© Wuhan University 2023

0 Introduction

Stochastic differential equations (SDEs) have been utilized to model various phenomena, such as asset price, SIS epidemic and population dynamics. Analytical solutions can rarely be found for nonlinear SDEs, whereas numerical solutions may be helpful. For SDEs with super-linearly growing coefficients, implicit Euler-Maruyama (EM) methods^[1-3] have been proposed. In general, the implicit schemes need to solve a nonlinear system at each iteration, and hence requires more computational efforts.

On the contrary, the explicit methods own simple algebraic structure, cheap computational costs and acceptable convergence rate. The explicit EM approximate solution to nonlinear SDEs may diverge to infinity in finite time^[4]. Therefore some modified EM methods have been proposed to numerically solve nonlinear SDEs, such as the stopped EM method^[5], the tamed EM method^[6] and the tamed Milstein method^[7]. Especially, Mao^[8,9] invented the truncated Euler-Maruyama scheme (TEM for short) with strong convergence theory, which stimulates many researchers' interest. There are extensive literatures with the TEM^[10-14]. These research results are important contributions to numerical approximation theory of SDEs. However, we find that the results normally require the drift and diffusion coefficients satisfy the one-sided linear growth condition:

$〈\mathit{x},f(\mathit{x})〉+\frac{p-\mathrm{1}}{\mathrm{2}}|g(\mathit{x}){|}^{\mathrm{2}}\le K\left(\mathrm{1}+|\mathit{x}{|}^{\mathrm{2}}\right),\mathit{x}\in {\mathbb{R}}^{\overline{n}}$(1)

the one-sided Lipschitz condition:

$(\mathit{x}-\mathit{y})(f(\mathit{x})-f(\mathit{y}))\le L_{\mathrm{1}}{(\mathit{x}-\mathit{y})}^{\mathrm{2}}$(2)

and the polynomial growth condition:

$|f(\mathit{x})-f(\mathit{y}){|}^{\mathrm{2}}\le H\left(\mathrm{1}+|\mathit{x}{|}^{\gamma }+|\mathit{y}{|}^{\gamma }\right)|\mathit{x}-\mathit{y}{|}^{\mathrm{2}}$(3)

The one-sided Lipschitz condition (2) and polynomial growth condition (3) are frequently assumed in order to establish the strong convergence rates of the implicit EM schemes for highly nonlinear SDEs^[15,16]. In this paper, we shall remove conditions (2) and (3). We only need that the drift coefficient $f$ satisfies the one-sided polynomial growth condition to guarantee the strong convergence rates of the underlying numerical solutions, which is much less restrictive than (2) and (3).

The main purpose of this paper is to establish new criteria on the strong convergence rates of the truncated approximation when the drift coefficient is one-sided polynomially growing whereas the diffusion coefficient is linearly growing, polynomial growing or Hölder continuous. For SDEs with one-sided polynomial growing drift and diffusion coefficients, the strong convergence rate is one half.

The next section introduces basic notations and the truncated Euler-Maruyama method. After that the strong convergence rate for SDEs with superlinearly growing drift and diffusion coefficients at time $T$ was established. Section 2 proves the path-dependent strong convergence rate for SDEs with super-linearly growing drift and linearly growing diffusion coefficients over a finite time interval $[\mathrm{0},T]$.

1 SDEs with Polynomial Growing Coefficients

Throughout this paper, unless otherwise specified, let $|\mathit{x}|$ be the Euclidean norm in $\mathit{x}\in {\mathbb{R}}^{\overline{n}}$. If $\mathit{A}$ is a vector or matrix, its transpose is denoted by ${\mathit{A}}^{\mathrm{T}}$. If $\mathit{A}$ is a matrix, its trace norm is denoted by $|\mathit{A}|=\sqrt[]{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\left({\mathit{A}}^{\mathrm{T}}\mathit{A}\right)}$, while its operator norm is denoted by $\Vert \mathit{A}\Vert =\mathrm{s}\mathrm{u}\mathrm{p}\{|\mathit{A}\mathit{x}|:|\mathit{x}|=\mathrm{1}\}$. Let $\left(\mathrm{\Omega },\mathrm{ℱ},{\left\{{\mathrm{ℱ}}_t\right\}}_{t\ge \mathrm{0}},P\right)$ be a complete probability space with a filtration ${\left\{{\mathrm{ℱ}}_t\right\}}_{t\ge \mathrm{0}}$, satisfying the usual conditions (i.e., it is increasing and right continuous and ${\mathrm{ℱ}}_{\mathrm{0}}$ contains all $P$-null sets). Let ${\mathbb{R}}_{+}=\left[\mathrm{0},+\mathrm{\infty }\right),$$a\vee b=\mathrm{m}\mathrm{a}\mathrm{x}\{a,b\}$ and $a\wedge b=\mathrm{m}\mathrm{i}\mathrm{n}\{a,b\}$. Let $\mathbb{N}$ be the set of all natural integers.

Consider an $\overline{n}$-dimensional stochastic differential system

$\mathrm{d}\mathit{x}(t)=f(\mathit{x}(t))\mathrm{d}t+g(\mathit{x}(t))\mathrm{d}\mathit{w}(t)$(4)

on $t\ge \mathrm{0}$ with initial data ${\mathit{x}}_{\mathrm{0}}\in {\mathbb{R}}^{\overline{n}}$, and $f:{\mathbb{R}}^{\overline{n}}\to {\mathbb{R}}^{\overline{n}},g:{\mathbb{R}}^{\overline{n}}\to {\mathbb{R}}^{\overline{n}\times \overline{m}}$ are Borel-measurable.

Assumption 1 (local Lipschitz condition) For each real number $R\ge \mathrm{1}$, there is a positive constant $k_R$ such that

$\left|f\left({\mathit{x}}_{\mathrm{1}}\right)-f\left({\mathit{x}}_{\mathrm{2}}\right)\right|\vee \left|g\left({\mathit{x}}_{\mathrm{1}}\right)-g\left({\mathit{x}}_{\mathrm{2}}\right)\right|\le k_R\left|{\mathit{x}}_{\mathrm{1}}-{\mathit{x}}_{\mathrm{2}}\right|$(5)

for ${\mathit{x}}_i\in {\mathbb{R}}^{\overline{n}}$ with $\left|{\mathit{x}}_i\right|\le R,i=\mathrm{1,2}$.

Assumption 2 (one-sided polynomial growth condition) There exist $n\in \mathbb{N},p\ge \mathrm{2}$, and positive constants $a_{\mathrm{0}},a,\alpha ,a_i,{\alpha }_i,i=\mathrm{1,2},\dots ,n$ such that

$〈\mathit{x},f(\mathit{x})〉+\frac{p-\mathrm{1}}{\mathrm{2}}|g(\mathit{x}){|}^{\mathrm{2}}\le a_{\mathrm{0}}|\mathit{x}{|}^{\mathrm{2}}+{\displaystyle \sum _{i=\mathrm{1}}^n}{a}_i|\mathit{x}{|}^{{\alpha }_i+\mathrm{1}}-a|\mathit{x}{|}^{\alpha +\mathrm{2}}$(6)

for $\mathit{x}\in {\mathbb{R}}^{\overline{n}}$.

In this paper, we only require that the drift coefficient $f$ satisfies Assumptions 1 and 2. We remove the one-sided Lipschitz condition (2) in Refs. [8,9] and the polynomial growth condition (3) in Ref. [17]. The latter is a vital assumption in establishing the strong convergence rate of the implicit EM scheme for highly nonlinear SDEs.

It is easy to prove that there is a unique global solution to Eq. (4) under Assumptions 1 and 2 with $\alpha \ge \mathrm{2}{\alpha }_i$ and $a>{\displaystyle \sum _{i=\mathrm{1}}^n}{a}_i$ (see Ref. [17]). Let $R>\mathrm{0}$ be an arbitrary number and $\mathit{x}(t)$ be the solution of Eq. (4), define

${\sigma }_R=\mathrm{i}\mathrm{n}\mathrm{f}\{t\ge \mathrm{0}:|\mathit{x}(t)|\ge R\}$(7)

It is easy to show that there exists a positive constant $c_p$ such that

$E|\mathit{x}(t){|}^p\le c_p,P\left\{{\sigma }_R\le T\right\}\le \frac{c_p}{R^p},$

where $c_p$ represents a generic positive constant, whose value varies with each appearance throughout the paper.

The following result plays a key role in subsequent sections (For proof, please refer to page 425 of Ref. [3]).

Lemma 1^[3]Define a polynomial function $\phi (x)$ of a nonnegative real argument $x$ by $\phi (x)=a{x}^p+b{x}^{\gamma +p}-{\displaystyle \sum _{l=\mathrm{1}}^n} c_l{x}^{{\gamma }_l+p},$ where $a,b,p,\gamma ,c_l,{\gamma }_l(\mathrm{1}\le l\le n)$ are nonnegative real numbers satisfying $\mathrm{0}<{\gamma }_{\mathrm{1}}\le {\gamma }_{\mathrm{2}}\le \cdots \le {\gamma }_n\le \gamma$ and $b\ge c={\displaystyle \sum _{l=\mathrm{1}}^n} c_l$. If $a>p_{\mathrm{0}}c$, then $\phi (x)\ge \left(a-p_{\mathrm{0}}c\right)x^p,$ where the nonnegative constant $p_{\mathrm{0}}$ is

$p_{\mathrm{0}}=\{\begin{array}{ll}\mathrm{0},& {\gamma }_{\mathrm{1}}=\gamma \\ \left(\gamma -{\gamma }_{\mathrm{1}}\right){\left(\frac{{\gamma }_{\mathrm{1}}^{{\gamma }_{\mathrm{1}}}}{{\gamma }^{\gamma }}\right)}^{\frac{\mathrm{1}}{\gamma -{\gamma }_l}},& {\gamma }_{\mathrm{1}}\ne \gamma \end{array}$(8)

We shall now introduce the discrete truncated EM scheme. Choose a strictly increasing continuous function $\mu :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ such that $\mu (u)\to \mathrm{\infty }$ as $u\to \mathrm{\infty }$ and

$\underset{|\mathit{x}|\le u}{\mathrm{s}\mathrm{u}\mathrm{p}} |f(\mathit{x})|\vee |g(\mathit{x})|\le \mu (u),\forall u\ge \mathrm{1}$(9)

Denote by ${\mu }^{-\mathrm{1}}$ the inverse function of $\mu$, and ${\mu }^{-\mathrm{1}}:[\mu (\mathrm{0}),+\mathrm{\infty })\to {\mathbb{R}}_{+}$. Choose a number ${\mathrm{\Delta }}^{\mathrm{*}}\in (\mathrm{0,1})$, a strictly decreasing function $h:\left(\mathrm{0},{\mathrm{\Delta }}^{\mathrm{*}}\right)\to (\mathrm{0},+\mathrm{\infty })$ such that

$h\left({\mathrm{\Delta }}^{\mathrm{*}}\right)>\mu (\mathrm{1}),\underset{\mathrm{\Delta }\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}} h(\mathrm{\Delta })=\mathrm{\infty },{\mathrm{\Delta }}^{\frac{\mathrm{1}}{\mathrm{4}}}h(\mathrm{\Delta })\le \mathrm{1},\forall \mathrm{\Delta }\in \left(\mathrm{0},{\mathrm{\Delta }}^{\mathrm{*}}\right)$(10)

Let the step size $\mathrm{\Delta }\in \left(\mathrm{0},{\mathrm{\Delta }}^{\mathrm{*}}\right)$ be a fraction of $T$, namely $\mathrm{\Delta }=T/M$ for some integer $M$. Define a mapping $\pi :{\mathbb{R}}^{\overline{n}}\to \left\{\mathit{x}\in {\mathbb{R}}^{\overline{n}}:|\mathit{x}|\le {\mu }^{-\mathrm{1}}(h(\mathrm{\Delta }))\right\}$ by $\pi (\mathit{x})=\left(|\mathit{x}|\wedge {\mu }^{-\mathrm{1}}(h(\mathrm{\Delta }))\right)\frac{\mathit{x}}{|\mathit{x}|}$, where $\mathit{x}/|\mathit{x}|=\mathrm{0}$ when $\mathit{x}=\mathrm{0}$. Define the truncated functions

$f_{\mathrm{\Delta }}(\mathit{x})=f(\pi (\mathit{x})),g_{\mathrm{\Delta }}(\mathit{x})=g(\pi (\mathit{x}))$

It is easy to see that $\left|f_{\mathrm{\Delta }}(\mathit{x})\right|\vee \left|g_{\mathrm{\Delta }}(\mathit{x})\right|\le \mu \left({\mu }^{-\mathrm{1}}(h(\mathrm{\Delta }))\right)=h(\mathrm{\Delta }),\forall \mathit{x}\in {\mathbb{R}}^{\overline{n}}$.

Obviously, the truncated functions $f_{\mathrm{\Delta }}$ and $g_{\mathrm{\Delta }}$ are bounded, although $f$ and $g$ may not be.

Denote $t_k=k\mathrm{\Delta }$. The discrete-time truncated EM numerical solution ${\mathit{X}}_{\mathrm{\Delta }}\left(t_k\right)\left(\approx \mathit{x}\left(t_k\right)\right)$ is defined by

${\mathit{X}}_{\mathrm{\Delta }}\left(t_{k+\mathrm{1}}\right)={\mathit{X}}_{\mathrm{\Delta }}\left(t_k\right)+f_{\mathrm{\Delta }}\left({\mathit{X}}_{\mathrm{\Delta }}\left(t_k\right)\right)\mathrm{\Delta }+g_{\mathrm{\Delta }}\left({\mathit{X}}_{\mathrm{\Delta }}\left(t_k\right)\right)\mathrm{\Delta }{\mathit{w}}_{t_k},k=\mathrm{0,1},\mathrm{2},\cdots$(11)

where $\mathrm{\Delta }{\mathit{w}}_{t_k}=\mathit{w}\left(t_{k+\mathrm{1}}\right)-\mathit{w}\left(t_k\right)$. The increments $\mathrm{\Delta }{\mathit{w}}_{t_k}$ are independent $N(\mathrm{0},\mathrm{\Delta })$-distributed Gaussian random variables ${\mathrm{ℱ}}_{t_k}$-measurable at the mesh points $t_k$. Define two continuous-time truncated EM solutions as

${\overline{\mathit{x}}}_{\mathrm{\Delta }}\left(t\right)={\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{\infty }}}{\mathit{X}}_{\Delta }\left(t_k\right)I_{\left[t_k,t_{k+\mathrm{1}}\right)}\left(t\right),t\ge \mathrm{0}$(12)

${\mathit{x}}_{\mathrm{\Delta }}\left(t\right)={\mathit{x}}_{\mathrm{0}}+{\int }_{\mathrm{0}}^t f_{\mathrm{\Delta }}\left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\mathrm{d}s+{\int }_{\mathrm{0}}^t g_{\mathrm{\Delta }}\left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\mathrm{d}\mathit{w}(s),t\ge \mathrm{0}$(13)

where $I_{\left[t_k,t_{k+\mathrm{1}}\right)}(t)=\{\begin{array}{ll}\mathrm{1},& t\in \left[t_k,t_{k+\mathrm{1}}\right)\\ \mathrm{0},& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}$ is the indicator function. Clearly, ${\mathit{X}}_{\mathrm{\Delta }}\left(t_k\right)={\overline{\mathit{x}}}_{\mathrm{\Delta }}\left(t_k\right)=$${\mathit{x}}_{\mathrm{\Delta }}\left(t_k\right)$ for all $k\ge \mathrm{0}$.

The truncated functions preserve the Khasminskii-type condition nicely^[18]. They cannot preserve Assumption 2 exactly but piecewisely, as described in the following lemma.

Lemma 2   Let Assumption 2 hold. Denote$\tilde{a}=\left({\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{\infty }}}{a}_i+a_{\mathrm{0}}\right)/\mathrm{2}$, then for every$\Delta \in \left(\mathrm{0},{\Delta }^{*}\right)$and any$\mathit{x}\in {\mathbb{R}}^{\overline{n}}$, wehave

${\mathit{x}}^{\mathrm{T}}{f}_{\mathrm{\Delta }}(\mathit{x})+\frac{p-\mathrm{1}}{\mathrm{2}}{\left|g_{\mathrm{\Delta }}(\mathit{x})\right|}^{\mathrm{2}}\le \{\begin{array}{ll}{a}_{\mathrm{0}}|\mathit{x}{|}^{\mathrm{2}}+{\displaystyle \sum _{i=\mathrm{1}}^n}{a}_i|\mathit{x}{|}^{{\alpha }_i+\mathrm{1}}-a|\mathit{x}{|}^{\alpha +\mathrm{2}},& |\mathit{x}|\le {\mu }^{-\mathrm{1}}(h(\mathrm{\Delta }))\\ \frac{\mathrm{1}}{\mathrm{2}}{a}_{\mathrm{0}}|\pi (\mathit{x}){|}^{\mathrm{4}}+\tilde{a}|\mathit{x}{|}^{\mathrm{2}}+{\displaystyle \sum _{i=\mathrm{1}}^n}\frac{a_i}{\mathrm{2}}|\pi (\mathit{x}){|}^{\mathrm{2}{\alpha }_i+\mathrm{2}}-a|\pi (\mathit{x}){|}^{\alpha +\mathrm{2}},& |\mathit{x}|>{\mu }^{-\mathrm{1}}(h(\mathrm{\Delta }))\end{array}$(14)

Proof   Since $\mu (\cdot )$ is increasing, $h(\cdot )$ is decreasing and $h\left({\mathrm{\Delta }}^{\mathrm{*}}\right)>\mu (\mathrm{1})$, we have ${\mu }^{-\mathrm{1}}\left(h\left({\mathrm{\Delta }}^{\mathrm{*}}\right)\right)>\mathrm{1}$ and ${\mu }^{-\mathrm{1}}(h(\mathrm{\Delta }))>\mathrm{1}$ for $\mathrm{\Delta }\in \left(\mathrm{0},{\mathrm{\Delta }}^{\mathrm{*}}\right)$. For any $\mathit{x}\in {\mathbb{R}}^{\overline{n}}$ with $|\mathit{x}|\le {\mu }^{-\mathrm{1}}(h(\mathrm{\Delta }))$, it is clear that $\pi (\mathit{x})=\mathit{x}$ and hence

${\mathit{x}}^{\mathrm{T}}{f}_{\mathrm{\Delta }}(\mathit{x})+\frac{p-\mathrm{1}}{\mathrm{2}}{\left|g_{\mathrm{\Delta }}(\mathit{x})\right|}^{\mathrm{2}}={\mathit{x}}^{\mathrm{T}}f(\mathit{x})+\frac{p-\mathrm{1}}{\mathrm{2}}|g(\mathit{x}){|}^{\mathrm{2}}\le a_{\mathrm{0}}|\mathit{x}{|}^{\mathrm{2}}+{\displaystyle \sum _{i=\mathrm{1}}^n}{a}_i|\mathit{x}{|}^{{\alpha }_i+\mathrm{1}} -a|\mathit{x}{|}^{\alpha +\mathrm{2}}$(15)

For any $\mathit{x}\in {\mathbb{R}}^{\overline{n}}$ with $|\mathit{x}|>{\mu }^{-\mathrm{1}}(h(\mathrm{\Delta }))$, we have $\mathit{x}=\frac{|\mathit{x}|\pi (\mathit{x})}{{\mu }^{-\mathrm{1}}(h(\mathrm{\Delta }))},|\pi (\mathit{x})|<|\mathit{x}|$ and hence

${\mathit{x}}^{\mathrm{T}}{f}_{\mathrm{\Delta }}(\mathit{x})+\frac{p-\mathrm{1}}{\mathrm{2}}{\left|g_{\mathrm{\Delta }}(\mathit{x})\right|}^{\mathrm{2}}={\mathit{x}}^{\mathrm{T}}f(\pi (\mathit{x}))+\frac{p-\mathrm{1}}{\mathrm{2}}|g(\pi (\mathit{x})){|}^{\mathrm{2}}=\pi {(\mathit{x})}^{\mathrm{T}}f(\pi (\mathit{x}))+\frac{p-\mathrm{1}}{\mathrm{2}}|g(\pi (\mathit{x})){|}^{\mathrm{2}}+\left(\frac{|\mathit{x}|}{{\mu }^{-\mathrm{1}}(h(\mathrm{\Delta }))}-\mathrm{1}\right)\pi {(\mathit{x})}^{\mathrm{T}}f(\pi (\mathit{x})).$

By Assumption 2 and the inequality $ab\le \frac{a^{\mathrm{2}}+b^{\mathrm{2}}}{\mathrm{2}}$, we may compute

$\begin{array}{l}{\mathit{x}}^{\mathrm{T}}{f}_{\mathrm{\Delta }}(\mathit{x})+\frac{p-\mathrm{1}}{\mathrm{2}}{\left|g_{\mathrm{\Delta }}(\mathit{x})\right|}^{\mathrm{2}}\le a_{\mathrm{0}}|\pi (\mathit{x}){|}^{\mathrm{2}}+{\displaystyle \sum _{i=\mathrm{1}}^n} a_i|\pi (\mathit{x}){|}^{{\alpha }_i+\mathrm{1}}-a|\pi (\mathit{x}){|}^{\alpha +\mathrm{2}}+\left(\frac{|\mathit{x}|}{{\mu }^{-\mathrm{1}}(h(\mathrm{\Delta }))}-\mathrm{1}\right)\left({\displaystyle \sum _{i=\mathrm{1}}^n} a_i|\pi (\mathit{x}){|}^{{\alpha }_i+\mathrm{1}}+a_{\mathrm{0}}|\pi (\mathit{x}){|}^{\mathrm{2}}\right)\\ =-a|\pi (\mathit{x}){|}^{\alpha +\mathrm{2}}+\frac{|\mathit{x}|}{{\mu }^{-\mathrm{1}}(h(\mathrm{\Delta }))}\left({\displaystyle \sum _{i=\mathrm{1}}^n} a_i|\pi (\mathit{x}){|}^{{\alpha }_i+\mathrm{1}}+a_{\mathrm{0}}|\pi (\mathit{x}){|}^{\mathrm{2}}\right)\\ \le -a|\pi (\mathit{x}){|}^{\alpha +\mathrm{2}}+{\displaystyle \sum _{i=\mathrm{1}}^n} \frac{a_i}{\mathrm{2}}\left(|\pi (\mathit{x}){|}^{\mathrm{2}{\alpha }_i+\mathrm{2}}+|\mathit{x}{|}^{\mathrm{2}}\right)+\frac{a_{\mathrm{0}}}{\mathrm{2}}\left(|\pi (\mathit{x}){|}^{\mathrm{4}}+|\mathit{x}{|}^{\mathrm{2}}\right).\end{array}$

This completes the proof.

Lemma 3   Fix$T>\mathrm{0}$. Let Assumptions 1 and 2 hold. Then for any$\Delta \in \left(\mathrm{0},{\Delta }^{*}\right)$and$p>\mathrm{0}$, there exists $c_p$, a genericpositive constant dependent on$T,p,k_R$but independent of$\Delta$, such that

$E{\left|{\mathit{x}}_{\mathrm{\Delta }}(t)-{\overline{\mathit{x}}}_{\mathrm{\Delta }}(t)\right|}^p\le c_p{\mathrm{\Delta }}^{\frac{p}{\mathrm{2}}}(h{(\mathrm{\Delta }))}^p,\mathrm{0}\le t\le T.$

Proof   For $\forall t\in [\mathrm{0},T]$, there exists a unique nonnegative integer $k$ such that $t\in \left[t_k,t_{k+\mathrm{1}}\right)$. Since $\left|f_{\mathrm{\Delta }}\left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\right|\vee$$\left|g_{\mathrm{\Delta }}\left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\right|\le h(\mathrm{\Delta })$, we have

$\begin{array}{l}E{\left|{\mathit{x}}_{\mathrm{\Delta }}(t)-{\overline{\mathit{x}}}_{\mathrm{\Delta }}(t)\right|}^p\le E{\left|{\int }_{t_k}^t f_{\mathrm{\Delta }}\left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\mathrm{d}s+{\int }_{t_k}^t g_{\mathrm{\Delta }}\left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\mathrm{d}\mathit{w}(s)\right|}^p\\ \le c_p\left({\mathrm{\Delta }}^{p-\mathrm{1}}E{\int }_{t_k}^t {\left|f_{\mathrm{\Delta }}\left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\right|}^p\mathrm{d}s+{\mathrm{\Delta }}^{p/\mathrm{2}-\mathrm{1}}{\int }_{t_k}^t {\left|g_{\mathrm{\Delta }}\left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\right|}^p\mathrm{d}s\right)\le c_p{\mathrm{\Delta }}^{\frac{p}{\mathrm{2}}}(h{(\mathrm{\Delta }))}^p\end{array}$

by Eqs. (12) and (13) and the Hölder inequality. For $\mathrm{0}<\overline{p}<\mathrm{2}$, the Lyapunov inequality gives

$E{\left|\mathit{x}(t)-{\overline{\mathit{x}}}_{\mathrm{\Delta }}(t)\right|}^{\overline{p}}\le {\left(E{\left|\mathit{x}(t)-{\overline{\mathit{x}}}_{\mathrm{\Delta }}(t)\right|}^p\right)}^{\frac{\overline{p}}{p}}\le {\left(c_p{\mathrm{\Delta }}^{\frac{p}{\mathrm{2}}}(h{(\mathrm{\Delta }))}^p\right)}^{\frac{\overline{p}}{p}}=c_p{\mathrm{\Delta }}^{\frac{\overline{p}}{\mathrm{2}}}(h{(\mathrm{\Delta }))}^{\overline{p}}.$

This proof is completed.

Let $R>\mathrm{0}$ be an arbitrary number and ${\mathit{x}}_{\mathrm{\Delta }}(t)$ be the continuous-time truncated EM solution defined by Eq. (13), define

${\rho }_R=\mathrm{i}\mathrm{n}\mathrm{f}\left\{t\ge \mathrm{0}:\left|{\mathit{x}}_{\mathrm{\Delta }}(t)\right|\ge R\right\}$(16)

Lemma 4   Let Assumption 2 hold, $\alpha \ge \mathrm{2}{\alpha }_i\vee \mathrm{2}(i=\mathrm{1},\cdots ,n)$and$a>{\displaystyle \sum _{i=\mathrm{1}}^n} a_i+a_{\mathrm{0}}/\mathrm{2}$. Then for$\Delta \in \left(\mathrm{0},{\Delta }^{*}\right)$and$p>\mathrm{0}$, there exists a positive constant$c_p$such that

$\underset{\mathrm{0}<\mathrm{\Delta }<{\mathrm{\Delta }}^{\mathrm{*}}}{\mathrm{s}\mathrm{u}\mathrm{p}} \underset{\mathrm{0}\le t\le T}{\mathrm{s}\mathrm{u}\mathrm{p}} E{\left|{\mathit{x}}_{\mathrm{\Delta }}(t)\right|}^p\le c_p,\forall T>\mathrm{0}.$

Moreover,

$P\left\{{\rho }_R\le T\right\}\le \frac{c_p}{R^p}.$

Proof   We prove the results are true for $p\ge \mathrm{2}$ first. For any $s>\mathrm{0}$, there exists a unique nonnegative integer $k$ such that $s\in \left[t_k,t_{k+\mathrm{1}}\right)$. For $s\le {\rho }_R$, we see that $\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|\le R$ by definition of ${\rho }_R$; for $s\le {\rho }_R$, we deduce that $\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|\le R$ as well due to ${\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\equiv {\overline{\mathit{x}}}_{\mathrm{\Delta }}\left(t_k\right)={\mathit{x}}_{\mathrm{\Delta }}\left(t_k\right)$ for $s\in \left[t_k,t_{k+\mathrm{1}}\right)$. The Itô formula then gives

$\begin{array}{l}E{\left|{\mathit{x}}_{\mathrm{\Delta }}\left(t\wedge {\rho }_R\right)\right|}^p\le E{\left|{\mathit{x}}_{\mathrm{0}}\right|}^p+E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} p{\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|}^{p-\mathrm{2}}\left({\mathit{x}}_{\mathrm{\Delta }}{(s)}^{\mathrm{T}}{f}_{\mathrm{\Delta }}\left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)+\frac{p-\mathrm{1}}{\mathrm{2}}{\left|g_{\mathrm{\Delta }}\left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\right|}^{\mathrm{2}}\right)\mathrm{d}s\\ \le E{\left|{\mathit{x}}_{\mathrm{0}}\right|}^{\mathrm{2}}+{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} p{\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|}^{p-\mathrm{2}}\left({\overline{\mathit{x}}}_{\mathrm{\Delta }}{(s)}^{\mathrm{T}}{f}_{\mathrm{\Delta }}\left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)+\frac{p-\mathrm{1}}{\mathrm{2}}{\left|g_{\mathrm{\Delta }}\left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\right|}^{\mathrm{2}}\right)\mathrm{d}s+{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} p{\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|}^{p-\mathrm{2}}{\left({\mathit{x}}_{\mathrm{\Delta }}(s)-{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)}^{\mathrm{T}}{f}_{\mathrm{\Delta }}\left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\mathrm{d}s.\end{array}$

Depending on whether $\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|\le {\mu }^{-\mathrm{1}}(h(\mathrm{\Delta }))$, the rest of the proof falls into two cases:

Case 1: For any ${\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\in {\mathbb{R}}^{\overline{n}}$ with $\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|\le {\mu }^{-\mathrm{1}}(h(\mathrm{\Delta }))$, by Lemma 2, we have

$\begin{array}{l}E{\left|{\mathit{x}}_{\mathrm{\Delta }}\left(t\wedge {\rho }_R\right)\right|}^p\le E{\left|{\mathit{x}}_{\mathrm{0}}\right|}^p+E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} p{\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|}^{p-\mathrm{2}}\left(a_{\mathrm{0}}{\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|}^{\mathrm{2}}+{\displaystyle \sum _{i=\mathrm{1}}^n} a_i{\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|}^{{\alpha }_i+\mathrm{1}}-a{\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|}^{\alpha +\mathrm{2}}\right)\mathrm{d}s\\ +E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} p{\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|}^{p-\mathrm{2}}{\left({\mathit{x}}_{\mathrm{\Delta }}(s)-{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)}^{\mathrm{T}}{f}_{\mathrm{\Delta }}\left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\mathrm{d}s\\ \le E{\left|{\mathit{x}}_{\mathrm{0}}\right|}^p+\mathrm{2}p{a}_{\mathrm{0}}E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} {\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|}^{p-\mathrm{2}}{\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|}^{\mathrm{2}}\mathrm{d}s\\ +E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} p{\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|}^{p-\mathrm{2}}\left(-a_{\mathrm{0}}{\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|}^{\mathrm{2}}+{\displaystyle \sum _{i=\mathrm{1}}^n} a_i{\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|}^{{\alpha }_i+\mathrm{1}}-a{\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|}^{\alpha +\mathrm{2}}\right)\mathrm{d}s\\ +E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} p{\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|}^{p-\mathrm{2}}{\left({\mathit{x}}_{\mathrm{\Delta }}(s)-{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)}^{\mathrm{T}}{f}_{\mathrm{\Delta }}\left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\mathrm{d}s.\end{array}$

According to Lemma $\mathrm{1},\alpha \ge \mathrm{2}{\alpha }_i$ and $a>{\displaystyle \sum _{i=\mathrm{1}}^n} a_i$, there is a constant $c_{\mathrm{0}}$ such that

$a_{\mathrm{0}}{\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|}^{\mathrm{2}}-{\displaystyle \sum _{i=\mathrm{1}}^n} a_i{\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|}^{{\alpha }_i+\mathrm{1}}+a{\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|}^{\alpha +\mathrm{2}}\ge c_{\mathrm{0}}{\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|}^{\mathrm{2}}.$

This, together with the Young's inequality $a^{p-\mathrm{2}}b\le \frac{p-\mathrm{2}}{p}{a}^p+\frac{\mathrm{2}}{p}{b}^{p/\mathrm{2}}$ for $a,b\ge \mathrm{0}$, implies

$\begin{array}{l}E{\left|{\mathit{x}}_{\mathrm{\Delta }}\left(t\wedge {\rho }_R\right)\right|}^p\le E{\left|{\mathit{x}}_{\mathrm{0}}\right|}^p+\mathrm{2}{a}_{\mathrm{0}}E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} \left[(p-\mathrm{2}){\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|}^p+\mathrm{2}{\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|}^p\right]\mathrm{d}s+(p-\mathrm{2})E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} {\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|}^p\mathrm{d}s\\ +\mathrm{2}E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} {\left|{\mathit{x}}_{\mathrm{\Delta }}(s)-{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|}^{p/\mathrm{2}}{\left|f_{\mathrm{\Delta }}\left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\right|}^{p/\mathrm{2}}\mathrm{d}s\\ \le E{\left|{\mathit{x}}_{\mathrm{0}}\right|}^{\mathrm{2}}+\left(a_{\mathrm{0}}+\mathrm{1}\right)(p-\mathrm{2})E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} {\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|}^p\mathrm{d}s+\mathrm{4}{a}_{\mathrm{0}}E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} {\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|}^p\mathrm{d}s+\mathrm{2}h{(\mathrm{\Delta })}^{p/\mathrm{2}}E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} {\left|{\mathit{x}}_{\mathrm{\Delta }}(s)-{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|}^{p/\mathrm{2}}\mathrm{d}s.\end{array}$

By the Lyapunov inequality and Lemma 3, the above estimate becomes

$E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} {\left|{\mathit{x}}_{\mathrm{\Delta }}(s)-{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|}^{p/\mathrm{2}}\mathrm{d}s\le {\int }_{\mathrm{0}}^t {\left(E{\left|{\mathit{x}}_{\mathrm{\Delta }}\left(s\wedge {\rho }_R\right)-{\overline{\mathit{x}}}_{\mathrm{\Delta }}\left(s\wedge {\rho }_R\right)\right|}^p\right)}^{\mathrm{1}/\mathrm{2}}\mathrm{d}s\le c_{p}h{(\mathrm{\Delta })}^{p/\mathrm{2}}{\mathrm{\Delta }}^{p/\mathrm{4}}.$

Noticing that $\underset{\mathrm{0}\le t\le T}{\mathrm{s}\mathrm{u}\mathrm{p}} E{\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|}^p\le \underset{\mathrm{0}\le t\le T}{\mathrm{s}\mathrm{u}\mathrm{p}} E{\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|}^p$,

$\underset{\mathrm{0}\le t\le T}{\mathrm{s}\mathrm{u}\mathrm{p}} E{\left|{\mathit{x}}_{\mathrm{\Delta }}\left(t\wedge {\rho }_R\right)\right|}^p\le E{\left|{\mathit{x}}_{\mathrm{0}}\right|}^{\mathrm{2}}+\left[\mathrm{2}\left(a_{\mathrm{0}}+\mathrm{1}\right)p-\mathrm{2}\right]{\int }_{\mathrm{0}}^T \underset{\mathrm{0}\le v\le s}{\mathrm{s}\mathrm{u}\mathrm{p}} E{\left|{\mathit{x}}_{\mathrm{\Delta }}\left(v\wedge {\rho }_R\right)\right|}^p\mathrm{d}s+c_{p}h{(\mathrm{\Delta })}^p{\mathrm{\Delta }}^{p/\mathrm{4}}.$

Using the Gronwall inequality , let $R\to \mathrm{\infty }$, the Fatou lemma gives

$\underset{\mathrm{0}\le \mathrm{\Delta }\le {\mathrm{\Delta }}^{\mathrm{*}}}{\mathrm{l}\mathrm{i}\mathrm{m}} \underset{\mathrm{0}\le t\le T}{\mathrm{s}\mathrm{u}\mathrm{p}} E{\left|{\mathit{x}}_{\mathrm{\Delta }}(t)\right|}^p\le \left(E{\left|{\mathit{x}}_{\mathrm{0}}\right|}^{\mathrm{2}}+c_{p}h{(\mathrm{\Delta })}^p{\mathrm{\Delta }}^{p/\mathrm{4}}\right){\mathrm{e}}^{\left(\mathrm{2}\left(a_{\mathrm{0}}+\mathrm{1}\right)p-\mathrm{2}\right)T}\equiv c_p.$

Case 2: For any ${\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\in {\mathbb{R}}^{\overline{n}}$ with $\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|>{\mu }^{-\mathrm{1}}(h(\mathrm{\Delta }))$, we obtain by Lemma 2 :

$\begin{array}{l}E{\left|{\mathit{x}}_{\mathrm{\Delta }}\left(t\wedge {\rho }_R\right)\right|}^p\le E{\left|{\mathit{x}}_{\mathrm{0}}\right|}^p+p\tilde{a}E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} {\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|}^{p-\mathrm{2}}{\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|}^{\mathrm{2}}\mathrm{d}s\\ +E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} p{\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|}^{p-\mathrm{2}}\left[\frac{a_{\mathrm{0}}}{\mathrm{2}}{\left|\pi \left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\right|}^{\mathrm{4}}-a{\left|\pi \left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\right|}^{\alpha +\mathrm{2}}+{\displaystyle \sum _{i=\mathrm{1}}^n} \frac{a_i}{\mathrm{2}}{\left|\pi \left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\right|}^{\mathrm{2}{\alpha }_i+\mathrm{2}}\right]\mathrm{d}s\\ +E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} p{\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|}^{p-\mathrm{2}}{\left({\mathit{x}}_{\mathrm{\Delta }}(s)-{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)}^{\mathrm{T}}{f}_{\mathrm{\Delta }}\left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\mathrm{d}s.\end{array}$

Note that ${\left|\pi \left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\right|}^{\mathrm{2}}\le \left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|$, then

$\begin{array}{l}E{\left|{\mathit{x}}_{\mathrm{\Delta }}\left(t\wedge {\rho }_R\right)\right|}^p\le E{\left|{\mathit{x}}_{\mathrm{0}}\right|}^p+E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} p{\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|}^{p-\mathrm{2}}{\left({\mathit{x}}_{\mathrm{\Delta }}(s)-{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)}^{\mathrm{T}}{f}_{\mathrm{\Delta }}\left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\mathrm{d}s-\frac{a_{\mathrm{0}}}{\mathrm{2}}{\left|\pi \left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\right|}^{\mathrm{4}}\\ -E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} p{\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|}^{p-\mathrm{2}}\left[a_{\mathrm{0}}{\left|\pi \left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\right|}^{\mathrm{2}}+a{\left|\pi \left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\right|}^{\alpha +\mathrm{2}}-{\displaystyle \sum _{i=\mathrm{1}}^n} \frac{a_i}{\mathrm{2}}{\left|\pi \left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\right|}^{\mathrm{2}{\alpha }_i+\mathrm{2}}\right]\mathrm{d}s\\ +p{a}_{\mathrm{0}}E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} {\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|}^{p-\mathrm{2}}{\left|\pi \left({\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right)\right|}^{\mathrm{2}}\mathrm{d}s+p\tilde{a}E{\int }_{\mathrm{0}}^{t\wedge {\rho }_R} {\left|{\mathit{x}}_{\mathrm{\Delta }}(s)\right|}^{p-\mathrm{2}}{\left|{\overline{\mathit{x}}}_{\mathrm{\Delta }}(s)\right|}^{\mathrm{2}}\mathrm{d}s.\end{array}$