section 2.

Recently, time-changed semimartingales and their extensions have become prominent tools in modeling anomalous diffusion phenomena across disciplines such as cell biology, finance, hydrology, and physics (cf. Umarov et al.’s recent monograph [2]). Seminal contributions by Kobayashi [3] introduced stochastic integration theory for these processes, deriving a time-changed Itô formula applicable to the first hitting time of a $\beta$-stable subordinator ($\beta \in (0,1)$). This breakthrough catalyzed investigations into SDEs driven by Lévy noise or time-changed Brownian motion, for example, ref. [4,5] for harnack inequalities; ref. [6,7] for the numerical approximation scheme; and ref. [8,9] for averaging principles.

Given the general absence of explicit solutions for such SDEs, numerical methods become crucial when modeling real-world phenomena with uncertainty. To our knowledge, ref. [10] presents the first study on finite-time strong convergence of direct discretization schemes for time-changed SDEs. Therein, the authors employed the duality principle from [3] to develop the Euler–Maruyama (EM) method. Subsequent work by Deng [11] and Liu [12] extended this approach to semi-implicit and truncated EM methods respectively, addressing equations with superlinearly growing coefficients while establishing strong convergence and convergence rates. These four contributions fundamentally rely on the duality principle introduced by Kobayashi [3], which links the time-changed SDE (2)

$$\mathrm{d}X\left(t\right)=h\left(E\right(t),X(t\left)\right)\mathrm{d}E\left(t\right)+g\left(E\right(t),X(t\left)\right)\mathrm{d}B\left(E\right(t\left)\right),t\in [0,\mathrm{T}],$$

(2)

with the classical SDE (3)

$$\mathrm{d}Y\left(t\right)=h\left(E\right(t),Y(t\left)\right)\mathrm{d}t+g\left(E\right(t),Y(t\left)\right)\mathrm{d}B\left(t\right),t\in [0,\mathrm{T}],$$

(3)

in the following manner: if $Y\left(t\right)$ solves (3), then $X\left(t\right):=Y\left(E_t\right)$ solves (2), while if $X\left(t\right)$ solves (2), then $Y\left(t\right):=X\left(D_t\right)$ solves (3), where D is the original subordinator.

Crucially, the duality principle fails when both drifts f and h are present simultaneously. This occurs because the corresponding non-time-changed SDE (2) would contain a D-driven term, rendering Y dependent on E and invalidating the duality principle for (3). Alternatively, ref. [13] examines the EM approximate for time-changed SDE (4):

$$\mathrm{d}X\left(t\right)=f\left(E\right(t\left)\right)\mathrm{d}t+h\left(E\right(t),X(t\left)\right)\mathrm{d}E\left(t\right)+g\left(E\right(t),X(t\left)\right)\mathrm{d}B\left(E\right(t\left)\right),t\in [0,\mathrm{T}].$$

(4)

and ref. [14] proposed an EM scheme for time-changed MV-SDEs (5):

$$\mathrm{d}X\left(t\right)=f\left(E\left(t\right)\right)\mathrm{d}t+h(E\left(t\right),X\left(t\right),{\mu }_t^X)\mathrm{d}E\left(t\right)+g(E\left(t\right),X\left(t\right),{\mu }_t^X)\mathrm{d}B\left(E\left(t\right)\right),t\in [0,\mathrm{T}].$$

(5)

Compared to [13] and [14], this work establishes, to the best of our knowledge, the first strong convergence results and convergence rates for the tamed EM method applied to the time-changed MV-NSDE (1), under significantly weaker assumptions on the drift coefficient. Specifically, we require only a one-sided local Lipschitz condition and allow for super-linear growth, thereby lifting the constraints on the coefficients prevalent in existing literature. This extension significantly broadens the applicability of the tamed EM scheme to model complex systems involving mean-field interactions and stochastic time change, such as the migration dynamics of cell populations or animal groups, where individual interactions (mean-field effects) and environmentally driven random time-scale transformations (e.g., induced by seasonal shifts or abrupt environmental changes) can be naturally captured by the time-changed MV-NSDE (1).

This paper makes two primary contributions:

• We establish strong convergence of the tamed EM method over finite time horizons, obtaining a $\gamma /2$-order convergence rate with respect to step size. Integration with propagation of chaos yields an overall convergence rate for the scheme.
• For the drift coefficient h, we employ a one-sided local Lipschitz condition (versus the one-sided Lipschitz condition in [14]) and super-linear growth condition (replacing the linear growth condition in [14]).

The rest of the paper is organized as follows. In Section 2, we introduce some necessary preliminaries for the Wasserstein metric and the time-changed MV-NSDEs. In Section 3, chaos propagation and strong convergence have been proven under one-sided local Lipschitz conditions. In Section 4, by replacing the one-sided local Lipschitz condition with a global condition, we further derive an explicit convergence rate for the numerical scheme. In Section 5, two numerical examples validate the theoretical results.

2. Preliminaries

We fix notation used throughout this work. For $x,y\in {\mathbb{R}}^d$, denote the Euclidean norm by $|·|$ and inner product by $\langle x,y\rangle$. The transpose of any vector or matrix A is $A^{\top }$, while for matrices, the trace norm is $|A|=\sqrt{trace\left(A^{\top }A\right)}$. Define $\mathbb{R}=(-\infty ,\infty )$, ${\mathbb{R}}^{+}=[0,\infty )$ and $\mathbb{N}=\{1,2,3,\dots \}$. For $u,v\in \mathbb{R}$, set $u\wedge v=min\{u,v\}$ and $u\vee v=max\{u,v\}$.

In the paper, C denotes a generic positive constant, whose value may change for different usage. Similarly, $C_p$ denotes the generic positive constant depending on parameter p. If it is necessary, the dependence of constants on parameters will be indicated.

2.1. Empirical Measure and Wasserstein Metric

Let $\mathcal{P}\left(\mathbb{R}\right)$ denote all probability measures on $(\mathbb{R},\mathcal{B}(\mathbb{R}\left)\right)$, where $\mathcal{B}\left(\mathbb{R}\right)$ is the Borel $\sigma$-algebra over $\mathbb{R}$. For any $p>0$, a measure $\mu \in \mathcal{P}\left(\mathbb{R}\right)$ belongs to ${\mathcal{P}}_p\left(\mathbb{R}\right)$ if it possesses a finite p-th moment, i.e.,

$${\mathcal{P}}_p\left(\mathbb{R}\right)=\left\{\mu \in \mathcal{P}\left(\mathbb{R}\right):{\int }_{{\mathbb{R}}^d}{|x|}^p\mu \left(\mathrm{d}x\right)<\infty \right\}.$$

For $\mu ,v\in {\mathcal{P}}_p\left(\mathbb{R}\right)$, the Wasserstein distance between $\mu$ and v is defined by

$${\mathbb{W}}_p(\mu ,v)=\underset{\pi \in \mathcal{C}(\mu ,v)}{inf}\left\{{\left({\int }_{\mathbb{R}\times \mathbb{R}}{|x-y|}^p\pi (\mathrm{d}x,\mathrm{d}y)\right)}^{{\displaystyle \frac{1}{p}}}\right\},$$

where is the $\mathcal{C}(\mu ,\nu )$ denotes the collection of all couplings between $\mu$ and $\nu$. For a random variable X, denote ${\mathcal{L}}_X$ the law of X. In particular, if $\mu ={\mathcal{L}}_X$ and $v={\mathcal{L}}_Y$, then

$$\begin{array}{c}\hfill {\mathbb{W}}_1(\mu ,v)\le {\int }_{\mathbb{R}\times \mathbb{R}}|x-y|{\mathcal{L}}_{X,Y}(\mathrm{d}x,\mathrm{d}y)=\mathbb{E}|X-Y|,\end{array}$$

(6)

$$\begin{array}{c}\hfill {\mathbb{W}}_2^2(\mu ,v)\le \mathbb{E}{|X-Y|}^2,\end{array}$$

(7)

where ${\mathcal{L}}_{X,Y}$ represents the joint distribution of random variable $(X,Y)$. As ${\mathbb{W}}_p$ is a metric, for ${\mu }_1,{\mu }_2,{\mu }_3\in {\mathcal{P}}_p\left(\mathbb{R}\right)$, we always have (see [15])

$$\begin{array}{c}\hfill {\mathbb{W}}_p({\mu }_1,{\mu }_2)\le {\mathbb{W}}_p({\mu }_1,{\mu }_3)+{\mathbb{W}}_p({\mu }_3,{\mu }_2).\end{array}$$

(8)

Given a random process ${\left\{X_t\right\}}_{t\ge 0}$, assuming N independent and identically distributed (i.i.d.) samples $X_s^1$, $X_s^2$, …, $X_s^N$ are observed at time s, and the corresponding empirical measure is defined as

$${\widehat{\mu }}_s^{X,N}:={\displaystyle \frac{1}{N}}\sum _{j=1}^N{\delta }_{X_s^j},$$

where ${\delta }_{X_s^j}$ is the Dirac measure concentrated at sample point $X_s^j$. Empirical measurement approximates the true probability distribution ${\mu }_s^X$ of the random variable $X_s$ by weighted average of sample points (see [16]). Hence, the Wasserstein distance between empirical measures ${\widehat{\mu }}_s^{X,N}$ and ${\widehat{\mu }}_s^{Y,N}$ satisfies

$$\begin{array}{c}\hfill {\mathbb{W}}_p^p({\widehat{\mu }}_s^{X,N},{\widehat{\mu }}_s^{Y,N})\le {\displaystyle \frac{1}{N}}\sum _{j=1}^N{|X_s^j-Y_s^j|}^p.\end{array}$$

(9)

Unlike standard SDEs, time-changed MV-NSDEs require approximating ${\mu }_t$ at each time step. When ${\mu }_t$ in (1) is known a priori, these equations reduce to classical time-changed SDEs with time-dependent coefficients. Since this is typically not feasible, we approximate (1) via an N-dimensional stochastic interacting particle system.

Let $i=1,2,\dots ,N$ and consider N particles $X^{i,N}$ satisfying

$$\mathrm{d}[X_t^{i,N}-G\left(X_t^{i,N}\right)]=f\left(E_t\right)\mathrm{d}t+h(E_t,X_t^{i,N},{\mu }_t^{X,N})\mathrm{d}{E}_t+g(E_t,X_t^{i,N},{\mu }_t^{X,N})\mathrm{d}{B}_{E_t}^i,t\in [0,T],$$

(10)

with the independent identical initial value $X_0^{i,N}=X_0^i$, where ${\left\{B_{E_t}^i\right\}}_{1\le i\le N}$ are independent Brownian motion and ${\mu }_t^{X,N}$ means the empirical distribution corresponding to $X_t^{1,N}$, $X_t^{2,N}$, …, $X_t^{N,N}$, that is

$${\mu }_t^{X,N}:={\displaystyle \frac{1}{N}}\sum _{j=1}^N{\delta }_{X_t^{j,N}},$$

where ${\delta }_{X_t^{j,N}}$ is the Dirac measure at point $X_t^{j,N}$.

Propagation of chaos governs the convergence behavior as $N\to \infty$. We further investigate the stochastic non-interacting particle systems

$$\mathrm{d}[X_t^i-G\left(X_t^i\right)]=f\left(E_t\right)\mathrm{d}t+h(E_t,X_t^i,{\mu }_t^i)\mathrm{d}{E}_t+g(E_t,X_t^i,{\mu }_t^i)\mathrm{d}{B}_{E_t}^i,t\in [0,T],$$

(11)

with initial value $X_0^i=X_0$, where ${\mu }_t^i:={\mathcal{L}}_{X_t^i}$. Since $X^i$ are independent for all $i=1,2,\dots ,N$, we have ${\mu }_t^i={\mu }_t^X$ for all i.

Since ${\{X_s^j-X_s^{j,N}\}}_{\{1\le j\le N\}}$ are identically distributed, from (9), we obtain

$$\begin{array}{c}\hfill \mathbb{E}{\mathbb{W}}_p^p({\widehat{\mu }}_s^{X,N},{\mu }_s^{X,N})\le \mathbb{E}\left[{\displaystyle \frac{1}{N}}\sum _{j=1}^N{|X_s^j-X_s^{j,N}|}^p\right]=\mathbb{E}{|X_s^j-X_s^{j,N}|}^p.\end{array}$$

(12)

From this fact (see [17], Theorem 5.8), there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \mathbb{E}{\mathbb{W}}_1({\mu }_s,{\widehat{\mu }}_s^{X,N})\le C{N}^{-{\displaystyle \frac{1}{4}}},\mathbb{E}{\mathbb{W}}_2^2({\mu }_s,{\widehat{\mu }}_s^{X,N})\le C{N}^{-{\displaystyle \frac{1}{2}}}.\end{array}$$

(13)

From (9), (12) and (13), we obtain

$$\begin{array}{cc}\hfill & \mathbb{E}{\mathbb{W}}_1({\mu }_s,{\mu }_s^{X,N})\le \mathbb{E}{\mathbb{W}}_1({\mu }_s,{\widehat{\mu }}_s^{X,N})+\mathbb{E}{\mathbb{W}}_1({\widehat{\mu }}_s^{X,N},{\mu }_s^{X,N})\le \mathbb{E}|X_s^j-X_s^{j,N}|+C{N}^{-{\displaystyle \frac{1}{4}}},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \mathbb{E}{\mathbb{W}}_2^2({\mu }_s,{\mu }_s^{X,N})\le C\mathbb{E}{|X_s^j-X_s^{j,N}|}^2+C{N}^{-{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

(14)

2.2. Time-Changed Brownian Motion

Consider a complete probability space $({\Omega }_B,{\mathcal{F}}^B,{\mathbb{P}}_B)$ endowed with a filtration ${\left\{{\mathcal{F}}_t^B\right\}}_{t\ge 0}$ satisfying the standard hypotheses: right continuity and inclusion of all ${\mathbb{P}}_B$-null sets in ${\mathcal{F}}_0^B$. Separately, let ${\left\{D\left(t\right)\right\}}_{t\ge 0}$ be a subordinator (a non-decreasing Lévy process with right-continuous paths and left limits) initialized at zero. This process resides on another complete probability space $({\Omega }_D,{\mathcal{F}}^D,{\mathbb{P}}_D)$ equipped with filtration ${\left\{{\mathcal{F}}_t^D\right\}}_{t\ge 0}$ that also fulfills the standard hypotheses. Define ${\mathbb{E}}_B$ and ${\mathbb{E}}_D$ as the expectation operators corresponding to probability measures ${\mathbb{P}}_B$ and ${\mathbb{E}}_D$, respectively.

A subordinator process $D\left(t\right)$ is termed a β-stable subordinator when strictly increasing, denoted $D_{\beta }\left(t\right)$, with Laplace transform characterization:

$$\mathbb{E}\left[exp(-\lambda D_{\beta }\left(t\right))\right]=exp(-t{\lambda }^{\beta }),\lambda >0,\beta \in (0,1).$$

(15)

Given an adapted $\beta$-stable subordinator $D_{\beta }\left(t\right)$, its generalized inverse process

$$E_t\left(\beta \right):=inf\left\{s>0:D_{\beta }\left(s\right)>t\right\}$$

(16)

represents the first hitting time. This time change $E_t\left(\beta \right)$ is continuous and non-decreasing, yet non-Markovian (see [1,18]).

Consider a standard Brownian motion $B_t$ independent of $E_t\left(\beta \right)$. The filtration is constructed as

$${\mathcal{F}}_t=\bigcap _{s>t}\left({\sigma }_1\left(B_r:0\le r\le s\right)\vee {\sigma }_2\left(E_r\left(\beta \right):r\ge 0\right)\right),$$

where ${\sigma }_1\vee {\sigma }_2$ indicates the $\sigma$-algebra generated by ${\sigma }_1\cup {\sigma }_2$. Consequently, the time-changed process $B_{E_t\left(\beta \right)}$ forms a square integrable martingale relative to ${\left\{{\mathcal{G}}_t\right\}}_{t\ge 0}$ with ${\mathcal{G}}_t={\mathcal{F}}_{E_t\left(\beta \right)}$.

The product probability space $\mathcal{B}$ is defined by

$$(\Omega ,\mathcal{F},\mathbb{P}):=\left({\Omega }_B\times {\Omega }_D,{\mathcal{F}}^B\otimes {\mathcal{F}}^D,{\mathbb{P}}_B\times {\mathbb{P}}_D\right).$$

Denoting by $\mathbb{E}$ the expectation under $\mathbb{P}$, we have the identity

$$\mathbb{E}[·]={\mathbb{E}}_D\left[{\mathbb{E}}_B[·]\right]={\mathbb{E}}_B\left[{\mathbb{E}}_D[·]\right].$$

Direct simulation of (10) is generally infeasible, necessitating numerical schemes like the tamed Euler–Maruyama method. Fixing a uniform step size $\delta \in (0,1)$, we approximate $E_t$ over $[0,T]$ (for arbitrary $T>0$) following [19,20] (see Figure 1). This entails simulating a subordinator path $D_t$ with independent stationary increments: Initialize $D_0=0$ and compute iteratively

$$D_{i\delta }=D_{(i-1)\delta }+Z_i,i\in \mathbb{N},$$

where ${\left\{Z_i\right\}}_{i\in \mathbb{N}}$ are i.i.d. with $Z_i\stackrel{d}{=}{\mathrm{D}}_{\delta }$. The sequence $\left\{Z_i\right\}$ is generated via [21] (Chapter 6), terminating when $T\in [D_{M\delta },D_{(M+1)\delta })$ for some integer M.

Next, the approximate $E_t^{\delta }$ to $E_t$ is generated by

$$E_t^{\delta }:=(min\{m\in \mathbb{N};D_{m\delta }>t\}-1)\delta .$$

It is clear that for $i=0,1,2,\dots ,M$, when $s\in [D_{i\delta },D_{(i+1)\delta })$

$$E_s^{\delta }=i\delta .$$

The next Lemma 1 will be used as the approximation error of $E_t^{\delta }$ to $E_t$, whose proof can be found in [10,20].

Lemma 1.

Almost surely,

$$E_r-\delta \le E_r^{\delta }\le E_r,$$

holds for all $r>0$.

Now, for $m=0,1,2,\dots ,M$, let

$${\tau }_m^{\delta }=D_{m\delta },$$

and let

$$n_t=max\left\{m\in \mathbb{N}\cup \left\{0\right\};{\tau }_m^{\delta }\le t\right\},\mathrm{for}t\ge 0.$$

Then it is easy to see that for any $t\ge 0$

$${\tau }_{n_t}^{\delta }\le t<{\tau }_{n_t+1}^{\delta }.$$

(17)

Define ${\delta }_1:=max\{\delta ,(s-{\tau }_{n_s}^{\delta })\}$. It can be easily observed that for any $c<0$

$${\delta }_1^c\le min\{{\delta }^c,{(s-{\tau }_{n_s}^{\delta })}^c\}.$$

We adapt the approach in [13,22] for discrete and continuous-time approximations. Exploiting the independence of B and E, we discretize B at grid points $\{0,\delta ,2\delta ,\dots ,M\delta \}$. Define a discrete-time process ${\left\{X_{{\tau }_m^{\delta }}^{i,\delta }\right\}}_{m\in \left\{0,1,2,\dots ,M\right\}}$ by setting $X_0^{i,\delta }=X_0^{i,N}=X_0^i$ and

$$\begin{array}{cc}\hfill X_{{\tau }_{m+1}^{\delta }}^{i,\delta }:=& G\left(X_{{\tau }_{m+1}^{\delta }}^{i,\delta }\right)-G\left(X_{{\tau }_m^{\delta }}^{i,\delta }\right)+X_{{\tau }_m^{\delta }}^{i,\delta }+f\left(E_{{\tau }_m^{\delta }}^{\delta }\right)({\tau }_{m+1}^{\delta }-{\tau }_m^{\delta })+h_{\delta }(E_{{\tau }_m^{\delta }}^{\delta },X_{{\tau }_m^{\delta }}^{i,\delta },{\mu }_{{\tau }_m^{\delta }}^{\delta })\delta \hfill \\ \hfill & +g(E_{{\tau }_m^{\delta }}^{\delta },X_{{\tau }_m^{\delta }}^{i,\delta },{\mu }_{{\tau }_m^{\delta }}^{\delta })(B_{E_{{\tau }_{m+1}^{\delta }}^{\delta }}^i-B_{E_{{\tau }_m^{\delta }}^{\delta }}^i),\hfill \end{array}$$

(18)

where

$$h_{\delta }(E_{{\tau }_m^{\delta }}^{\delta },X_{{\tau }_m^{\delta }}^{i,\delta },{\mu }_{{\tau }_m^{\delta }}^{\delta })={\displaystyle \frac{h}{1+{\delta }_1^{\gamma /2}|h|}}.$$

It can be easily observed that

$$h_{\delta }\le min\{{\delta }_1^{-\gamma /2},|h|\},{\delta }_1^{-\gamma /2}\le min\{{\delta }^{-\gamma /2},{(s-{\tau }_{n_s}^{\delta })}^{-\gamma /2}\},$$

(19)

and define the continuous-time process ${\left\{X_t^{i,\delta }\right\}}_{t\in [0,T]}$ via continuous interpolation: when $t\in [{\tau }_m^{\delta },{\tau }_{m+1}^{\delta })$,

$$\begin{array}{cc}\hfill X_t^{i,\delta }:=& G\left(X_t^{i,\delta }\right)-G\left(X_{{\tau }_m^{\delta }}^{i,\delta }\right)+X_{{\tau }_m^{\delta }}^{i,\delta }+{\int }_{{\tau }_m^{\delta }}^{t}f\left(E_{{\tau }_m^{\delta }}^{\delta }\right)\mathrm{d}s+{\int }_{{\tau }_m^{\delta }}^t{h}_{\delta }(E_{{\tau }_m^{\delta }}^{\delta },X_{{\tau }_m^{\delta }}^{i,\delta },{\mu }_{{\tau }_m^{\delta }}^{\delta })\mathrm{d}{E}_s\hfill \\ \hfill & +{\int }_{{\tau }_m^{\delta }}^{t}g(E_{{\tau }_m^{\delta }}^{\delta },X_{{\tau }_m^{\delta }}^{i,\delta },{\mu }_{{\tau }_m^{\delta }}^{\delta })\mathrm{d}{B}_{E_s}^i.\hfill \end{array}$$

(20)

Using (17) and (18) and the identity

$$X_t^{i,\delta }-X_0^i=(X_t^{i,\delta }-X_{{\tau }_{n_t}^{\delta }}^{i,\delta })+\sum _{s=0}^{n_t-1}(X_{{\tau }_{s+1}^{\delta }}^{i,\delta }-X_{{\tau }_s^{\delta }}^{i,\delta }),$$

we can express $X_t^{i,\delta }-X_0^i$ as

$$\begin{array}{cc}\hfill \sum _{s=0}^{n_t-1}& G\left(X_{{\tau }_{s+1}^{\delta }}^{i,\delta }\right)-G\left(X_{{\tau }_s^{\delta }}^{i,\delta }\right)+f\left(E_{{\tau }_s^{\delta }}\right)({\tau }_{s+1}^{\delta }-{\tau }_s^{\delta })+h_{\delta }(E_{{\tau }_s^{\delta }},X_{{\tau }_s^{\delta }}^{i,\delta },{\mu }_{{\tau }_s^{\delta }}^{\delta })\delta \hfill \\ \hfill & +g(E_{{\tau }_s^{\delta }},X_{{\tau }_s^{\delta }}^{i,\delta },{\mu }_{{\tau }_s^{\delta }}^{\delta })(B_{(s+1)\delta }^i-B_{s\delta }^i)+(X_t^{i,\delta }-X_{{\tau }_{n_t}^{\delta }}^{i,\delta }).\hfill \end{array}$$

Using (20), we reformulate the latter expression as

$$\begin{array}{cc}\hfill X_t^{i,\delta }:=& G\left(X_t^{i,\delta }\right)-G\left(X_0^i\right)+X_0^i+{\int }_0^{t}f\left(E_{{\tau }_{n_s}^{\delta }}\right)\mathrm{d}s+{\int }_0^t{h}_{\delta }(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })\mathrm{d}{E}_s\hfill \\ \hfill & +{\int }_0^{t}g(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })\mathrm{d}{B}_{E_s}^i,\hfill \end{array}$$

(21)

where ${\mu }_s^{\delta }:={\displaystyle \frac{1}{N}}{\sum }_{j=1}^N{\delta }_{X_s^{j,\delta }}$.

2.3. Assumption

Assumption 1.

In order to derive the main results, we require that the coefficients G, f, h and g satisfy the following assumptions. Regarding convergence, we propose the following hypotheses.

Hypothesis 1.

Let $f:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ be Borel-measurable functions and $f\left(0\right)=0$. There exists positive constant $K_f$ and $\gamma \in (0,1]$ such that for all $\phi \in {\mathcal{P}}_2\left(\mathbb{R}\right)$, $\in \mathbb{R}$ and $t_1,t_2\in {\mathbb{R}}^{+}$

$$|f\left(t_1\right)-f\left(t_2\right)|\vee |h(t_1,x,\phi )-h(t_2,x,\phi )|\vee |g(t_1,x,\phi )-g(t_2,x,\phi )|\le K_f{|t_1-t_2|}^{\gamma }.$$

(22)

Regarding well-posedness, we propose the following hypothesis

Hypothesis 2.

Let $G:\mathbb{R}\to {\mathbb{R}}^{+}$ be Borel-measurable functions and $G\left(0\right)=0$. For any $\varphi ,\psi \in \mathbb{R}$, there exists a constant $v\in (0,1)$ such that

$$|G(\varphi )-G(\psi )|\le v|\varphi -\psi |.$$

(23)

Hypothesis 3.

Let $h:{\mathbb{R}}^{+}\times \mathbb{R}\times {\mathcal{P}}_2\left(\mathbb{R}\right)\to \mathbb{R}$ be Borel-measurable and continuous functions. For any $R>1$, a positive constant a and all $|\varphi |\vee |\psi |\le R$, there exists a positive constant $K_R\le alogR$ such that for any $s\in {\mathbb{R}}^{+}$, $\varphi ,\psi \in \mathbb{R}$ and $\phi \in {\mathcal{P}}_2\left(\mathbb{R}\right)$

$$(\varphi -\psi )(h(s,\varphi ,\phi )-h(s,\psi ,\phi ))\le K_R{|\varphi -\psi |}^2.$$

(24)

Hypothesis 4.

Let $g:{\mathbb{R}}^{+}\times \mathbb{R}\times {\mathcal{P}}_2\left(\mathbb{R}\right)\to \mathbb{R}$ be Borel-measurable and continuous functions. There exists positive constant $K_g$ such that for any $s\in {\mathbb{R}}^{+}$, $\varphi ,\psi \in \mathbb{R}$ and $\phi \in {\mathcal{P}}_2\left(\mathbb{R}\right)$

$$|g(s,\varphi ,\phi )-g(s,\psi ,\phi )|\le K_g|\varphi -\psi |.$$

(25)

Hypothesis 5.

There exists constant $K_1>0$ such that for any $s\in {\mathbb{R}}^{+}$, $\varphi \in \mathbb{R}$ and ${\phi }_1,{\phi }_2\in {\mathcal{P}}_2\left(\mathbb{R}\right)$

$$\begin{array}{c}\hfill |h(s,\varphi ,{\phi }_1)-h(s,\varphi ,{\phi }_2)|\vee |g(s,\varphi ,{\phi }_1)-g(s,x,{\phi }_2)|\le K_1{\mathbb{W}}_1({\phi }_1,{\phi }_2).\end{array}$$

(26)

Hypothesis 6.

There exists constant $L_h>0$ and $p>1$ such that for any $s\in {\mathbb{R}}^{+}$, $\varphi \in \mathbb{R}$ and $\phi \in {\mathcal{P}}_2\left(\mathbb{R}\right)$

$$\begin{array}{c}\hfill xh(s,\varphi ,\phi )\le L_1{(1+|\varphi |}^2+{\mathbb{W}}_1^2(\phi ,{\delta }_0)).\end{array}$$

(27)

$$\begin{array}{c}\hfill |h(s,\varphi ,\phi )|\le L_h{(1+|\varphi |}^p+{\mathbb{W}}_1(\phi ,{\delta }_0)).\end{array}$$

(28)

Hypothesis 7.

There exists constant $L_g>0$ such that for any $s\in {\mathbb{R}}^{+}$, $\varphi \in \mathbb{R}$ and $\phi \in {\mathcal{P}}_2\left(\mathbb{R}\right)$

$$\begin{array}{c}\hfill |g(s,\varphi ,\phi )|\le L_g(1+|\varphi |+{\mathbb{W}}_1(\phi ,{\delta }_0)).\end{array}$$

(29)

Hypothesis 8.

If $X_t$ is a right continuous with left limit and ${\mathcal{G}}_t$-adapted process, then

$$f\left(E_t\right),h(E_t,X_t,{\mu }_t),g(E_t,X_t,{\mu }_t)\in \mathcal{L}\left({\mathcal{G}}_t\right),$$

where $\mathcal{L}\left({\mathcal{G}}_t\right)$ denotes the class of left continuous with right limit.

Under Assumption 1, we establish the well-posedness of (1).

Lemma 2.

Let Assumption 1 hold. Then, the time-changed neutral stochastic differential Equation (1) has a unique strong solution which is a continuous ${\mathcal{F}}_{E_t}$-semimartingale.

Proof. 

The proof of existence and uniqueness is almost similar to [3,11,23,24], so it is omitted here. □

3. Strong Convergence

In this section, we will study the numerical solution of (1) based on the previous section. Firstly, we have constructed the N-dimensional interacting particle system (10), and we prove that the N-dimensional interacting particle system (10) converges to the non-interacting particle system (11). Then, we use the tamed EM method (21) to approximate the N-dimensional interacting particle system (10). Strong convergence has been proven. When the local one-sided Lipschitz condition is replaced by the global one-sided Lipschitz condition, the convergence order is obtained.

3.1. A New Time-Changed Gronwall’s Inequality

This section aims to develop the mean square convergence rate of (21) in any finite time $[0,T]$. To this end, the following lemma connects different types of time-changed integrals.

Lemma 3.

([3]) Let $D_t$ be a β-stable subordinator with inverse $E_t$. If $E_t$ is ${\mathcal{G}}_t$-measurable and $f,g$ are ${\mathcal{G}}_t$-measurable integrable functions, then for any $s\ge 0$,

$${\int }_0^{s}f\left(r\right)\mathrm{d}{E}_r+{\int }_0^{s}g\left(r\right)\mathrm{d}{B}_{E_r}={\int }_0^{E_s}f\left(D\right(r-\left)\right)\mathrm{d}r+{\int }_0^{E_s}g\left(D\right(r-\left)\right)\mathrm{d}{E}_r.$$

Now, we establish a novel Gronwall inequality essential for our analysis.

Proposition 1.

Let $D_t$ be a subordinator with infinite Lévy measure and its inverse is $E_t$. Consider ${\mathcal{G}}_t$-adapted functions $f,b,\sigma :\Omega \times {\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ that are integrable with respect to s and $E_s$. If $k(·):{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is monotonic and non-decreasing, then the following inequality

$$f\left(s\right)\le k\left(s\right)+{\int }_0^{s}b\left(r\right)u\left(r\right)\mathrm{d}r+{\int }_0^s\sigma \left(r\right)u\left(r\right)\mathrm{d}{E}_r,s\ge 0,a.s.,$$

(30)

implies

$$f\left(s\right)\le k\left(s\right)exp\left\{{\int }_0^{s}b\left(r\right)\mathrm{d}r+{\int }_0^s\sigma \left(r\right)\mathrm{d}{E}_r\right\},s\ge 0,a.s.$$

Proof. 

Let

$$y\left(s\right):=k\left(s\right)+{\int }_0^{s}b\left(r\right)f\left(r\right)\mathrm{d}r+{\int }_0^s\sigma \left(r\right)f\left(r\right)\mathrm{d}{E}_r,s\ge 0.$$

(31)

Since f, b and $\sigma$ are positive and k is non-decreasing, the function $y\left(s\right)$ defined in (32) is non-decreasing. Moreover, from (30) and (31),

$$f\left(s\right)\le y\left(s\right),s\ge 0,$$

which implies that

$$y\left(s\right)\le k\left(s\right)+{\int }_0^{s}b\left(r\right)y\left(r\right)\mathrm{d}r+{\int }_0^s\sigma \left(r\right)y\left(r\right)\mathrm{d}{E}_r,s\ge 0.$$

Applying the connections among different kinds of time-changed integrals (see the Lemma 3) yields

$$y\left(s\right)\le k\left(s\right)+{\int }_0^{s}b\left(r\right)y\left(r\right)\mathrm{d}r+{\int }_0^{E_s}\sigma \left(D(r-)\right)y\left(D(r-)\right)dr,s\ge 0.$$

(32)

For $s\ge 0$, $D(s-)$ is defined as

$$D(s-)=inf\{r,E_r\ge s\},$$

yielding the fundamental properties:

$$E_{D(s-)}=s\mathrm{and}D(E_s-)\le s.$$

(33)

Let $\varphi \in [0,\infty )$, then it holds from (32) and (33) that

$$\begin{array}{cc}\hfill y\left(D\right(\varphi -\left)\right)\le & k\left(D(\varphi -)\right)+{\int }_0^{D(\varphi -)}b\left(r\right)y\left(r\right)\mathrm{d}r+{\int }_0^{E_{D(\varphi -)}}\sigma \left(D(r-)\right)y\left(D(r-)\right)\mathrm{d}r\hfill \\ \hfill =& k\left(D(\varphi -)\right)+{\int }_0^{D(\varphi -)}b\left(r\right)y\left(r\right)\mathrm{d}r+{\int }_0^{\varphi }\sigma \left(D(r-)\right)y\left(D(r-)\right)\mathrm{d}r.\hfill \end{array}$$

From the retarded Gronwall-like inequality (see Proposition 1 in [25]), we get

$$f\left(D(\varphi -)\right)\le y\left(D(\varphi -)\right)\le k\left(D(\varphi -)\right)\exp \left\{{\int }_0^{D(\varphi -)}b\left(r\right)\mathrm{d}r+{\int }_0^{\varphi }\sigma \left(D(r-)\right)\mathrm{d}r\right\}.$$

For any $s\ge 0$, we easily know that there is $\varphi \in [0,+\infty )$ such that $D(\varphi -)=s$ provided E is strictly increasing in a certain neighborhood of s. Since $k\left(s\right)$ is non-decreasing, applying (33) and the connections among different kinds of time-changed integrals (see Lemma 3), we can get

$$f\left(s\right)\le k\left(s\right)\exp \left\{{\int }_0^{s}b\left(r\right)\mathrm{d}r+{\int }_0^{E_s}\sigma \left(D(r-)\right)\mathrm{d}r\right\}=k\left(s\right)\exp \left\{{\int }_0^{s}b\left(r\right)\mathrm{d}r+{\int }_0^s\sigma \left(r\right)\mathrm{d}{E}_r\right\}.$$

If E is a constant $\varphi$ in a certain neighborhood $U(s;\epsilon )$ of s, then $s-\epsilon =D(\varphi -)$. Since $k\left(s\right)$ is non-decreasing, by (32), we can get

$$f(s-\epsilon )\le y(s-\epsilon )\le k(s-\epsilon )\exp \left\{{\int }_0^{s-\epsilon }b\left(r\right)\mathrm{d}r+{\int }_0^{E_{s-\epsilon }}\sigma \left(D(r-)\right)\mathrm{d}r\right\}.$$

Then, for any $s\ge 0$, we have by the connections among different kinds of time-changed integrals (see the Lemma 5.1)

$$f\left(s\right)\le k\left(s\right)\exp \left\{{\int }_0^{s}b\left(r\right)\mathrm{d}r+{\int }_0^{E_s}\sigma \left(D(r-)\right)\mathrm{d}r\right\}=k\left(s\right)\exp \left\{{\int }_0^{s}b\left(r\right)\mathrm{d}r+{\int }_0^s\sigma \left(r\right)\mathrm{d}{E}_r\right\}.$$

The proof is complete. □

3.2. Interacting Particle Systems

Denote

$${\Omega }_R^i:=\left\{\omega \in \Omega :\underset{0\le t\le T}{sup}|X_t^i|\vee \underset{0\le t\le T}{sup}|X_t^{i,N}|\ge R\right\}.$$

By Chebyshev’s inequality,

$$\begin{array}{c}\hfill \mathbb{P}\left({\Omega }_R^i\right)\le \left(\mathbb{E}\left[\underset{0\le t\le T}{sup}{|X_t^i|}^q\right]\vee \mathbb{E}\left[\underset{0\le t\le T}{sup}{|X_t^{i,N}|}^q\right]\right){\displaystyle \frac{1}{R^q}}\le {\displaystyle \frac{C_q}{R^q}}.\end{array}$$

(34)

Lemma 4.

(The weighted Young inequality) For any $a,b>0$, there exists a constant $\theta \in (0,1)$ such that

$$\begin{array}{cc}\hfill {(a+b)}^p& \le {\left(\theta \right)}^{1-p}{a}^p+{(1-\theta )}^{1-p}{b}^p.\hfill \end{array}$$

Proposition 2.

Suppose that Assumption 1 holds. Let ${\left\{X_t^{i,N}\right\}}_{t\in [0,T]}$ be the solution to (10) and ${\left\{X_t^i\right\}}_{t\in [0,T]}$ be the solution to (11) with ${\mathbb{E}}_B{|X_0^i|}^p<\infty$. Then for any $p>0$, there exists a constant $C_p$ such that

$$\begin{array}{c}\hfill \mathbb{E}\left[\underset{0\le t\le T}{sup}{|X_t^i|}^p\right]\le C_p,\mathbb{E}\left[\underset{0\le t\le T}{sup}{|X_t^{i,N}|}^p\right]\le C_p,\end{array}$$

where $C_p$ is a positive constant independent of N.

Proof. 

For $p\ge 2$, by using the time-changed Itô’s formula to ${[1+|x|}^2{]}^{p/2}$ yields

$$\begin{array}{cc}\hfill & [1+|X_r^i-G\left(X_r^i\right){{|}^2]}^{{\displaystyle \frac{p}{2}}}\hfill \\ \hfill & =[1+|X_0^i-G\left(X_0^i\right){{|}^2]}^{{\displaystyle \frac{p}{2}}}\hfill \\ & +p{\int }_0^r[1+|X_s^i-G\left(X_s^i\right){{|}^2]}^{{\displaystyle \frac{p-2}{2}}}(X_s^i-G\left(X_s^i\right))f\left(E_s\right)\mathrm{d}s\hfill \\ & +p{\int }_0^r[1+|X_s^i-G\left(X_s^i\right){{|}^2]}^{{\displaystyle \frac{p-2}{2}}}(X_s^i-G\left(X_s^i\right))h(E_s,X_s^i,{\mu }_s^i)\mathrm{d}{E}_s\hfill \\ & +{\displaystyle \frac{p}{2}}{\int }_0^r[1+|X_s^i-G\left(X_s^i\right){|}^2{]}^{{\displaystyle \frac{p-2}{2}}}{|g(E_s,X_s^i,{\mu }_s^i)|}^2\mathrm{d}{E}_s\hfill \\ & +{\displaystyle \frac{p(p-2)}{2}}{\int }_0^r[1+|X_s^i-G\left(X_s^i\right){|}^2{]}^{{\displaystyle \frac{p-4}{2}}}|X_s^i-G\left(X_s^i\right){|}^2{|g(E_s,X_s^i,{\mu }_s^i)|}^2\mathrm{d}{E}_s\hfill \\ & +p{\int }_0^r[1+|X_s^i-G\left(X_s^i\right){{|}^2]}^{{\displaystyle \frac{p-2}{2}}}〈X_s^i-G\left(X_s^i\right),g(E_s,X_s^i,{\mu }_s^i)〉\mathrm{d}{B}_{E_s}^i\hfill \\ \hfill & =:J_1\left(r\right)+J_2\left(r\right)+J_3\left(r\right)+J_4\left(r\right)+J_5\left(r\right)+J_6\left(r\right).\hfill \end{array}$$

(35)

For every integer $m\ge 1$, define the stopping time ${\tau }_m=inf\{t\ge 0:|X_t^i|\ge m\}\wedge T$. We will estimate $J_1\left(r\right)$, $J_2\left(r\right)$, $J_3\left(r\right)$, $J_4\left(r\right)$, $J_5\left(r\right)$ and $J_6\left(r\right)$ one by one. For $J_1\left(r\right)$, by using (23) and Lemma 4 with $\theta =1/2$, we obtain that for any $0\le r\le t\le T$

$$\begin{array}{cc}\hfill J_1\left(r\right)& \le {\mathbb{E}}_B[1+2|X_0^i{|}^2+2|G\left(X_0^i\right){{|}^2]}^{{\displaystyle \frac{p}{2}}}\hfill \\ \hfill & \le {\mathbb{E}}_B[1+2(1+v^2)|X_0^i{{|}^2]}^{{\displaystyle \frac{p}{2}}}\hfill \\ \hfill & \le C+C{\mathbb{E}}_B{|X_0^i|}^p\hfill \\ \hfill & \le C.\hfill \end{array}$$

For $J_2\left(r\right)$, applying (23) and (22) and the Fubini theorem, we have

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le r\le t\wedge {\tau }_m}{sup}{J}_2\left(r\right)\right]& \le {\displaystyle \frac{p}{2}}{\mathbb{E}}_B{\int }_0^{t\wedge {\tau }_m}[1+|X_s^i-G\left(X_s^i\right){|}^2{]}^{{\displaystyle \frac{p-2}{2}}}(|X_s^i-G\left(X_s^i\right){|}^2+|f\left(E_s\right){|}^2)\mathrm{d}s\hfill \\ \hfill & \le {\displaystyle \frac{p}{2}}{\mathbb{E}}_B{\int }_0^{t\wedge {\tau }_m}[1+|X_s^i-G\left(X_s^i\right){|}^2{]}^{{\displaystyle \frac{p-2}{2}}}(|X_s^i-G\left(X_s^i\right){|}^2+K_f^2|E_T{|}^{2\gamma })\mathrm{d}s\hfill \\ \hfill & \le C{\mathbb{E}}_B{\int }_0^{t\wedge {\tau }_m}[1+|X_s^i-G\left(X_s^i\right){|}^2{]}^{{\displaystyle \frac{p-2}{2}}}(1+|X_s^i-G\left(X_s^i\right){|}^2)\mathrm{d}s\hfill \\ \hfill & \le C{\mathbb{E}}_B{\int }_0^{t\wedge {\tau }_m}[1+|X_s^i-G\left(X_s^i\right){{|}^2]}^{{\displaystyle \frac{p}{2}}}\mathrm{d}s\hfill \\ \hfill & \le C{\mathbb{E}}_B{\int }_0^{t\wedge {\tau }_m}[1+(2+2v^2)|X_s^i{{|}^2]}^{{\displaystyle \frac{p}{2}}}\mathrm{d}s\hfill \\ \hfill & \le C{\int }_0^{t\wedge {\tau }_m}{\mathbb{E}}_B[1+|X_s^i{{|}^2]}^{{\displaystyle \frac{p}{2}}}\mathrm{d}s.\hfill \end{array}$$

For $J_3\left(r\right)$, applying (23) and (27), the Young inequality and Fubini theorem, we can get

$$\begin{array}{cc}\hfill & {\mathbb{E}}_B\left[\underset{0\le r\le t\wedge {\tau }_m}{sup}{J}_3\left(r\right)\right]\hfill \\ \hfill & \le p{\mathbb{E}}_B{\int }_0^{t\wedge {\tau }_m}[1+(2+2v^2)|X_s^i{|}^2{]}^{{\displaystyle \frac{p-2}{2}}}(1+v)L_1(1+|X_s^i{|}^2+{\mathbb{W}}_1^2({\mu }_s^i,{\delta }_0))\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\mathbb{E}}_B{\int }_0^{t\wedge {\tau }_m}[1+|X_s^i{|}^2{]}^{{\displaystyle \frac{p-2}{2}}}(1+|X_s^i{|}^2)\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\int }_0^{t\wedge {\tau }_m}{\mathbb{E}}_B[1+|X_s^i{{|}^2]}^{{\displaystyle \frac{p}{2}}}\mathrm{d}{E}_s.\hfill \end{array}$$

For $J_4\left(r\right)$, applying (23) and (29), the Young inequality and Fubini theorem, we can get

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le r\le t\wedge {\tau }_m}{sup}{J}_4\left(r\right)\right]& \le {\displaystyle \frac{p}{2}}{\mathbb{E}}_B{\int }_0^{t\wedge {\tau }_m}[1+(2+2v^2)|X_s^i{|}^2{]}^{{\displaystyle \frac{p-2}{2}}}{L}_g^2(1+|X_s^i{|}^2+{\mathbb{W}}_1^2({\mu }_s^i,{\delta }_0))\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\mathbb{E}}_B{\int }_0^{t\wedge {\tau }_m}[1+|X_s^i{|}^2{]}^{{\displaystyle \frac{p-2}{2}}}(1+|X_s^i{|}^2)\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\int }_0^{t\wedge {\tau }_m}{\mathbb{E}}_B[1+|X_s^i{{|}^2]}^{{\displaystyle \frac{p}{2}}}\mathrm{d}{E}_s.\hfill \end{array}$$

For $J_5\left(r\right)$, applying (23) and (29), the Young inequality and Fubini theorem, we get

$$\begin{array}{cc}\hfill & {\mathbb{E}}_B\left[\underset{0\le r\le t\wedge {\tau }_m}{sup}{J}_5\left(r\right)\right]\hfill \\ \hfill & \le C{\mathbb{E}}_B{\int }_0^{t\wedge {\tau }_m}[1+(2+2v^2)|X_s^i{|}^2{]}^{{\displaystyle \frac{p-4}{2}}}(2+2v^2)(1+|X_s^i{|}^2)L_g^2(1+|X_s^i{|}^2+{\mathbb{W}}_1^2({\mu }_s^i,{\delta }_0))\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\mathbb{E}}_B{\int }_0^{t\wedge {\tau }_m}[1+|X_s^i{|}^2{]}^{{\displaystyle \frac{p-4}{2}}}(1+|X_s^i{{|}^2)}^2\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\int }_0^{t\wedge {\tau }_m}{\mathbb{E}}_B[1+|X_s^i{{|}^2]}^{{\displaystyle \frac{p}{2}}}\mathrm{d}{E}_s.\hfill \end{array}$$

For $J_6\left(r\right)$, applying (23) and (29), the Young inequality, Fubini theorem and Burkhold-Davis-Gundy’s inequality, we have

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le r\le t\wedge {\tau }_m}{sup}{J}_6\left(r\right)\right]& \le C{\mathbb{E}}_B{\left[{\int }_0^{t\wedge {\tau }_m}[1+(2+2v^2)|X_s^i{|}^2{]}^{p-2}|X_s^i{|}^2{|g(E_s,X_s^i,{\mu }_s^i)|}^2\mathrm{d}{E}_s\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le C{\mathbb{E}}_B{\left[{\int }_0^{t\wedge {\tau }_m}[1+|X_s^i{|}^2{]}^{p-1}{L}_g^2(1+|X_s^i{|}^2+{\mathbb{W}}_1^2({\mu }_s^i,{\delta }_0))\mathrm{d}{E}_s\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le C{\mathbb{E}}_B{\left[{\int }_0^{t\wedge {\tau }_m}[1+|X_s^i{{|}^2]}^p\mathrm{d}{E}_s\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le C+C{\mathbb{E}}_B\left[\underset{0\le r\le t\wedge {\tau }_m}{sup}{|X_r^i|}^p\right]+C{\mathbb{E}}_B{\int }_0^{t\wedge {\tau }_m}[1+|X_s^i{{|}^2]}^{{\displaystyle \frac{p}{2}}}\mathrm{d}{E}_s.\hfill \end{array}$$

From (23) and Lemma 4, we get

$$\begin{array}{cc}\hfill |X_t^i{|}^p& \le {\left({\displaystyle \frac{1}{1-v}}\right)}^{p-1}|X_t^i-G\left(X_t^i\right){|}^p+{\left({\displaystyle \frac{1}{v}}\right)}^{p-1}|G\left(X_t^i\right){|}^2\hfill \\ \hfill & \le {\left({\displaystyle \frac{1}{1-v}}\right)}^{p-1}|X_t^i-G\left(X_t^i\right){|}^p+v|X_t^i{|}^2.\hfill \end{array}$$

From $v\in (0,1)$, we have

$$\begin{array}{cc}\hfill |X_t^{i,\delta }{|}^p& \le C|X_t^{i,\delta }-G\left(X_t^{i,\delta }\right){|}^p.\hfill \end{array}$$

Taking ${\mathbb{E}}_D$ on both sides, by Proposition 1, we get

$$\begin{array}{cc}\hfill \mathbb{E}\left[\underset{0\le r\le t\wedge {\tau }_m}{sup}{|X_r^i|}^p\right]& \le \mathbb{E}\left[\underset{0\le r\le t\wedge {\tau }_m}{sup}{|X_r^i-G\left(X_r^i\right)|}^p\right]\hfill \\ \hfill & \le \mathbb{E}\left[\underset{0\le r\le t\wedge {\tau }_m}{sup}[1+|X_r^i-G\left(X_r^i\right){{|}^2]}^{{\displaystyle \frac{p}{2}}}\right]\hfill \\ \hfill & \le C+C{\int }_0^{t\wedge {\tau }_m}\mathbb{E}\left[\underset{0\le u\le s}{sup}{|X_u^n|}^p\right]\mathrm{d}s+C{\int }_0^{t\wedge {\tau }_m}\mathbb{E}\left[\underset{0\le u\le s}{sup}{|X_u^n|}^p\right]\mathrm{d}{E}_s.\hfill \end{array}$$

(36)

By using (36), we have

$$\begin{array}{c}\hfill m^p\mathbb{P}({\tau }_m\le t)=\mathbb{E}\left(|X_{{\tau }_m}^i{|}^p{I}_{\{{\tau }_m\le t\}}\right)\le \mathbb{E}{|X_{t\wedge {\tau }_m}^i|}^p\le C_p,\end{array}$$

which implies the series

$$\begin{array}{c}\hfill \sum _{m=1}^{\infty }\mathbb{P}({\tau }_m\le t)\le \sum _{m=1}^{\infty }{m}^{-p}{C}_p<\infty .\end{array}$$

According to the Borel–Cantelli lemma, we see that for almost all $\omega \in \Omega$, there exists an integer $m_0$ such that if $m\ge m_0$, ${\tau }_m>t$. By the arbitrariness of t, we obtain $\underset{m\to \infty }{lim}{\tau }_m=T$, a.s. By choosing $t=T$ and letting $m\to \infty$ on both sides (36), applying Fatou’s lemma, we obtain that

$$\begin{array}{c}\hfill \mathbb{E}\left[\underset{0\le r\le T}{sup}{|X_r^i|}^p\right]=\mathbb{E}\left[\underset{m\to \infty }{lim}\underset{0\le r\le T\wedge {\tau }_m}{sup}{|X_r^i|}^p\right]\le \underset{m\to \infty }{lim}\mathbb{E}\left[\underset{0\le r\le T\wedge {\tau }_m}{sup}{|X_r^i|}^p\right]\le C_p.\end{array}$$

For $p\in (0,2)$, by using the Lyapunov inequality, we can obtain that for any $0<p<q$,

$$\begin{array}{c}\hfill {\left[\mathbb{E}\left[\underset{0\le r\le T}{sup}{|X_r^i|}^p\right]\right]}^{{\displaystyle \frac{1}{p}}}\le {\left[\mathbb{E}\left[\underset{0\le r\le T}{sup}{|X_r^i|}^q\right]\right]}^{{\displaystyle \frac{1}{q}}}\le {\left(C_q\right)}^{{\displaystyle \frac{1}{q}}}.\end{array}$$

In summary, for any $p>0$, there exist a positive constant $C_p$ which is independent of n, such that

$$\begin{array}{c}\hfill \mathbb{E}\left[\underset{0\le t\le T}{sup}{|X_t^i|}^p\right]\le C_p.\end{array}$$

The proof establishing uniform boundedness of $\mathbb{E}|X_t^{i,N}{|}^2$ follows analogously to that for $X_t^i$, and is therefore omitted. This completes the proof. □

The next Theorem shows the propagation of chaos.

Theorem 1.

Suppose that Assumption 1 holds. Let ${\left\{X_t^{i,N}\right\}}_{t\in [0,T]}$ be the solution to (10) and ${\left\{X_t^i\right\}}_{t\in [0,T]}$ be the solution to (11) with $\mathbb{E}|X_0^i{|}^p<\infty$, for any p. Then, for any $T>0$,

$$\begin{array}{c}\hfill \underset{N\to \infty }{lim}\underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le s\le T}{sup}{|X_s^i-X_s^{i,N}|}^2\right]=0.\end{array}$$

(37)

If we further assume that h satisfies one-side global Lipschitz condition, then there exists a positive constant C such that

$$\begin{array}{c}\hfill \underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le s\le T}{sup}{|X_s^i-X_s^{i,N}|}^2\right]\le C{N}^{-{\displaystyle \frac{1}{2}}},\end{array}$$

(38)

with C may be dependent on $v,K_1,K_g,K_R,L_1,L_h,L_g,T,E_T$, but independent of N and R.

Proof. 

For every fixed $1\le i\le N$, define $Z_t^{i,N}:=[(X_t^i-G\left(X_t^i\right))-(X_t^{i,N}-G\left(X_t^{i,N}\right))]$. By using the time-changed Itô’s formula to $|Z_t^{i,N}{|}^2$, we obtain that for any $t\in [0,T]$,

$$\begin{array}{cc}\hfill |Z_t^{i,N}{|}^2& =2{\int }_0^t{Z}_s^{i,N}(h(E_s,X_s^i,{\mu }_s^i)-h(E_s,X_s^{i,N},{\mu }_s^{X,N}))\mathrm{d}{E}_s\hfill \\ & +{\int }_0^t|g(E_s,X_s^i,{\mu }_s^i)-g(E_s,X_s^{i,N},{\mu }_s^{X,N}){|}^2\mathrm{d}{E}_s\hfill \\ & +2{\int }_0^t{Z}_s^{i,N}(g(E_s,X_s^i,{\mu }_s^i)-g(E_s,X_s^{i,N},{\mu }_s^{X,N}))\mathrm{d}{B}_{E_s}^i\hfill \\ \hfill & =:I_1\left(t\right)+I_2\left(t\right)+I_3\left(t\right).\hfill \end{array}$$

(39)

Note, for any $0\le s\le t\le T$,

$$\begin{array}{c}\hfill |Z_s^{i,N}|\le |X_s^i-X_s^{i,N}|+|G\left(X_s^i\right)-G\left(X_s^{i,N}\right)|\le (1+v)|X_s^i-X_s^{i,N}|=:(1+v)|Y_s^{i,N}|.\end{array}$$

(40)

For $I_1\left(t\right)$, we obtain that for any $t\in [0,T]$,

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}{I}_1\left(t\right)\right]& \le 2{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}{\int }_0^t{Z}_s^{i,N}(h(E_s,X_s^i,{\mu }_s^i)-h(E_s,X_s^i,{\mu }_s^{X,N}))\mathrm{d}{E}_s\right]\hfill \\ \hfill & +2{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}{\int }_0^t{Z}_s^{i,N}(h(E_s,X_s^i,{\mu }_s^{X,N})-h(E_s,X_s^{i,N},{\mu }_s^{X,N}))I_{\Omega /{\Omega }_R^i}\mathrm{d}{E}_s\right]\hfill \\ \hfill & +2{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}{\int }_0^t{Z}_s^{i,N}(h(E_s,X_s^i,{\mu }_s^{X,N})-h(E_s,X_s^{i,N},{\mu }_s^{X,N}))I_{{\Omega }_R^i}\mathrm{d}{E}_s\right]\hfill \\ \hfill & \le C{\mathbb{E}}_B{\int }_0^T|Y_s^{i,N}||h(E_s,X_s^i,{\mu }_s^i)-h(E_s,X_s^i,{\mu }_s^{X,N})|\mathrm{d}{E}_s\hfill \\ \hfill & +C{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}{\int }_0^t{Y}_s^{i,N}(h(E_s,X_s^i,{\mu }_s^{X,N})-h(E_s,X_s^{i,N},{\mu }_s^{X,N}))I_{\Omega /{\Omega }_R^i}\mathrm{d}{E}_s\right]\hfill \\ \hfill & +C{\mathbb{E}}_B{\int }_0^T|Y_s^{i,N}|\left(|h(E_s,X_s^i,{\mu }_s^{X,N})|+|h(E_s,X_s^{i,N},{\mu }_s^{X,N})|\right)I_{{\Omega }_R^i}\mathrm{d}{E}_s]\hfill \\ \hfill & =:I_{11}+I_{12}+I_{13}.\hfill \end{array}$$

(41)

By using (14) and (26), the elementary inequality and the Fubini theorem, we have

$$\begin{array}{cc}\hfill I_{11}& \le C{\int }_0^T{\mathbb{E}}_B|Y_s^{i,N}{|}^2\mathrm{d}{E}_s+C{\int }_0^T{\mathbb{E}}_B|h(E_s,X_s^i,{\mu }_s^i)-h(E_s,X_s^i,{\mu }_s^{X,N}){|}^2\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\int }_0^T{\mathbb{E}}_B|Y_s^{i,N}{|}^2\mathrm{d}{E}_s+C{K}_1^2{\int }_0^T{\mathbb{E}}_B{\mathbb{W}}_1^2({\mu }_s^{X,N},{\mu }_s^i)\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\int }_0^T{\mathbb{E}}_B|Y_s^{i,N}{|}^2\mathrm{d}{E}_s+C{K}_1^2{\int }_0^T{\mathbb{E}}_B|Y_s^{i,N}{|}^2+C{N}^{-{\displaystyle \frac{1}{2}}}\mathrm{d}{E}_s\hfill \\ \hfill & \le C{N}^{-{\displaystyle \frac{1}{2}}}+C{\int }_0^T{\mathbb{E}}_B|Y_s^{i,N}{|}^2\mathrm{d}{E}_s.\hfill \end{array}$$

(42)

By using (24) and the Fubini theorem, we get

$$\begin{array}{c}\hfill I_{12}\le C{K}_R{\int }_0^T{\mathbb{E}}_B|X_s^i-X_s^{i,N}{|}^2\mathrm{d}{E}_s.\end{array}$$

(43)

By using (28), the elementary inequality, the Cauchy-Schwarz inequality and the Fubini theorem, we have

$$\begin{array}{cc}\hfill I_{13}& \le C{\mathbb{E}}_B{\int }_0^T|Y_s^{i,N}|\left(|h(E_s,X_s^i,{\mu }_s^{X,N})|+|h(E_s,X_s^{i,N},{\mu }_s^{X,N})|\right)I_{{\Omega }_R^i}\mathrm{d}{E}_s]\hfill \\ \hfill & \le C{\int }_0^T{L}_h{\mathbb{E}}_B\left[2+|X_s^i{|}^l+{|X_s^{i,N}|}^l+2{\mathbb{W}}_1({\mu }_s^{X,N},{\delta }_0)\right]I_{{\Omega }_R^i}\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\int }_0^T{\left[{\mathbb{E}}_B{\left[2+|X_s^i{|}^l+{|X_s^{i,N}|}^l+2{\mathbb{W}}_1({\mu }_s^{X,N},{\delta }_0)\right]}^2\right]}^{{\displaystyle \frac{1}{2}}}{\left[{\mathbb{E}}_B{I}_{{\Omega }_R^i}^2\right]}^{{\displaystyle \frac{1}{2}}}\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\int }_0^T{\left[20+5{\mathbb{E}}_B|X_s^i{|}^{2l}+5{\mathbb{E}}_B{|X_s^{i,N}|}^{2l}+20{\mathbb{E}}_B{\mathbb{W}}_1^2({\mu }_s^{X,N},{\delta }_0)\right]}^{{\displaystyle \frac{1}{2}}}{\left[{\mathbb{E}}_B{I}_{{\Omega }_R^i}\right]}^{{\displaystyle \frac{1}{2}}}\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\left[\mathbb{P}\left({\Omega }_R^i\right)\right]}^{{\displaystyle \frac{1}{2}}}\hfill \end{array}$$

(44)

Hence, by using (41)–(44), we obtain

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}{I}_1\left(t\right)\right]& \le (C+C{K}_R){\int }_0^T{\mathbb{E}}_B|Y_s^{i,N}|\mathrm{d}{E}_s+C{N}^{-{\displaystyle \frac{1}{2}}}+C{\left[\mathbb{P}\left({\Omega }_R^i\right)\right]}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

(45)

For $I_2\left(t\right)$, By using (14), (25), (26) and the Fubini theorem, we get

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}{I}_2\left(t\right)\right]& \le C{\mathbb{E}}_B{\int }_0^T|g(E_s,X_s^i,{\mu }_s^i)-g(E_s,X_s^i,{\mu }_s^{X,N}){|}^2\mathrm{d}{E}_s\hfill \\ \hfill & +C{\mathbb{E}}_B{\int }_0^T|g(E_s,X_s^i,{\mu }_s^{X,N})-g(E_s,X_s^{i,N},{\mu }_s^{X,N}){|}^2\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\mathbb{E}}_B{\int }_0^T{K}_1^2{\mathbb{E}}_B{\mathbb{W}}_1^2({\mu }_s^{X,N},{\mu }_s^i)\mathrm{d}{E}_s+C{\mathbb{E}}_B{\int }_0^T{K}_g^2|X_s^i-X_s^{i,N}{|}^2\mathrm{d}{E}_s\hfill \\ \hfill & \le C{N}^{-{\displaystyle \frac{1}{2}}}+C{\int }_0^T{\mathbb{E}}_B|X_s^i-X_s^{i,N}{|}^2\mathrm{d}{E}_s.\hfill \end{array}$$

(46)

For $I_3\left(t\right)$, By using (46), the Burkholder–Davis–Gundy inequality and the Fubini theorem, we have

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}{I}_3\left(t\right)\right]& \le C{\mathbb{E}}_B{\left[\underset{0\le t\le T}{sup}{\int }_0^t|Z_s^{i,N}{|}^2|g(E_s,X_s^i,{\mu }_s^i)-g(E_s,X_s^{i,N},{\mu }_s^{X,N}){|}^2\mathrm{d}{E}_s\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le C{\mathbb{E}}_B{\left[{\int }_0^T|Z_s^{i,N}{|}^2|g(E_s,X_s^i,{\mu }_s^i)-g(E_s,X_s^{i,N},{\mu }_s^{X,N}){|}^2\mathrm{d}{E}_s\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|Z_s^{i,N}{|}^2\right]+C{\mathbb{E}}_B{\int }_0^T|g(E_s,X_s^i,{\mu }_s^i)-g(E_s,X_s^{i,N},{\mu }_s^{X,N}){|}^2\mathrm{d}{E}_s\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|Z_s^{i,N}{|}^2\right]+C{N}^{-{\displaystyle \frac{1}{2}}}+C{\int }_0^T{\mathbb{E}}_B|X_s^i-X_s^{i,N}{|}^2\mathrm{d}{E}_s.\hfill \end{array}$$

(47)

Hence, by using (45), (46) and (47), we obtain

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}{|Z_t^{i,N}|}^2\right]& \le (C+C{K}_R){\int }_0^T{\mathbb{E}}_B|Y_s^{i,N}|\mathrm{d}{E}_s+C{N}^{-{\displaystyle \frac{1}{2}}}+C{\left[\mathbb{P}\left({\Omega }_R^i\right)\right]}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

(48)

From (23) and the weighted Young inequality, we have

$$\begin{array}{c}\hfill |X_t^i-X_t^{i,N}{|}^2\le {\displaystyle \frac{1}{1-v}}|Z_t^{i,N}{|}^2+{\displaystyle \frac{1}{v}}|G\left(X_t^i\right)-G\left(X_t^{i,N}\right){|}^2\le {\displaystyle \frac{1}{1-v}}|Z_t^{i,N}{|}^2+v|X_t^i-X_t^{i,N}{|}^2.\end{array}$$

Taking ${\mathbb{E}}_D$ on both sides, by Proposition 1 and (34), we get

$$\begin{array}{c}\hfill \mathbb{E}\left[\underset{0\le t\le T}{sup}|X_t^i-X_t^{i,N}{|}^2\right]\le \mathbb{E}\left[\underset{0\le t\le T}{sup}{|Z_t^{i,N}|}^2\right]\le \left(C{N}^{-{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{C}{R^{q/2}}}\right)e^{(C+C{K}_R)E_T}.\end{array}$$

(49)

Note that $K_R\le alogR$. Choosing $q>4aC{E}_T$. Let $R\to \infty$ and $N\to \infty$, we have

$$\begin{array}{cc}\hfill \underset{R\to \infty }{lim}\underset{N\to \infty }{lim}\underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le t\le T}{sup}|X_t^i-X_t^{i,N}{|}^2\right]& \le \underset{R\to \infty }{lim}\underset{N\to \infty }{lim}\left(C{N}^{-{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{C}{R^{q/2}}}\right)C{R}^{aC{E}_T}\hfill \\ \hfill & \le \underset{R\to \infty }{lim}C{R}^{-aC{E}_T}\hfill \\ \hfill & =0.\hfill \end{array}$$

Moreover, if $K_R$ is independent of R, in other words, h satisfies one-side global Lipschitz condition. From (49), we have

$$\begin{array}{c}\hfill \underset{R\to \infty }{lim}\underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le t\le T}{sup}|X_t^i-X_t^{i,N}{|}^2\right]\le \underset{R\to \infty }{lim}\left(C{N}^{-{\displaystyle \frac{1}{2}}}+{\displaystyle \frac{C}{R^{q/2}}}\right)C\le C{N}^{-{\displaystyle \frac{1}{2}}},\end{array}$$

where $C>0$ may be dependent on $v,K_1,K_g,K_R,L_1,L_h,L_g,T,E_T$, but independent of N and R. The proof is complete. □

3.3. Tamed Euler–Maruyama Method

Proposition 3.

Suppose that Assumption 1 holds and ${\left\{X_t^{i,\delta }\right\}}_{t\in [0,T]}$ is the solution to (21) with $\mathbb{E}|X_0^i{|}^2<\infty$. Then, for some positive constant C independent of N and δ,

$$\begin{array}{c}\hfill \underset{\delta >0}{sup}\underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le t\le T}{sup}{|X_t^{i,\delta }|}^2\right]\le C.\end{array}$$

(50)

Proof. 

For every fixed i, applying Itô formula again yields that

$$\begin{array}{cc}\hfill |X_t^{i,\delta }-G\left(X_t^{i,\delta }\right){|}^2& =|X_0^i-G\left(X_0^i\right){|}^2+2{\int }_0^t(X_s^{i,\delta }-G\left(X_s^{i,\delta }\right))f\left(E_{{\tau }_{n_s}^{\delta }}\right)\mathrm{d}s\hfill \\ \hfill & +2{\int }_0^t(X_s^{i,\delta }-G\left(X_s^{i,\delta }\right))h_{\delta }(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })\mathrm{d}{E}_s\hfill \\ \hfill & +{\int }_0^t|g(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }){|}^2\mathrm{d}{E}_s\hfill \\ \hfill & +2{\int }_0^t(X_s^{i,\delta }-G\left(X_s^{i,\delta }\right))g(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })\mathrm{d}{B}_{E_s}^i\hfill \\ \hfill & =:A_1\left(t\right)+A_2\left(t\right)+A_3\left(t\right)+A_4\left(t\right)+A_5\left(t\right).\hfill \end{array}$$

(51)

We will estimate $A_1\left(t\right)$, $A_2\left(t\right)$, $A_3\left(t\right)$, $A_4\left(t\right)$ and $A_5\left(t\right)$ one by one. For $A_1\left(t\right)$, from (23), we have

$$\begin{array}{c}\hfill {\mathbb{E}}_B\left[A_1\left(t\right)\right]\le 2{\mathbb{E}}_B|X_0^i{|}^2+2{\mathbb{E}}_B|G\left(X_0^i\right){|}^2\le 2{\mathbb{E}}_B|X_0^i{|}^2+2v^2{\mathbb{E}}_B|X_0^i{|}^2\le C.\end{array}$$

For $A_2\left(t\right)$, from (23) and (22) and the Fubini theorem, we get

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[A_2\left(t\right)\right]\le & {\mathbb{E}}_B{\int }_0^t|X_s^{i,\delta }-G\left(X_s^{i,\delta }\right){|}^2+|f\left(E_{{\tau }_{n_s}^{\delta }}\right){|}^2\mathrm{d}s\hfill \\ \hfill \le & {\mathbb{E}}_B{\int }_0^t(2+2v^2)|X_s^{i,\delta }{|}^2+K_f^2|E_{{\tau }_{n_s}^{\delta }}{|}^{2\gamma }\mathrm{d}s\hfill \\ \hfill \le & C+C{\int }_0^t{\mathbb{E}}_B|X_s^{i,\delta }{|}^2\mathrm{d}s.\hfill \end{array}$$

For $A_3\left(t\right)$, from (19), (23), (27) and the Fubini theorem, we have

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[A_3\left(t\right)\right]\le & 2{\mathbb{E}}_B{\int }_0^t(X_{{\tau }_{n_s}^{\delta }}^{i,\delta }-G\left(X_{{\tau }_{n_s}^{\delta }}^{i,\delta }\right))h_{\delta }(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })\mathrm{d}{E}_s\hfill \\ \hfill & +2{\mathbb{E}}_B{\int }_0^t(X_s^{i,\delta }-X_{{\tau }_{n_s}^{\delta }}^{i,\delta }-G\left(X_s^{i,\delta }\right)+G\left(X_{{\tau }_{n_s}^{\delta }}^{i,\delta }\right))h_{\delta }(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })\mathrm{d}{E}_s\hfill \\ \hfill \le & 2(1+v)L_1{\int }_0^t{\mathbb{E}}_B\left(1+|X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^2+{\mathbb{W}}_1^2({\mu }_{{\tau }_{n_s}^{\delta }}^{\delta },{\delta }_0)\right)\mathrm{d}{E}_s\hfill \\ \hfill & +2{\int }_0^t|X_s^{i,\delta }-X_{{\tau }_{n_s}^{\delta }}^{i,\delta }-G\left(X_s^{i,\delta }\right)+G\left(X_{{\tau }_{n_s}^{\delta }}^{i,\delta }\right)||h_{\delta }(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })|\mathrm{d}{E}_s\hfill \\ \hfill \le & C+C{\int }_0^t{\mathbb{E}}_B\underset{0\le r\le s}{sup}|X_s^{i,\delta }{|}^2\mathrm{d}{E}_s+2{\mathbb{E}}_B{\int }_0^t{\int }_{{\tau }_{n_s}^{\delta }}^{s}f\left(E_{{\tau }_{n_s}^{\delta }}\right)\mathrm{d}s{\delta }^{-{\displaystyle \frac{\gamma }{2}}}\mathrm{d}{E}_s\hfill \\ \hfill & +2{\mathbb{E}}_B{\int }_0^t{\int }_{{\tau }_{n_s}^{\delta }}^s|h_{\delta }(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })|\mathrm{d}{E}_s{\delta }^{-{\displaystyle \frac{\gamma }{2}}}\mathrm{d}{E}_s\hfill \\ \hfill & +2{\mathbb{E}}_B{\int }_0^t{\int }_{{\tau }_{n_s}^{\delta }}^s|g(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })|\mathrm{d}{B}_{E_s}^i{\delta }^{-{\displaystyle \frac{\gamma }{2}}}\mathrm{d}{E}_s\hfill \\ \hfill \le & C+C{\int }_0^t{\mathbb{E}}_B\underset{0\le r\le s}{sup}|X_s^{i,\delta }{|}^2\mathrm{d}{E}_s+2{\delta }^{-{\displaystyle \frac{\gamma }{2}}}{K}_f{|E_T|}^{\gamma }{\mathbb{E}}_B{\int }_0^t(s-{\tau }_{n_s}^{\delta })\mathrm{d}{E}_s\hfill \\ \hfill & +2{\mathbb{E}}_B{\int }_0^t\delta {\delta }^{-{\displaystyle \frac{\gamma }{2}}}{\delta }^{-{\displaystyle \frac{\gamma }{2}}}\mathrm{d}{E}_s\hfill \\ \hfill \le & C+C{\int }_0^t{\mathbb{E}}_B\underset{0\le r\le s}{sup}|X_s^{i,\delta }{|}^2\mathrm{d}{E}_s+2K_f{|E_T|}^{\gamma }T{\delta }^{1-{\displaystyle \frac{\gamma }{2}}}+2E_T{\delta }^{1-\gamma }\hfill \\ \hfill \le & C+C{\int }_0^t{\mathbb{E}}_B\underset{0\le r\le s}{sup}|X_s^{i,\delta }{|}^2\mathrm{d}{E}_s.\hfill \end{array}$$

In the present work, we employ the technical approach outlined in Theorem 4.1 of reference [22]:

$$\begin{array}{cc}\hfill C{\int }_0^t(s-{\tau }_{n_s}^{\delta }){\delta }^{-{\displaystyle \frac{\gamma }{2}}}\mathrm{d}{E}_s& \le C{\delta }^{-{\displaystyle \frac{\gamma }{2}}}\sum _{i=0}^{n_t-1}{\int }_{{\tau }_i^{\delta }}^{{\tau }_{i+1}^{\delta }}(s-{\tau }_i^{\delta })d{E}_s+C{\delta }^{-{\displaystyle \frac{\gamma }{2}}}{\int }_{{\tau }_{n_t}}^t(s-{\tau }_{n_t}^{\delta })d{E}_s\hfill \\ \hfill & \le C{\delta }^{-{\displaystyle \frac{\gamma }{2}}}\delta \left(\sum _{i=0}^{n_t-1}({\tau }_{i+1}^{\delta }-{\tau }_i^{\delta })+(t-{\tau }_{n_t}^{\delta })\right)\hfill \\ \hfill & \le CT{\delta }^{1-{\displaystyle \frac{\gamma }{2}}}.\hfill \end{array}$$

For $A_4\left(t\right)$, from (29) and the Fubini theorem, we have

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[A_4\left(t\right)\right]\le & {\mathbb{E}}_B{\int }_0^t|g(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }){|}^2\mathrm{d}{E}_s\hfill \\ \hfill \le & L_g^2{\mathbb{E}}_B{\int }_0^T\left(1+|X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^2+{\mathbb{W}}_1^2({\mu }_{{\tau }_{n_s}^{\delta }}^{\delta },{\delta }_0)\right)\mathrm{d}{E}_s\hfill \\ \hfill \le & C+C{\int }_0^t{\mathbb{E}}_B\underset{0\le r\le s}{sup}|X_s^{i,\delta }{|}^2\mathrm{d}{E}_s.\hfill \end{array}$$

For $A_5\left(t\right)$, from (23) and (29), Burkhold–Davis–Gundy’s inequality, and the Fubini theorem, we get

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}{A}_5\left(t\right)\right]\le & 2{\mathbb{E}}_B{\left[\underset{0\le t\le T}{sup}{\int }_0^t|X_s^{i,\delta }-G\left(X_s^{i,\delta }\right){|}^2|g(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }){|}^2\mathrm{d}{E}_s\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill \le & 2L_g{\mathbb{E}}_B{\left[{\int }_0^T(2+2v^2)|X_s^{i,\delta }{|}^2\left(1+|X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^2+{\mathbb{W}}_1^2({\mu }_{{\tau }_{n_s}^{\delta }}^{\delta },{\delta }_0)\right)\mathrm{d}{E}_s\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill \le & {\displaystyle \frac{1}{2}}{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|X_t^{i,\delta }{|}^2\right]+C{\int }_0^{T}1+{\mathbb{E}}_B\left[\underset{0\le r\le s}{sup}|X_s^{i,\delta }{|}^2\right]\mathrm{d}{E}_s.\hfill \end{array}$$

Hence, we have

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|X_t^{i,\delta }-G\left(X_t^{i,\delta }\right){|}^2\right]& \le C+C{\int }_0^t{\mathbb{E}}_B\left[\underset{0\le r\le s}{sup}|X_s^{i,\delta }{|}^2\right]\mathrm{d}s+C{\int }_0^t{\mathbb{E}}_B\left[]\underset{0\le r\le s}{sup}|X_s^{i,\delta }{|}^2\right]\mathrm{d}{E}_s\hfill \\ & +{\displaystyle \frac{1}{2}}{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|X_t^{i,\delta }{|}^2\right].\hfill \end{array}$$

(52)

From Lemma 4, we get

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|X_t^{i,\delta }{|}^2\right]& \le {\displaystyle \frac{1}{1-v}}{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|X_t^{i,\delta }-G\left(X_t^{i,\delta }\right){|}^2\right]+{\displaystyle \frac{1}{v}}{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|G\left(X_t^{i,\delta }\right){|}^2\right]\hfill \\ \hfill & \le {\displaystyle \frac{1}{1-v}}{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|X_t^{i,\delta }-G\left(X_t^{i,\delta }\right){|}^2\right]+v{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|X_t^{i,\delta }{|}^2\right].\hfill \end{array}$$

(53)

From (66) and (53), we have

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|X_t^{i,\delta }{|}^2\right]& \le C+C{\int }_0^t{\mathbb{E}}_B\left[\underset{0\le r\le s}{sup}|X_s^{i,\delta }{|}^2\right]\mathrm{d}s+C{\int }_0^t{\mathbb{E}}_B\left[\underset{0\le r\le s}{sup}|X_s^{i,\delta }{|}^2\right]\mathrm{d}{E}_s.\hfill \end{array}$$

(54)

From Proposition 1, take ${\mathbb{E}}_D$ on both side of (53), we get

$$\begin{array}{c}\hfill \mathbb{E}\left[\underset{0\le t\le T}{sup}|X_t^{i,\delta }{|}^2\right]\le C{e}^{CT+C{E}_T}\le C.\end{array}$$

where $C>0$ may be dependent on $\mathbb{E}|X_0^i{|}^2,v,K_f,L_1,L_g,T,E_T$, but independent of N and $\delta$.

Hence, we obtain that for any $t\in [0,T]$,

$$\begin{array}{c}\hfill \underset{\delta >0}{sup}\underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le t\le T}{sup}{|X_t^{i,\delta }|}^2\right]\le C.\end{array}$$

The proof is complete. □

Proposition 4.

Suppose that Assumption 1 holds and ${\left\{X_t^{i,\delta }\right\}}_{t\in [0,T]}$ is the solution to (21) with $\mathbb{E}|X_0^i{|}^2<\infty$. Then, for any $p\in (0,2]$, there exists a constant $C>0$, which is not depend on N and δ, such that for any $0\le s\le t\le T$,

$$\begin{array}{c}\hfill \underset{1\le i\le N}{sup}\underset{0\le s\le T}{sup}\mathbb{E}{|X_s^{i,\delta }-X_{{\tau }_{n_s}^{\delta }}^{i,\delta }|}^p\le C{(s-{\tau }_{n_s}^{\delta })}^p+C{\delta }^{{\displaystyle \frac{p}{2}}},\end{array}$$

(55)

$$\begin{array}{c}\hfill {\int }_0^t\mathbb{E}{|X_s^{i,\delta }-X_{{\tau }_{n_s}^{\delta }}^{i,\delta }|}^p\mathrm{d}{E}_s\le C{\delta }^{{\displaystyle \frac{p}{2}}},\end{array}$$

(56)

and

$$\begin{array}{c}\hfill \underset{1\le i\le N}{sup}\underset{0\le s\le T}{sup}\mathbb{E}|X_s^{i,\delta }-X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^p{|h_{\delta }(E_{{\tau }_{n_s}^{\delta }}^{\delta },X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })|}^p\le C.\end{array}$$

(57)

For any given $p>2$, suppose that the following condition is satisfied:

$$\begin{array}{c}\hfill \underset{\delta >0}{sup}\underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le s\le T}{sup}{\left|X_s^{i,\delta }\right|}^p\right]<\infty ,\end{array}$$

(58)

then the conclusions stated in (55) and (57) remain valid for this value of p.

Proof. 

For any $p>0$, applying the Burkhold–Davis–Gundy inequality and the Hölder inequality, we can get

$$\begin{array}{cc}\hfill & {\mathbb{E}}_B{|(X_s^{i,\delta }-G\left(X_s^{i,\delta }\right))-(X_{{\tau }_{n_s}^{\delta }}^{i,\delta }-G\left(X_{{\tau }_{n_s}^{\delta }}^{i,\delta }\right))|}^p\hfill \\ \hfill & \le 3^{p-1}{\mathbb{E}}_B|{\int }_{{\tau }_{n_s}^{\delta }}^{s}f\left(E_{{\tau }_{n_s}^{\delta }}\right)\mathrm{d}r{|}^p+3^{p-1}{\mathbb{E}}_B|{\int }_{{\tau }_{n_s}^{\delta }}^s{h}_{\delta }(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })\mathrm{d}{E}_r{|}^p\hfill \\ \hfill & +3^{p-1}{\mathbb{E}}_B|{\int }_{{\tau }_{n_s}^{\delta }}^{s}g(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })\mathrm{d}{B}_{E_r}^i{|}^p\hfill \\ \hfill & \le C{(s-{\tau }_{n_s}^{\delta })}^{p-1}{\mathbb{E}}_B{\int }_{{\tau }_{n_s}^{\delta }}^s|f\left(E_{{\tau }_{n_s}^{\delta }}\right){|}^p\mathrm{d}r+C{(E_s-E_{{\tau }_s^n})}^{p-1}{\mathbb{E}}_B{\int }_{{\tau }_{n_s}^{\delta }}^s|h_{\delta }(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }){|}^p\mathrm{d}{E}_r\hfill \\ \hfill & +C{\mathbb{E}}_B|{\int }_{{\tau }_{n_s}^{\delta }}^s|g(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }){|}^2\mathrm{d}{E}_r{|}^{{\displaystyle \frac{p}{2}}}\hfill \\ \hfill & \le C{(s-{\tau }_{n_s}^{\delta })}^p+C{\delta }^{p(1-{\displaystyle \frac{\gamma }{2}})}+C{\mathbb{E}}_B|{\int }_{{\tau }_{n_s}^{\delta }}^s|g(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }){|}^2\mathrm{d}{E}_r{|}^{{\displaystyle \frac{p}{2}}}.\hfill \end{array}$$

(59)

When $p=2$, it follows from Proposition 3 that

$$\begin{array}{cc}\hfill & {\mathbb{E}}_B{|(X_s^{i,\delta }-G\left(X_s^{i,\delta }\right))-(X_{{\tau }_{n_s}^{\delta }}^{i,\delta }-G\left(X_{{\tau }_{n_s}^{\delta }}^{i,\delta }\right))|}^2\hfill \\ \hfill & \le C{(s-{\tau }_{n_s}^{\delta })}^2+C{\delta }^{2-\gamma }+C{\mathbb{E}}_B{\int }_{{\tau }_{n_s}^{\delta }}^s|g(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }){|}^2\mathrm{d}{E}_r\hfill \\ \hfill & \le C{(s-{\tau }_{n_s}^{\delta })}^2+C{\delta }^{2-\gamma }+C{\int }_{{\tau }_{n_s}^{\delta }}^{s}1+{\mathbb{E}}_B|X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^2\mathrm{d}{E}_r\hfill \\ \hfill & \le C{(s-{\tau }_{n_s}^{\delta })}^2+C{\delta }^{2-\gamma }+C\delta \underset{{\tau }_{n_s}^{\delta }\le r\le s}{sup}(1+{\mathbb{E}}_B|X_r^{i,\delta }{|}^2)\hfill \\ \hfill & \le C{(s-{\tau }_{n_s}^{\delta })}^2+C\delta .\hfill \end{array}$$

(60)

From the weighted Young inequality, we get

$$\begin{array}{cc}\hfill |X_s^{i,\delta }-X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^2& \le {\displaystyle \frac{1}{1-v}}{|(X_s^{i,\delta }-G\left(X_s^{i,\delta }\right))-(X_{{\tau }_{n_s}^{\delta }}^{i,\delta }-G\left(X_{{\tau }_{n_s}^{\delta }}^{i,\delta }\right))|}^2+{\displaystyle \frac{1}{v}}|G\left(X_s^{i,\delta }\right)-G\left(X_{{\tau }_{n_s}^{\delta }}^{i,\delta }\right){|}^2\hfill \\ \hfill & \le {\displaystyle \frac{1}{1-v}}{|(X_s^{i,\delta }-G\left(X_s^{i,\delta }\right))-(X_{{\tau }_{n_s}^{\delta }}^{i,\delta }-G\left(X_{{\tau }_{n_s}^{\delta }}^{i,\delta }\right))|}^2+v|X_s^{i,\delta }-X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^2.\hfill \end{array}$$

(61)

Hence, for $p=2$, there exists a constant $C>0$, which is independent of N and $\delta$, such that

$$\begin{array}{cc}\hfill & \underset{1\le i\le N}{sup}\underset{0\le s\le T}{sup}\mathbb{E}{|X_s^{i,\delta }-X_{{\tau }_{n_s}^{\delta }}^{i,\delta }|}^2\le C{(s-{\tau }_{n_s}^{\delta })}^2+C\delta .\hfill \end{array}$$

Furthermore, we employ the technical approach outlined in Theorem 4.1 of reference [22]:

$$\begin{array}{cc}\hfill {\int }_0^t\mathbb{E}{|X_s^{i,\delta }-X_{{\tau }_{n_s}^{\delta }}^{i,\delta }|}^p\mathrm{d}{E}_s& \le {\int }_0^{t}C{(s-{\tau }_{n_s}^{\delta })}^2\mathrm{d}{E}_s+{\int }_0^{t}C\delta \mathrm{d}{E}_s\hfill \\ \hfill & \le C\sum _{i=0}^{n_t-1}{\int }_{{\tau }_i^{\delta }}^{{\tau }_{i+1}^{\delta }}{(s-{\tau }_i^{\delta })}^{2}d{E}_s+{\int }_{{\tau }_{n_t}}^t{(s-{\tau }_{n_t}^{\delta })}^{2}d{E}_s+C{E}_T\delta \hfill \\ \hfill & \le C\delta \left(\sum _{i=0}^{n_t-1}{({\tau }_{i+1}^{\delta }-{\tau }_i^{\delta })}^2+{(t-{\tau }_{n_t}^{\delta })}^2\right)+C\delta \hfill \\ \hfill & \le C\delta T\left(\sum _{i=0}^{n_t-1}({\tau }_{i+1}^{\delta }-{\tau }_i^{\delta })+(t-{\tau }_{n_t}^{\delta })\right)+C\delta \hfill \\ \hfill & \le C\delta T^2+C\delta \hfill \\ \hfill & \le C\delta .\hfill \end{array}$$

For $p\in (0,2]$, applying Lyapunov’s inequality, (55) and (56) follow. When $p>2$ and (58) holds, it follows from (59) that

$$\begin{array}{cc}\hfill \mathbb{E}|X_s^{i,\delta }-X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^p& \le C{(s-{\tau }_{n_s}^{\delta })}^p+C{\delta }^{q(1-{\displaystyle \frac{\gamma }{2}})}+C{\mathbb{E}}_B|{\int }_{{\tau }_{n_s}^{\delta }}^s|g(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }){|}^2\mathrm{d}{E}_r{|}^{{\displaystyle \frac{p}{2}}}\hfill \\ \hfill & \le C{(s-{\tau }_{n_s}^{\delta })}^p+C{\delta }^{q(1-{\displaystyle \frac{\gamma }{2}})}+C{\mathbb{E}}_B|\delta \underset{{\tau }_{n_s}^{\delta }\le r\le s}{sup}(1+{\mathbb{E}}_B|X_r^{i,\delta }{|}^2){|}^{{\displaystyle \frac{p}{2}}}\hfill \\ \hfill & \le C{(s-{\tau }_{n_s}^{\delta })}^p+C{\delta }^{p(1-{\displaystyle \frac{\gamma }{2}})}+C{\delta }^{{\displaystyle \frac{p}{2}}}\hfill \\ \hfill & \le C{(s-{\tau }_{n_s}^{\delta })}^p+C{\delta }^{{\displaystyle \frac{p}{2}}}.\hfill \end{array}$$

Furthermore, if (58) holds, since (19), we can easily derive that

$$\begin{array}{cc}\hfill \mathbb{E}|X_s^{i,\delta }-X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^p{\left|h_{\delta }(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })\right|}^p& \le (C{(s-{\tau }_{n_s}^{\delta })}^p+C{\delta }^{{\displaystyle \frac{p}{2}}}){\delta }_1^{-{\displaystyle \frac{p\gamma }{2}}}\hfill \\ \hfill & \le C{(s-{\tau }_{n_s}^{\delta })}^{p-{\displaystyle \frac{p\gamma }{2}}}+C{\delta }^{{\displaystyle \frac{p}{2}}-{\displaystyle \frac{p\gamma }{2}}}\hfill \\ \hfill & \le C.\hfill \end{array}$$

The proof is complete. □

Proposition 5.

Suppose that Assumption 1 holds and ${\left\{X_t^{i,\delta }\right\}}_{t\in [0,T]}$ is the solution to (21) with $\mathbb{E}|X_0^i{|}^q<\infty$. Then, there exists a constant $C>0$ such that for any $t\in [0,T]$,

$$\begin{array}{c}\hfill \underset{\delta >0}{sup}\underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le t\le T}{sup}{\left|X_t^{i,\delta }\right|}^p\right]\le C.\end{array}$$

(62)

where constant C is independent of N and δ.

Proof. 

As we have proved before, (62) holds for $p=2$ and (55), (57) holds for $p=2$; if we set $q=2p=4$, by (51) and Hölder’s inequality, we obtain

$$\begin{array}{cc}\hfill |X_t^{i,\delta }-G\left(X_t^{i,\delta }\right){|}^q& =\left[\right|X_0^i-G\left(X_0^i\right){|}^2+2{\int }_0^t(X_s^{i,\delta }-G\left(X_s^{i,\delta }\right))f\left(E_{{\tau }_{n_s}^{\delta }}\right)\mathrm{d}s\hfill \\ \hfill & +2{\int }_0^t(X_s^{i,\delta }-G\left(X_s^{i,\delta }\right))h_{\delta }(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })\mathrm{d}{E}_s\hfill \\ \hfill & +{\int }_0^t|g(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }){|}^2\mathrm{d}{E}_s\hfill \\ \hfill & +2{\int }_0^t(X_s^{i,\delta }-G\left(X_s^{i,\delta }\right))g(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })\mathrm{d}{B}_{E_s}^i{]}^{{\displaystyle \frac{q}{2}}}\hfill \\ \hfill & \le C|X_0^i{|}^q+C{\int }_0^t|X_s^{i,\delta }-G\left(X_s^{i,\delta }\right){|}^{{\displaystyle \frac{q}{2}}}|f\left(E_{{\tau }_{n_s}^{\delta }}\right){|}^{{\displaystyle \frac{q}{2}}}\mathrm{d}s\hfill \\ \hfill & +C{\int }_0^t(X_s^{i,\delta }-G\left(X_s^{i,\delta }\right))h_{\delta }(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }){|}^{{\displaystyle \frac{q}{2}}}\mathrm{d}{E}_s\hfill \\ \hfill & +C{\int }_0^t|g(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }){|}^q\mathrm{d}{E}_s\hfill \\ \hfill & +C{\left[{\int }_0^t|X_s^{i,\delta }-G\left(X_s^{i,\delta }\right)|\left|g(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })\right|\mathrm{d}{B}_{E_s}^i\right]}^{{\displaystyle \frac{q}{2}}}\hfill \\ \hfill & =:C|X_0^i{|}^q+O_1\left(t\right)+O_2\left(t\right)+O_3\left(t\right)+O_4\left(t\right).\hfill \end{array}$$

(63)

We will estimate $O_1\left(t\right)$, $O_2\left(t\right)$, $O_3\left(t\right)$ and $O_4\left(t\right)$ one by one. For $O_2\left(t\right)$, from (23), (22) and the Fubini theorem, we have

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[O_1\left(t\right)\right]\le & {\mathbb{E}}_B{\int }_0^t|X_s^{i,\delta }-G\left(X_s^{i,\delta }\right){|}^q+|f\left(E_{{\tau }_{n_s}^{\delta }}\right){|}^q\mathrm{d}s\hfill \\ \hfill \le & C{\mathbb{E}}_B{\int }_0^t|X_s^{i,\delta }{|}^q+K_f^q|E_{{\tau }_{n_s}^{\delta }}{|}^{q\gamma }\mathrm{d}s\hfill \\ \hfill \le & C+C{\int }_0^t{\mathbb{E}}_B|X_s^{i,\delta }{|}^q\mathrm{d}s.\hfill \end{array}$$

For $O_2\left(t\right)$, from (19), (23), (27) and the Fubini theorem, we get

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[O_2\left(t\right)\right]\le & C{\int }_0^t{\mathbb{E}}_B\left(1+|X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^q+{\mathbb{W}}_1^q({\mu }_{{\tau }_{n_s}^{\delta }}^{\delta },{\delta }_0)\right)\mathrm{d}{E}_s\hfill \\ \hfill & +C{\int }_0^t|X_s^{i,\delta }-X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^{{\displaystyle \frac{q}{2}}}|h_{\delta }(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }){|}^{{\displaystyle \frac{q}{2}}}\mathrm{d}{E}_s\hfill \\ \hfill \le & C+C{\int }_0^t{\mathbb{E}}_B\left(\underset{0\le r\le s}{sup}|X_s^{i,\delta }{|}^q\right)\mathrm{d}{E}_s.\hfill \end{array}$$

For $O_3\left(t\right)$, from (29) and the Fubini theorem, we have

$$\begin{array}{c}\hfill {\mathbb{E}}_B\left[O_3\left(t\right)\right]\le L_g^2{\mathbb{E}}_B{\int }_0^T\left(1+|X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^q+{\mathbb{W}}_1^q({\mu }_{{\tau }_{n_s}^{\delta }}^{\delta },{\delta }_0)\right)\mathrm{d}{E}_s\le C+C{\int }_0^t{\mathbb{E}}_B\underset{0\le r\le s}{sup}|X_s^{i,\delta }{|}^q\mathrm{d}{E}_s.\end{array}$$

For $O_4\left(t\right)$, from (23), (29), Burkhold-Davis-Gundy’s inequality, Hölder’s inequality and the Fubini theorem, we get

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}{O}_4\left(t\right)\right]& \le 2{\mathbb{E}}_B{\left[\underset{0\le t\le T}{sup}{\int }_0^t|X_s^{i,\delta }-G\left(X_s^{i,\delta }\right){|}^2|g(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }){|}^2\mathrm{d}{E}_s\right]}^{{\displaystyle \frac{q}{4}}}\hfill \\ & \le C{\mathbb{E}}_B{\left[{\int }_0^T(2+2v^2)|X_s^{i,\delta }{|}^2\left(1+|X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^2+{\mathbb{W}}_1^2({\mu }_{{\tau }_{n_s}^{\delta }}^{\delta },{\delta }_0)\right)\mathrm{d}{E}_s\right]}^{{\displaystyle \frac{q}{4}}}\hfill \\ & \le {\displaystyle \frac{1}{2}}{\mathbb{E}}_B\underset{0\le t\le T}{sup}|X_t^{i,\delta }{|}^q+C{\mathbb{E}}_B{\left[{\int }_0^{T}1+\underset{0\le r\le s}{sup}|X_s^{i,\delta }{|}^2\mathrm{d}{E}_s\right]}^{{\displaystyle \frac{q}{2}}}\hfill \\ & \le {\displaystyle \frac{1}{2}}{\mathbb{E}}_B\underset{0\le t\le T}{sup}|X_t^{i,\delta }{|}^q+C{\int }_0^{T}1+{\mathbb{E}}_B\underset{0\le r\le s}{sup}|X_s^{i,\delta }{|}^q\mathrm{d}{E}_s.\hfill \end{array}$$

Hence, we have

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|X_t^{i,\delta }-G\left(X_t^{i,\delta }\right){|}^q\right]& \le C+C{\int }_0^t{\mathbb{E}}_B\left[\underset{0\le r\le s}{sup}|X_s^{i,\delta }{|}^q\right]\mathrm{d}s+C{\int }_0^t{\mathbb{E}}_B\left[\underset{0\le r\le s}{sup}|X_s^{i,\delta }{|}^q\right]\mathrm{d}{E}_s\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|X_t^{i,\delta }{|}^q\right].\hfill \end{array}$$

(64)

From the weighted Young inequality, we get

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|X_t^{i,\delta }{|}^q\right]& \le {\left({\displaystyle \frac{1}{1-v}}\right)}^{q-1}{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|X_t^{i,\delta }-G\left(X_t^{i,\delta }\right){|}^q\right]+{\left({\displaystyle \frac{1}{v}}\right)}^{q-1}{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|G\left(X_t^{i,\delta }\right){|}^2\right]\hfill \\ \hfill & \le {\left({\displaystyle \frac{1}{1-v}}\right)}^{q-1}{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|X_t^{i,\delta }-G\left(X_t^{i,\delta }\right){|}^q\right]+v{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|X_t^{i,\delta }{|}^2\right].\hfill \end{array}$$

(65)

From (64) and (65), we have

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|X_t^{i,\delta }{|}^q\right]& \le C+C{\int }_0^T{\mathbb{E}}_B\left[\underset{0\le r\le s}{sup}|X_s^{i,\delta }{|}^q\right]\mathrm{d}s+C{\int }_0^T{\mathbb{E}}_B\left[\underset{0\le r\le s}{sup}|X_s^{i,\delta }{|}^q\right]\mathrm{d}{E}_s.\hfill \end{array}$$

(66)

From Proposition 1, take ${\mathbb{E}}_D$ on both sides of (53), and we have

$$\begin{array}{c}\hfill \mathbb{E}\left[\underset{0\le t\le T}{sup}|X_t^{i,\delta }{|}^q\right]\le C{e}^{CT+C{E}_T}\le C.\end{array}$$

where $C>0$ may be dependent on $\mathbb{E}|X_0^i{|}^q,v,K_f,L_1,L_g,T,E_T$, but independent of N and $\delta$.

Hence, we obtain that for any $t\in [0,T]$

$$\begin{array}{c}\hfill \underset{\delta >0}{sup}\underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le t\le T}{sup}{\left|X_t^{i,\delta }\right|}^q\right]\le C.\end{array}$$

Thus, (62) holds for $p=4$. From Proposition 4, (55), (57) holds for $p=4$. If we set $q=2p=8$, by the same procedure, we get that (62) holds for $p=8$. The iteration leads us to that (62) holds for the even order. By the Lyapunov inequality, we obtain that (62) holds for $p>0$. The proof is complete. □

4. Mean Square Convergence Rate

Assumption 2.

Hypothesis 3′

Let $h:{\mathbb{R}}^{+}\times \mathbb{R}\times {\mathcal{P}}_2\left(\mathbb{R}\right)\to \mathbb{R}$ be Borel-measurable and continuous functions. For any $R>1$, a positive constant a and all $\left|x\right|\vee \left|y\right|\le R$, there exists a positive constant $K_R\le alogR$ such that for all $t\in {\mathbb{R}}^{+}$, $x,y\in \mathbb{R}$ and $\mu \in {\mathcal{P}}_2\left(\mathbb{R}\right)$,

$${|h(t,x,\mu )-h(t,y,\mu )|}^2\le K_R{|x-y|}^2.$$

(67)

Hypothesis 3″

Let $h:{\mathbb{R}}^{+}\times \mathbb{R}\times {\mathcal{P}}_2\left(\mathbb{R}\right)\to \mathbb{R}$ be Borel-measurable and continuous functions. There exists positive constant K and $l>1$ such that for all $t\in {\mathbb{R}}^{+}$, $x,y\in \mathbb{R}$ and $\mu \in {\mathcal{P}}_2\left(\mathbb{R}\right)$,

$$(x-y)(h(t,x,\mu )-h(t,y,\mu ))\le {K|x-y|}^2,$$

(68)

and

$$|h(t,x,\mu )-h(t,y,\mu )|\le K(1+|x{|}^l+\left|y{|}^l\right)|x-y|.$$

(69)

As we have proved before, for any $p>0$, there exists a constant $C_p>0$ such that

$$\begin{array}{c}\hfill \mathbb{E}\left[\underset{0\le s\le T}{sup}{\left|X_s^{i,N}\right|}^p\right]\le C_p,\mathbb{E}\left[\underset{0\le s\le T}{sup}{\left|X_s^{i,\delta }\right|}^p\right]\le C_p.\end{array}$$

Denote

$${\Omega }_R^{i,\delta }:=\left\{\omega \in \Omega :\underset{0\le t\le T}{sup}|X_t^{i,N}|\vee \underset{0\le t\le T}{sup}\left|X_t^{i,\delta }\right|\ge R\right\}.$$

By Chebyshev’s inequality,

$$\begin{array}{c}\hfill \mathbb{P}\left({\Omega }_R^{i,\delta }\right)\le \left(\mathbb{E}\left[\underset{0\le t\le T}{sup}{\left|X_t^{i,N}\right|}^q\right]\vee \mathbb{E}\left[\underset{0\le t\le T}{sup}{\left|X_t^{i,\delta }\right|}^q\right]\right){\displaystyle \frac{1}{R^q}}\le {\displaystyle \frac{C_q}{R^q}}.\end{array}$$

(70)

Proposition 6.

Suppose that Assumption 1 holds. Let ${\left\{X_t^{i,N}\right\}}_{t\in [0,T]}$ be the solution to (10) and ${\left\{X_t^{i,\delta }\right\}}_{t\in [0,T]}$ be the solution to (21) with $\mathbb{E}|X_0^i{|}^p<\infty$, for any p. If we further assume that h satisfies the local Lipschitz condition (67), then for any $T>0$,

$$\begin{array}{c}\hfill \underset{N\to \infty }{lim}\underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le s\le T}{sup}{|X_s^{i,N}-X_s^{i,\delta }|}^2\right]=0.\end{array}$$

(71)

If we further assume that h satisfies (68) and (69), then there exists a positive constant C such that

$$\begin{array}{c}\hfill \underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le s\le T}{sup}{|X_s^{i,N}-X_s^{i,\delta }|}^2\right]\le C{\delta }^{\gamma },\end{array}$$

(72)

where C iis independent of δ, N and R.

Proof. 

For every fixed $1\le i\le N$, define $Z_t^{i,\delta }:=[(X_t^{i,N}-G\left(X_t^{i,N}\right))-(X_t^{i,\delta }-G\left(X_t^{i,\delta }\right))]$. By using Itô’s formula to $|Z_t^{i,\delta }{|}^2$, we have

$$\begin{array}{cc}\hfill |Z_t^{i,\delta }{|}^2& =2{\int }_0^t{Z}_s^{i,\delta }(f\left(E_s\right)-f\left(E_{{\tau }_{n_s}^{\delta }}\right))\mathrm{d}s\hfill \\ & +2{\int }_0^t{Z}_s^{i,\delta }\left(h\left(E_s,X_s^{i,N},{\mu }_s^{X,N}\right)-h_{\delta }\left(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right)\right)\mathrm{d}{E}_s\hfill \\ & +{\int }_0^t|g\left(E_s,X_s^{i,N},{\mu }_s^{X,N}\right)-g\left(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right){|}^2\mathrm{d}{E}_s\hfill \\ & +2{\int }_0^t{Z}_s^{i,\delta }\left(g\left(E_s,X_s^{i,N},{\mu }_s^{X,N}\right)-g\left(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right)\right)\mathrm{d}{B}_{E_s}^i\hfill \\ \hfill & =:M_1\left(t\right)+M_2\left(t\right)+M_3\left(t\right)+M_4\left(t\right).\hfill \end{array}$$

(73)

Note, for any $0\le s\le t\le T$,

$$\begin{array}{c}\hfill \left|Z_s^{i,\delta }\right|\le |X_s^{i,N}-X_s^{i,\delta }|+|G\left(X_s^{i,N}\right)-G\left(X_s^{i,\delta }\right)|\le (1+v)|X_s^{i,N}-X_s^{i,\delta }|.\end{array}$$

(74)

For $M_1\left(t\right)$, by using (22) and (74), the elementary inequality and the Fubini theorem, we have

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}{M}_1\left(t\right)\right]& \le 2{\int }_0^T\left|Z_s^{i,\delta }\right||f\left(E_s\right)-f\left(E_{{\tau }_{n_s}^{\delta }}\right)|\mathrm{d}s\hfill \\ \hfill & \le {\int }_0^T|Z_s^{i,\delta }{|}^2+|f\left(E_s\right)-f\left(E_{{\tau }_{n_s}^{\delta }}\right){|}^2\mathrm{d}s\hfill \\ \hfill & \le {\int }_0^T{(1+v)}^2|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}s+{\int }_0^T{K}_f^2{\delta }^{2\gamma }\mathrm{d}s\hfill \\ \hfill & \le C{\delta }^{2\gamma }+C{\int }_0^T|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}s.\hfill \end{array}$$

(75)

For $M_2\left(t\right)$, we get

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}{M}_2\left(t\right)\right]& \le 2{\mathbb{E}}_B{\int }_0^T\left|Z_s^{i,\delta }\right||h\left(E_s,X_s^{i,N},{\mu }_s^{X,N}\right)-h\left(E_s,X_s^{i,N},{\mu }_s^{\delta }\right)|\mathrm{d}{E}_s\hfill \\ & +2{\mathbb{E}}_B{\int }_0^T\left|Z_s^{i,\delta }\right||h\left(E_s,X_s^{i,N},{\mu }_s^{\delta }\right)-h\left(E_s,X_s^{i,\delta },{\mu }_s^{\delta }\right)|\mathrm{d}{E}_s\hfill \\ & +2{\mathbb{E}}_B{\int }_0^T\left|Z_s^{i,\delta }\right||h\left(E_s,X_s^{i,\delta },{\mu }_s^{\delta }\right)-h\left(E_{{\tau }_{n_s}^{\delta }},X_s^{i,\delta },{\mu }_s^{\delta }\right)|\mathrm{d}{E}_s\hfill \\ & +2{\mathbb{E}}_B{\int }_0^T\left|Z_s^{i,\delta }\right||h\left(E_{{\tau }_{n_s}^{\delta }},X_s^{i,\delta },{\mu }_s^{\delta }\right)-h\left(E_{{\tau }_{n_s}^{\delta }},X_s^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right)|\mathrm{d}{E}_s\hfill \\ & +2{\mathbb{E}}_B{\int }_0^T\left|Z_s^{i,\delta }\right||h\left(E_{{\tau }_{n_s}^{\delta }},X_s^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right)-h\left(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right)|\mathrm{d}{E}_s\hfill \\ & +2{\mathbb{E}}_B{\int }_0^T\left|Z_s^{i,\delta }\right||h\left(E_{{\tau }_{n_s}^{\delta }}^{\delta },X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right)-h_{\delta }\left(E_{{\tau }_{n_s}^{\delta }}^{\delta },X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right)|\mathrm{d}{E}_s\hfill \\ & =:M_{21}+M_{22}+M_{23}+M_{24}+M_{25}+M_{26}.\hfill \end{array}$$

(76)

By using (67), (26), (74) and the Fubini theorem, we have

$$\begin{array}{cc}\hfill M_{21}& \le {\int }_0^T{\mathbb{E}}_B|Z_s^{i,\delta }{|}^2\mathrm{d}{E}_s+{\int }_0^T{\mathbb{E}}_B|h\left(E_s,X_s^{i,N},{\mu }_s^{X,N}\right)-h\left(E_s,X_s^{i,N},{\mu }_s^{\delta }\right){|}^2\mathrm{d}{E}_s\hfill \\ \hfill & \le {\int }_0^T{(1+v)}^2{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s+{\mathbb{E}}_B{\int }_0^T{K}_1^2{\mathbb{W}}_1^2({\mu }_s^{X,N},{\mu }_s^{\delta })\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s.\hfill \end{array}$$

By using (67) and (28), Proposition 5 and the Fubini theorem, we get

$$\begin{array}{cc}\hfill M_{22}\le & {\int }_0^T{\mathbb{E}}_B|Z_s^{i,\delta }{|}^2\mathrm{d}{E}_s+{\int }_0^T{\mathbb{E}}_B|h\left(E_s,X_s^{i,N},{\mu }_s^{\delta }\right)-h\left(E_s,X_s^{i,\delta },{\mu }_s^{\delta }\right){|}^2{I}_{\Omega /{\Omega }_R^{i,\delta }}\mathrm{d}{E}_s\hfill \\ \hfill & +{\int }_0^T{\mathbb{E}}_B|h\left(E_s,X_s^{i,N},{\mu }_s^{\delta }\right)-h\left(E_s,X_s^{i,\delta },{\mu }_s^{\delta }\right){|}^2{I}_{{\Omega }_R^{i,\delta }}\mathrm{d}{E}_s\hfill \\ \hfill \le & {\int }_0^T{(1+v)}^2{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s+K_R{\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s\hfill \\ \hfill & +C{\mathbb{E}}_B{\int }_0^T\left(|h\left(E_s,X_s^{i,N},{\mu }_s^{\delta }\right){|}^2+|h\left(E_s,X_s^{i,\delta },{\mu }_s^{\delta }\right){|}^2\right)I_{{\Omega }_R^{i,\delta }}\mathrm{d}{E}_s\hfill \\ \hfill \le & (C+K_R){\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s\hfill \\ & +C{\int }_0^T{L}_h^2{\mathbb{E}}_B\left(C+|X_s^{i,N}{|}^{2l}+|X_s^{i,\delta }{|}^{2l}\right)I_{{\Omega }_R^{i,\delta }}\mathrm{d}{E}_s\hfill \\ \hfill \le & (C+K_R){\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s\hfill \\ & +C{\int }_0^T{\left[{\mathbb{E}}_B\left(C+|X_s^{i,N}{|}^{4l}+|X_s^{i,\delta }{|}^{4l}\right)\right]}^{{\displaystyle \frac{1}{2}}}{\left[{\mathbb{E}}_B{I}_{{\Omega }_R^{i,\delta }}^2\right]}^{{\displaystyle \frac{1}{2}}}\mathrm{d}{E}_s\hfill \\ \hfill \le & (C+K_R){\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s+C{\left[\mathbb{P}\left({\Omega }_R^{i,\delta }\right)\right]}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

(77)

By using (22) and (74) and the Fubini theorem, we have

$$\begin{array}{cc}\hfill M_{23}& \le {\int }_0^T{\mathbb{E}}_B|Z_s^{i,\delta }{|}^2\mathrm{d}{E}_s+{\int }_0^T{\mathbb{E}}_B|h\left(E_s,X_s^{i,\delta },{\mu }_s^{\delta }\right)-h\left(E_{{\tau }_{n_s}^{\delta }},X_s^{i,\delta },{\mu }_s^{\delta }\right){|}^2\mathrm{d}{E}_s\hfill \\ \hfill & \le {\int }_0^T{(1+v)}^2{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s+C{\int }_0^T{K}_f^2{\delta }^{2\gamma }\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\delta }^{2\gamma }+C{\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s.\hfill \end{array}$$

By using (12), (26), (74), (56) and the Fubini theorem, we get

$$\begin{array}{cc}\hfill M_{24}& \le {\int }_0^T{\mathbb{E}}_B|Z_s^{i,\delta }{|}^2\mathrm{d}{E}_s+{\int }_0^T{\mathbb{E}}_B|h\left(E_{{\tau }_{n_s}^{\delta }},X_s^{i,\delta },{\mu }_s^{\delta }\right)-h\left(E_{{\tau }_{n_s}^{\delta }},X_s^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right){|}^2\mathrm{d}{E}_s\hfill \\ \hfill & \le {\int }_0^T{(1+v)}^2{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s+{\mathbb{E}}_B{\int }_0^T{K}_1^2{\mathbb{W}}_1^2({\mu }_s^{\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s+C{\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^2\mathrm{d}{E}_s\hfill \\ \hfill & \le C\delta +C{\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s.\hfill \end{array}$$

By using (67) and (28), Proposition 5 and the Fubini theorem, we have

$$\begin{array}{cc}\hfill M_{25}\le & {\int }_0^T{\mathbb{E}}_B|Z_s^{i,\delta }{|}^2\mathrm{d}{E}_s+{\int }_0^T{\mathbb{E}}_B|h\left(E_{{\tau }_{n_s}^{\delta }},X_s^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right)-h\left(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right){|}^2{I}_{\Omega /{\Omega }_R^{i,\delta }}\mathrm{d}{E}_s\hfill \\ \hfill & +{\int }_0^T{\mathbb{E}}_B|h\left(E_{{\tau }_{n_s}^{\delta }},X_s^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right)-h\left(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right){|}^2{I}_{{\Omega }_R^{i,\delta }}\mathrm{d}{E}_s\hfill \\ \hfill \le & {\int }_0^T{(1+v)}^2{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s+K_R{\int }_0^T{\mathbb{E}}_B|X_s^{i,\delta }-X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^2\mathrm{d}{E}_s\hfill \\ \hfill & +C{\mathbb{E}}_B{\int }_0^T\left(|h\left(E_{{\tau }_{n_s}^{\delta }},X_s^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right){|}^2+|h\left(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right){|}^2\right)I_{{\Omega }_R^{i,\delta }}\mathrm{d}{E}_s\hfill \\ \hfill \le & (C+K_R){\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s+C{K}_R\delta \hfill \\ & +C{\int }_0^T{L}_h^2{\mathbb{E}}_B\left(C+|X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^{2l}+|X_s^{i,\delta }{|}^{2l}\right)I_{{\Omega }_R^{i,\delta }}\mathrm{d}{E}_s\hfill \\ \hfill \le & (C+K_R){\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s+C{K}_R\delta +C{\left[\mathbb{P}\left({\Omega }_R^{i,\delta }\right)\right]}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

(78)

By using (12), (26), (74), (56) and the Fubini theorem, we get

$$\begin{array}{cc}\hfill M_{26}& \le {\int }_0^T{\mathbb{E}}_B|Z_s^{i,\delta }{|}^2\mathrm{d}{E}_s+{\int }_0^T{\mathbb{E}}_B|h\left(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right)-h_{\delta }\left(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right){|}^2\mathrm{d}{E}_s\hfill \\ \hfill & \le {\int }_0^T{(1+v)}^2{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s+{\mathbb{E}}_B{\int }_0^T|h-{\displaystyle \frac{h}{1+{\delta }_1^{\gamma /2}\left|h\right|}}{|}^2\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s+{\mathbb{E}}_B{\int }_0^T\left|\right|h|{\displaystyle \frac{{\delta }_1^{\gamma /2}\left|h\right|}{1+{\delta }_1^{\gamma /2}\left|h\right|}}{|}^2\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s+{\mathbb{E}}_B{\int }_0^T{\delta }_1^{\gamma }{\left|h(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })\right|}^4\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s+C{\int }_0^T{\delta }_1^{\gamma }{\mathbb{E}}_B(1+|X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^{4l})\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s+C{\int }_0^T{\delta }_1^{\gamma }\mathrm{d}{E}_s\hfill \\ \hfill & \le C{\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s+C{\delta }^{1\wedge \gamma }.\hfill \end{array}$$

Similarly, we here use the technique of Theorem 4.1 in [22]:

$$\begin{array}{c}\hfill C{\int }_0^T{\delta }_1^{\gamma }\mathrm{d}{E}_s\le max\{C{T}^{\gamma }\delta ,C{E}_T{\delta }^{\gamma }\}\le C{\delta }^{\gamma }.\end{array}$$

Hence, by using (41)–(44), we obtain

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}{M}_2\left(t\right)\right]\le & C\delta +C{\delta }^{2\gamma }+C{\delta }^{\gamma }+C{K}_R\delta +C{\left[\mathbb{P}\left({\Omega }_R^i\right)\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ & +(C+K_R){\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s\hfill \\ \hfill \le & C{\delta }^{\gamma }+C{K}_R\delta +C{\left[\mathbb{P}\left({\Omega }_R^i\right)\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ & +(C+K_R){\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s.\hfill \end{array}$$

(79)

For $M_3\left(t\right)$, we have

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}{M}_3\left(t\right)\right]& \le C{\mathbb{E}}_B{\int }_0^T|g\left(E_s,X_s^{i,N},{\mu }_s^{X,N}\right)-g\left(E_s,X_s^{i,N},{\mu }_s^{\delta }\right){|}^2\mathrm{d}{E}_s\hfill \\ & +C{\mathbb{E}}_B{\int }_0^T|g\left(E_s,X_s^{i,N},{\mu }_s^{\delta }\right)-g\left(E_s,X_s^{i,\delta },{\mu }_s^{\delta }\right){|}^2\mathrm{d}{E}_s\hfill \\ & +C{\mathbb{E}}_B{\int }_0^T|g\left(E_s,X_s^{i,\delta },{\mu }_s^{\delta }\right)-g\left(E_{{\tau }_{n_s}^{\delta }},X_s^{i,\delta },{\mu }_s^{\delta }\right){|}^2\mathrm{d}{E}_s\hfill \\ & +C{\mathbb{E}}_B{\int }_0^T|g\left(E_{{\tau }_{n_s}^{\delta }},X_s^{i,\delta },{\mu }_s^{\delta }\right)-g\left(E_{{\tau }_{n_s}^{\delta }},X_s^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right){|}^2\mathrm{d}{E}_s\hfill \\ & +C{\mathbb{E}}_B{\int }_0^T|g\left(E_{{\tau }_{n_s}^{\delta }},X_s^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right)-g\left(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right){|}^2\mathrm{d}{E}_s\hfill \\ & =:M_{31}+M_{32}+M_{33}+M_{34}+M_{35}.\hfill \end{array}$$

(80)

By using (12) and (26) and the Fubini theorem, we get

$$\begin{array}{c}\hfill M_{31}\le C{\mathbb{E}}_B{\int }_0^T{K}_1^2{\mathbb{W}}_1^2({\mu }_s^{X,N},{\mu }_s^{\delta })\mathrm{d}{E}_s\le C{\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s.\end{array}$$

By using (25) and (56) and the Fubini theorem, we have

$$\begin{array}{c}\hfill M_{32}\le C{\int }_0^T{K}_g^2{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s\le C{\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s.\end{array}$$

By using (22) and the Fubini theorem, we get

$$\begin{array}{c}\hfill M_{33}\le C{\int }_0^T{K}_f^2{\delta }^{2\gamma }\mathrm{d}{E}_s\le C{\delta }^{2\gamma }.\end{array}$$

By using (12), (26), (56) and the Fubini theorem, we have

$$\begin{array}{c}\hfill M_{34}\le C{\mathbb{E}}_B{\int }_0^T{K}_1^2{\mathbb{W}}_1^2({\mu }_s^{\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })\mathrm{d}{E}_s\le C{\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^2\mathrm{d}{E}_s\le C\delta .\end{array}$$

By using (25) and (56) and the Fubini theorem, we have

$$\begin{array}{c}\hfill M_{35}\le C{\int }_0^T{K}_g^2{\mathbb{E}}_B|X_s^{i,N}-X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^2\mathrm{d}{E}_s\le C\delta .\end{array}$$

Hence, we obtain

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}{M}_3\left(t\right)\right]& \le C\delta +C{\delta }^{2\gamma }+C{\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s.\hfill \end{array}$$

(81)

For $M_4\left(t\right)$, by using (46), the Burkholder–Davis–Gundy inequality and the Fubini theorem, we have

$$\begin{array}{cc}\hfill & {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}{M}_4\left(t\right)\right]\hfill \\ \hfill & \le 8\sqrt{2}{\mathbb{E}}_B{\left[\underset{0\le t\le T}{sup}{\int }_0^t\left|Z_s^{i,\delta }{|}^2\right|g\left(E_s,X_s^{i,N},{\mu }_s^{X,N}\right)-g\left(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right){|}^2\mathrm{d}{E}_s\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le 8\sqrt{2}{\mathbb{E}}_B{\left[{\int }_0^T\left|Z_s^{i,\delta }{|}^2\right|g\left(E_s,X_s^{i,N},{\mu }_s^{X,N}\right)-g\left(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right){|}^2\mathrm{d}{E}_s\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le 8\sqrt{2}{\mathbb{E}}_B{\left[|\underset{0\le t\le T}{sup}{Z}_t^{i,\delta }{|}^2{\int }_0^T|g\left(E_s,X_s^{i,N},{\mu }_s^{X,N}\right)-g\left(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right){|}^2\mathrm{d}{E}_s\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|Z_t^{i,\delta }{|}^2\right]+C{\mathbb{E}}_B{\int }_0^T|g\left(E_s,X_s^{i,N},{\mu }_s^{X,N}\right)-g\left(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right){|}^2\mathrm{d}{E}_s\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|Z_t^{i,\delta }{|}^2\right]+C\delta +C{\delta }^{2\gamma }+C{\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s.\hfill \end{array}$$

(82)

Hence, by using (75), (79), (81) and (82), we obtain

$$\begin{array}{cc}\hfill {\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}{\left|Z_t^{i,N}\right|}^2\right]\le & {\displaystyle \frac{1}{2}}{\mathbb{E}}_B\left[\underset{0\le t\le T}{sup}|Z_t^{i,\delta }{|}^2\right]+C{\delta }^{\gamma }+C{K}_R\delta +C{\left[\mathbb{P}\left({\Omega }_R^i\right)\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +C{\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}s+(C+K_R){\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s.\hfill \end{array}$$

(83)

From Lemma 4, we get

$$\begin{array}{cc}\hfill |X_t^{i,N}-X_t^{i,\delta }{|}^2& \le {\displaystyle \frac{1}{1-v}}|Z_t^{i,\delta }{|}^2+{\displaystyle \frac{1}{v}}|G\left(X_t^{i,N}\right)-G\left(X_t^{i,\delta }\right){|}^2\hfill \\ \hfill & \le {\displaystyle \frac{1}{1-v}}|Z_t^{i,\delta }{|}^2+v|X_t^{i,N}-X_t^{i,\delta }{|}^2.\hfill \end{array}$$

Taking ${\mathbb{E}}_D$ on both sides, by Proposition 1 and (70), we have

$$\begin{array}{c}\hfill \mathbb{E}\left[\underset{0\le t\le T}{sup}|X_t^{i,N}-X_t^{i,\delta }{|}^2\right]\le \mathbb{E}\left[\underset{0\le t\le T}{sup}{\left|Z_t^{i,\delta }\right|}^2\right]\le \left(C{\delta }^{\gamma }+C{K}_R\delta +{\displaystyle \frac{C}{R^{q/2}}}\right)e^{CT+(C+K_R)E_T}.\end{array}$$

(84)

Note that $K_R\le alogR$. Choosing $q>4aC{E}_T$. Let $R\to \infty$ and $\delta \to 0$, we get

$$\begin{array}{cc}\hfill \underset{R\to \infty }{lim}\underset{\delta \to 0}{lim}\underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le t\le T}{sup}|X_t^i-X_t^{i,N}{|}^2\right]& \le \underset{R\to \infty }{lim}\underset{\delta \to 0}{lim}\left(C{\delta }^{\gamma }+C{K}_R\delta +{\displaystyle \frac{C}{R^{q/2}}}\right)C{R}^{aC{E}_T}\hfill \\ \hfill & \le \underset{R\to \infty }{lim}C{R}^{-aC{E}_T}\hfill \\ \hfill & =0.\hfill \end{array}$$

Moreover, if (68) and (69) replace (67), the above estimation (77) can be modified as

$$\begin{array}{cc}\hfill M_{25}\le & {\int }_0^T{\mathbb{E}}_B|Z_s^{i,\delta }{|}^2\mathrm{d}{E}_s+{\int }_0^T{\mathbb{E}}_B|h\left(E_{{\tau }_{n_s}^{\delta }},X_s^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right)-h\left(E_{{\tau }_{n_s}^{\delta }},X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta }\right){|}^2\mathrm{d}{E}_s\hfill \\ \hfill \le & {\int }_0^T{(1+v)}^2{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s\hfill \\ & +K{\int }_0^T{\mathbb{E}}_B|X_s^{i,\delta }-X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^2\left(1+|X_s^{i,\delta }{|}^2+|X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^2\right)\mathrm{d}{E}_s\hfill \\ \hfill \le & C{\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s\hfill \\ & +CK{\int }_0^T{\left[{\mathbb{E}}_B|X_s^{i,\delta }-X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^4\right]}^{1/2}{\left[{\mathbb{E}}_B{\left(1+|X_s^{i,\delta }{|}^2+|X_{{\tau }_{n_s}^{\delta }}^{i,\delta }{|}^2\right)}^2\right]}^{1/2}\mathrm{d}{E}_s\hfill \\ \hfill \le & C{\int }_0^T{\mathbb{E}}_B|X_s^{i,N}-X_s^{i,\delta }{|}^2\mathrm{d}{E}_s+CK\delta .\hfill \end{array}$$

Similarly, we have

$$\begin{array}{c}\hfill \mathbb{E}\left[\underset{0\le t\le T}{sup}|X_t^{i,N}-X_t^{i,\delta }{|}^2\right]\le \mathbb{E}\left[\underset{0\le t\le T}{sup}{\left|Z_t^{i,\delta }\right|}^2\right]\le \left(C{\delta }^{\gamma }+CK\delta \right)e^{CT+(C+K)E_T}.\end{array}$$

(85)

Thus, it follows that

$$\begin{array}{cc}\hfill \underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le t\le T}{sup}|X_t^{i,N}-X_t^{i,\delta }{|}^2\right]& \le \left(C{\delta }^{\gamma }+CK\delta \right)C\le C{\delta }^{\gamma },\hfill \end{array}$$

where $C>0$ may be dependent on $v,K_1,K_f,K_R,K_g,L_1,L_h,L_g,T,E_T$, but independent of $\delta$, N and R. The proof is complete. □

Theorem 2.

Suppose that Assumption 1 holds. Let ${\left\{X_t^i\right\}}_{t\in [0,T]}$ be the solution to (11) and ${\left\{X_t^{i,\delta }\right\}}_{t\in [0,T]}$ be the solution to (21) with $\mathbb{E}|X_0^i{|}^p<\infty$, for any p. If we further assume that h satisfies the local Lipschitz condition (67), then for any $T>0$,

$$\begin{array}{c}\hfill \underset{N\to \infty }{lim}\underset{\delta \to 0}{lim}\underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le s\le T}{sup}{|X_s^i-X_s^{i,\delta }|}^2\right]=0.\end{array}$$

(86)

If we further assume that h satisfies (68) and (69), then there exists a constant $C>0$ such that

$$\begin{array}{c}\hfill \underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le s\le T}{sup}{|X_s^i-X_s^{i,\delta }|}^2\right]\le C{N}^{-{\displaystyle \frac{1}{2}}}+C{\delta }^{\gamma },\end{array}$$

(87)

where C is independent of δ, N and R.

Proof. 

From (37) and (71), we have

$$\begin{array}{cc}\hfill & \underset{N\to \infty }{lim}\underset{\delta \to 0}{lim}\underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le s\le T}{sup}{|X_s^i-X_s^{i,\delta }|}^2\right]\hfill \\ \hfill & \le \underset{N\to \infty }{lim}\underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le s\le T}{sup}{|X_s^i-X_s^{i,N}|}^2\right]+\underset{\delta \to 0}{lim}\underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le s\le T}{sup}{|X_s^{i,N}-X_s^{i,\delta }|}^2\right]\hfill \\ \hfill & =0.\hfill \end{array}$$

If we further assume that h satisfies (68) and (69), from (38) and (72), then there exists a constant C such that

$$\begin{array}{cc}\hfill \underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le s\le T}{sup}{|X_s^i-X_s^{i,\delta }|}^2\right]& \le \underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le s\le T}{sup}{|X_s^i-X_s^{i,N}|}^2\right]+\underset{1\le i\le N}{sup}\mathbb{E}\left[\underset{0\le s\le T}{sup}{|X_s^{i,N}-X_s^{i,\delta }|}^2\right]\hfill \\ \hfill & \le C{N}^{-{\displaystyle \frac{1}{2}}}+C{\delta }^{\gamma },\hfill \end{array}$$

where $C>0$ may be dependent on $v,K_1,K_f,K_R,K_g,L_1,L_h,L_g,T,E_T$, but independent of $\delta$, N and R. The proof is complete. □

5. Example

In this section, we construct three numerical experiments to validate the convergence rate established in preceding theoretical analyses. First, by fixing the particle count at N = 1000 and N = 20,000, we ensure the avoidance of ‘‘particle corruption’’ phenomena (as discussed in [24]). Subsequently, we rigorously demonstrate the strong convergence of the tamed EM method for time-changed MV-NSDEs. Finally, a systematic comparison is conducted between the tamed EM scheme and the classical EM method (see [14]) in the context of time-changed MV-SDEs, highlighting the tamed EM scheme performance characteristics. Throughout this section, we focus on the case that $E_t$ is an inverse $0.9$-stable subordinator.

Example 1.

Our analysis begins with scalar time-changed MV-NSDEs:

$$\mathrm{d}[X_t-sin{X}_t]=2E_t\mathrm{d}t+(2E_t+X_t-X_t^3+{\mu }_t^X)\mathrm{d}{E}_t+(E_t{X}_t+{\mu }_t^X)\mathrm{d}{B}_{E_t},t\in [0,1].$$

(88)

It is straightforward to verify compliance of $f,g$ and h with Assumption 1 holding for (88) with $\gamma =1$. For numerical implementation methodologies regarding the inverse subordinator E and time-changed Brownian motion $B_{E_t}$, we direct readers to [10,22,26]. As demonstrated in Figure 2, under the condition that B is a standard Brownian motion independent of the inverse subordinator E, the jump discontinuities of $B_{E_t}$ exhibit a one-to-one correspondence with those of $X_t$. Furthermore, whenever $B_{E_t}$ remains constant over a subinterval, $X_t$ simultaneously maintains a constant trajectory.

Then, for numerical validation, we perform simulations with particle counts N = 10,000 and N = 20,000, employing a time step $\delta =2^{-14}$. Figure 3 and Figure 4 confirm the absence of ‘‘particle corruption’’ phenomena (as discussed in [24]) throughout the entire path simulation, indicating that all particle trajectories exhibit stable and well-regularized behavior.

In order to detect the mean square convergence rate of the tamed EM method, the unavailable exact solution is identified with a numerical approximation generated by (21) with a fine step size $\delta =2^{-14}$. The other numerical approximations are calculated by (21) and applied to (88) with five different step sizes ${\delta }_0=2^{-11},2^{-10},2^{-9},2^{-8},2^{-7}$. For the given step size $\delta$, time horizon $T=1$, number of particles $N=20,000$, the ${\mathrm{L}}^1$ strong error is calculated by

$$Error(X_t,\delta ):={\displaystyle \frac{1}{N}}\sum _{i=1}^N|X_{\mathrm{T}}^{\delta }\left(w_i\right)-X_{\mathrm{T}}^{{\delta }_0}\left(w_i\right)|.$$

(89)

In Figure 5, the red line is the reference line with the slope of $1/2$. We can see that the slopes of the two curves appear to match well. A simple regression shows that the slope of the line for the tamed EM method is 0.4963, identifying Theorem 2 numerically.

Example 2.

We focus on the following time-changed MV-NSDEs:

$$\mathrm{d}[X_t-sin{X}_t]=2E_t^{\gamma }\mathrm{d}t+(2E_t^{\gamma }+2X_t-X_t^3+{\mu }_t^X)\mathrm{d}{E}_t+(E_t^{\gamma }+X_t+{\mu }_t^X)\mathrm{d}{B}_{E_t},t\in [0,1].$$

(90)

It is easy to verify that the Assumptions 1 hold for (88). Select $\gamma =0.4,0.6,0.8$, other settings are the same as in Example 1.

In Figure 6, Figure 7 and Figure 8, we can see that the slopes of the two curves appear to match well. A simple regression shows that the slope of the line for the tamed EM method are 0.2871, 0.3106, and 0.3967 (see

Table 1. Convergence results under different parameter combinations.

Parameter Combination	Convergence Rate	Step Sizes
$N=20,000,\gamma =1$	0.4963	${\delta }_0=2^{-11},2^{-10},2^{-9},2^{-8},2^{-7}$
$N=20,000,\gamma =0.4$	0.2871	${\delta }_0=2^{-11},2^{-10},2^{-9},2^{-8},2^{-7}$
$N=20,000,\gamma =0.6$	0.3106	${\delta }_0=2^{-11},2^{-10},2^{-9},2^{-8},2^{-7}$
$N=20,000,\gamma =0.8$	0.3967	${\delta }_0=2^{-11},2^{-10},2^{-9},2^{-8},2^{-7}$

), respectively, which numerically confirms Theorem 2.

Example 3.

Consider the following time-changed MV-NSDEs.

$$\mathrm{d}[X_t-asin{X}_t]=2E_t\mathrm{d}t+(2E_t+X_t-X_t^3+{\mu }_t^X)\mathrm{d}{E}_t+(E_t+X_t+{\mu }_t^X)\mathrm{d}{B}_{E_t},t\in [0,1].$$

(91)

In a very recent work [14], the authors studied the EM method for a class of time-changed MV-SDEs with linear drift and diffusion coefficients.

$$\begin{array}{cc}\hfill X_t^{i,\delta }:=& X_0^i+{\int }_0^{t}f\left(E_{{\tau }_{n_s}^{\delta }}^{\delta }\right)\mathrm{d}s+{\int }_0^{t}h(E_{{\tau }_{n_s}^{\delta }}^{\delta },X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })\mathrm{d}{E}_s+{\int }_0^{t}g(E_{{\tau }_{n_s}^{\delta }}^{\delta },X_{{\tau }_{n_s}^{\delta }}^{i,\delta },{\mu }_{{\tau }_{n_s}^{\delta }}^{\delta })\mathrm{d}{B}_{E_s}^i.\hfill \end{array}$$

(92)

Step size $\delta =2^{-14}$ and five different step sizes ${\delta }_0=2^{-11},2^{-10},2^{-9},2^{-8},2^{-7}$ are used to generate the unavailable exact solution and the other numerical approximations via (21) and (92), respectively. Figure 9 shows that when $a=0$, the tamed EM method is also applicable to the time-changed MV-SDEs in [14], and the convergence rate and accuracy are better.

6. Conclusions

This paper establishes convergence results for time-changed MV-NSDEs driven by two drift terms and one diffusion term, analyzed through interacting particle systems. Our main results show that the investigated method converges strongly in the mean square sense with order $\gamma /2$. These theoretical results are finally validated by two numerical experiments. Concerning that strong convergence rates for numerical approximations are particularly important to design efficient multilevel Monte Carlo approximation methods [27], there are two interesting ideas for our future work. One of them is to add the time variable t to the coefficient $f,g,h$ in (1), and the other is to develop numerical methods with higher-order convergence rates [28].