Convergence Rates for the Truncated Euler-Maruyama Method for Nonlinear Stochastic Differential Equations

Xuejing MENG; Linfeng LYU

doi:10.1051/wujns/2023285399

Open Access

Issue		Wuhan Univ. J. Nat. Sci. Volume 28, Number 5, October 2023


Page(s)		399 - 410
DOI		https://doi.org/10.1051/wujns/2023285399
Published online		10 November 2023

Wuhan University Journal of Natural Sciences, 2023, Vol.28 No.5, 399-410

Mathematics

CLC number: O241.8

Convergence Rates for the Truncated Euler-Maruyama Method for Nonlinear Stochastic Differential Equations

Xuejing MENG¹^,2 and Linfeng LYU³

¹ School of Statistics and Mathematics, Hubei University of Economics, Wuhan 430205, Hubei, China
² Hubei Center for Data and Analysis, Hubei University of Economics, Wuhan 430205, Hubei, China
³ School of Science, Wuhan University of Technology, Wuhan 430074, Hubei, China

Received: 16 April 2023

Abstract

In this paper, our main aim is to investigate the strong convergence rate of the truncated Euler-Maruyama approximations for stochastic differential equations with superlinearly growing drift coefficients. When the diffusion coefficient is polynomially growing or linearly growing, the strong convergence rate of arbitrarily close to one half is established at a single time T or over a time interval [0,T], respectively. In both situations, the common one-sided Lipschitz and polynomial growth conditions for the drift coefficients are not required. Two examples are provided to illustrate the theory.

Key words: truncated Euler-Maruyama method / strong convergence / moment boundedness

Biography: MENG Xuejing, female, Associate professor, research direction: stochastic differential equations and applications. E-mail:This email address is being protected from spambots. You need JavaScript enabled to view it.

© Wuhan University 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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:

$〈 x, f (x) 〉 + \frac{p - 1}{2} | g (x) |^{2} \leq K (1 + | x |^{2}), x \in R^{\bar{n}}$ Mathematical equation (1)

the one-sided Lipschitz condition:

$(x - y) (f (x) - f (y)) \leq L_{1} {(x - y)}^{2}$ Mathematical equation (2)

and the polynomial growth condition:

$| f (x) - f (y) |^{2} \leq H (1 + | x |^{γ} + | y |^{γ}) | x - y |^{2}$ Mathematical equation (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 $f$ Mathematical equation 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 $T$ Mathematical equation 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 $[0, T]$ .

1 SDEs with Polynomial Growing Coefficients

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

Consider an $\bar{n}$ Mathematical equation -dimensional stochastic differential system

$d x (t) = f (x (t)) d t + g (x (t)) d w (t)$ Mathematical equation (4)

on $t \geq 0$ Mathematical equation with initial data $x_{0} \in R^{\bar{n}}$ , and $f : R^{\bar{n}} \to R^{\bar{n}}, g : R^{\bar{n}} \to R^{\bar{n} \times \bar{m}}$ are Borel-measurable.

Assumption 1 (local Lipschitz condition) For each real number $R \geq 1$ Mathematical equation , there is a positive constant $k_{R}$ such that

$| f (x_{1}) - f (x_{2}) | \lor | g (x_{1}) - g (x_{2}) | \leq k_{R} | x_{1} - x_{2} |$ Mathematical equation (5)

for $x_{i} \in R^{\bar{n}}$ Mathematical equation with $| x_{i} | \leq R, i = 1,2$ .

Assumption 2 (one-sided polynomial growth condition) There exist $n \in N, p \geq 2$ Mathematical equation , and positive constants $a_{0}, a, α, a_{i}, α_{i}, i = 1,2, \dots, n$ such that

$〈 x, f (x) 〉 + \frac{p - 1}{2} | g (x) |^{2} \leq a_{0} | x |^{2} + \sum_{i = 1}^{n} a_{i} | x |^{α_{i} + 1} - a | x |^{α + 2}$ Mathematical equation (6)

for $x \in R^{\bar{n}}$ Mathematical equation .

In this paper, we only require that the drift coefficient $f$ Mathematical equation 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 $α \geq 2 α_{i}$ Mathematical equation and $a > \sum_{i = 1}^{n} a_{i}$ (see Ref. [17]). Let $R > 0$ be an arbitrary number and $x (t)$ be the solution of Eq. (4), define

$σ_{R} = i n f {t \geq 0 : | x (t) | \geq R}$ Mathematical equation (7)

It is easy to show that there exists a positive constant $c_{p}$ Mathematical equation such that

$E | x (t) |^{p} \leq c_{p}, P {σ_{R} \leq T} \leq \frac{c_{p}}{R^{p}},$ Mathematical equation

where $c_{p}$ Mathematical equation 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 $φ (x)$ Mathematical equation of a nonnegative real argument $x$ by $φ (x) = a x^{p} + b x^{γ + p} - \sum_{l = 1}^{n} c_{l} x^{γ_{l} + p},$ where $a, b, p, γ, c_{l}, γ_{l} (1 \leq l \leq n)$ are nonnegative real numbers satisfying $0 < γ_{1} \leq γ_{2} \leq \dots \leq γ_{n} \leq γ$ and $b \geq c = \sum_{l = 1}^{n} c_{l}$ Mathematical equation . If $a > p_{0} c$ , then $φ (x) \geq (a - p_{0} c) x^{p},$ where the nonnegative constant $p_{0}$ is

$p_{0} = {\begin{array}{l} 0, & γ_{1} = γ \\ (γ - γ_{1}) {(\frac{γ_{1}^{γ_{1}}}{γ^{γ}})}^{\frac{1}{γ - γ_{l}}}, & γ_{1} \neq γ \end{array}$ Mathematical equation (8)

We shall now introduce the discrete truncated EM scheme. Choose a strictly increasing continuous function $μ : R_{+} \to R_{+}$ Mathematical equation such that $μ (u) \to \infty$ as $u \to \infty$ and

$\underset{| x | \leq u}{s u p} | f (x) | \lor | g (x) | \leq μ (u), \forall u \geq 1$ Mathematical equation (9)

Denote by $μ^{- 1}$ Mathematical equation the inverse function of $μ$ , and $μ^{- 1} : [μ (0), + \infty) \to R_{+}$ . Choose a number $Δ^{*} \in (0,1)$ , a strictly decreasing function $h : (0, Δ^{*}) \to (0, + \infty)$ such that

$h (Δ^{*}) > μ (1), \underset{Δ \to 0}{l i m} h (Δ) = \infty, Δ^{\frac{1}{4}} h (Δ) \leq 1, \forall Δ \in (0, Δ^{*})$ Mathematical equation (10)

Let the step size $Δ \in (0, Δ^{*})$ Mathematical equation be a fraction of $T$ , namely $Δ = T / M$ for some integer $M$ . Define a mapping $π : R^{\bar{n}} \to {x \in R^{\bar{n}} : | x | \leq μ^{- 1} (h (Δ))}$ by $π (x) = (| x | \land μ^{- 1} (h (Δ))) \frac{x}{| x |}$ , where $x / | x | = 0$ when $x = 0$ . Define the truncated functions

$f_{Δ} (x) = f (π (x)), g_{Δ} (x) = g (π (x))$ Mathematical equation

It is easy to see that $| f_{Δ} (x) | \lor | g_{Δ} (x) | \leq μ (μ^{- 1} (h (Δ))) = h (Δ), \forall x \in R^{\bar{n}}$ Mathematical equation .

Obviously, the truncated functions $f_{Δ}$ Mathematical equation and $g_{Δ}$ are bounded, although $f$ and $g$ may not be.

Denote $t_{k} = k Δ$ Mathematical equation . The discrete-time truncated EM numerical solution $X_{Δ} (t_{k}) (\approx x (t_{k}))$ is defined by

$X_{Δ} (t_{k + 1}) = X_{Δ} (t_{k}) + f_{Δ} (X_{Δ} (t_{k})) Δ + g_{Δ} (X_{Δ} (t_{k})) Δ w_{t_{k}}, k = 0,1, 2, \dots$ Mathematical equation (11)

where $Δ w_{t_{k}} = w (t_{k + 1}) - w (t_{k})$ Mathematical equation . The increments $Δ w_{t_{k}}$ are independent $N (0, Δ)$ -distributed Gaussian random variables $ℱ_{t_{k}}$ -measurable at the mesh points $t_{k}$ . Define two continuous-time truncated EM solutions as

${\bar{x}}_{Δ} (t) = \sum_{i = 1}^{\infty} X_{Δ} (t_{k}) I_{[t_{k}, t_{k + 1})} (t), t \geq 0$ Mathematical equation (12)

$x_{Δ} (t) = x_{0} + \int_{0}^{t} f_{Δ} ({\bar{x}}_{Δ} (s)) d s + \int_{0}^{t} g_{Δ} ({\bar{x}}_{Δ} (s)) d w (s), t \geq 0$ Mathematical equation (13)

where $I_{[t_{k}, t_{k + 1})} (t) = {\begin{array}{l} 1, & t \in [t_{k}, t_{k + 1}) \\ 0, & o t h e r w i s e \end{array}$ Mathematical equation is the indicator function. Clearly, $X_{Δ} (t_{k}) = {\bar{x}}_{Δ} (t_{k}) =$ $x_{Δ} (t_{k})$ for all $k \geq 0$ .

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 $\tilde{a} = (\sum_{i = 1}^{\infty} a_{i} + a_{0}) / 2$ Mathematical equation , then for every $Δ \in (0, Δ^{*})$ and any $x \in R^{\bar{n}}$ , wehave

$x^{T} f_{Δ} (x) + \frac{p - 1}{2} {| g_{Δ} (x) |}^{2} \leq {\begin{array}{l} a_{0} | x |^{2} + \sum_{i = 1}^{n} a_{i} | x |^{α_{i} + 1} - a | x |^{α + 2}, & | x | \leq μ^{- 1} (h (Δ)) \\ \frac{1}{2} a_{0} | π (x) |^{4} + \tilde{a} | x |^{2} + \sum_{i = 1}^{n} \frac{a_{i}}{2} | π (x) |^{2 α_{i} + 2} - a | π (x) |^{α + 2}, & | x | > μ^{- 1} (h (Δ)) \end{array}$ Mathematical equation (14)

Proof Since $μ (\cdot)$ Mathematical equation is increasing, $h (\cdot)$ is decreasing and $h (Δ^{*}) > μ (1)$ , we have $μ^{- 1} (h (Δ^{*})) > 1$ and $μ^{- 1} (h (Δ)) > 1$ for $Δ \in (0, Δ^{*})$ . For any $x \in R^{\bar{n}}$ with $| x | \leq μ^{- 1} (h (Δ))$ , it is clear that $π (x) = x$ and hence

$x^{T} f_{Δ} (x) + \frac{p - 1}{2} {| g_{Δ} (x) |}^{2} = x^{T} f (x) + \frac{p - 1}{2} | g (x) |^{2} \leq a_{0} | x |^{2} + \sum_{i = 1}^{n} a_{i} | x |^{α_{i} + 1} - a | x |^{α + 2}$ Mathematical equation (15)

For any $x \in R^{\bar{n}}$ Mathematical equation with $| x | > μ^{- 1} (h (Δ))$ , we have $x = \frac{| x | π (x)}{μ^{- 1} (h (Δ))}, | π (x) | < | x |$ and hence

$x^{T} f_{Δ} (x) + \frac{p - 1}{2} {| g_{Δ} (x) |}^{2} = x^{T} f (π (x)) + \frac{p - 1}{2} | g (π (x)) |^{2} = π {(x)}^{T} f (π (x)) + \frac{p - 1}{2} | g (π (x)) |^{2} + (\frac{| x |}{μ^{- 1} (h (Δ))} - 1) π {(x)}^{T} f (π (x)) .$ Mathematical equation

By Assumption 2 and the inequality $a b \leq \frac{a^{2} + b^{2}}{2}$ Mathematical equation , we may compute

$\begin{array}{l} x^{T} f_{Δ} (x) + \frac{p - 1}{2} {| g_{Δ} (x) |}^{2} \leq a_{0} | π (x) |^{2} + \sum_{i = 1}^{n} a_{i} | π (x) |^{α_{i} + 1} - a | π (x) |^{α + 2} + (\frac{| x |}{μ^{- 1} (h (Δ))} - 1) (\sum_{i = 1}^{n} a_{i} | π (x) |^{α_{i} + 1} + a_{0} | π (x) |^{2}) \\ = - a | π (x) |^{α + 2} + \frac{| x |}{μ^{- 1} (h (Δ))} (\sum_{i = 1}^{n} a_{i} | π (x) |^{α_{i} + 1} + a_{0} | π (x) |^{2}) \\ \leq - a | π (x) |^{α + 2} + \sum_{i = 1}^{n} \frac{a_{i}}{2} (| π (x) |^{2 α_{i} + 2} + | x |^{2}) + \frac{a_{0}}{2} (| π (x) |^{4} + | x |^{2}) . \end{array}$ Mathematical equation

This completes the proof.

Lemma 3 Fix $T > 0$ Mathematical equation . Let Assumptions 1 and 2 hold. Then for any $Δ \in (0, Δ^{*})$ and $p > 0$ , there exists $c_{p}$ , a genericpositive constant dependent on $T, p, k_{R}$ but independent of $Δ$ , such that

$E {| x_{Δ} (t) - {\bar{x}}_{Δ} (t) |}^{p} \leq c_{p} Δ^{\frac{p}{2}} (h {(Δ))}^{p}, 0 \leq t \leq T .$ Mathematical equation

Proof For $\forall t \in [0, T]$ Mathematical equation , there exists a unique nonnegative integer $k$ such that $t \in [t_{k}, t_{k + 1})$ . Since $| f_{Δ} ({\bar{x}}_{Δ} (s)) | \lor$ $| g_{Δ} ({\bar{x}}_{Δ} (s)) | \leq h (Δ)$ , we have

$\begin{array}{l} E {| x_{Δ} (t) - {\bar{x}}_{Δ} (t) |}^{p} \leq E {| \int_{t_{k}}^{t} f_{Δ} ({\bar{x}}_{Δ} (s)) d s + \int_{t_{k}}^{t} g_{Δ} ({\bar{x}}_{Δ} (s)) d w (s) |}^{p} \\ \leq c_{p} (Δ^{p - 1} E \int_{t_{k}}^{t} {| f_{Δ} ({\bar{x}}_{Δ} (s)) |}^{p} d s + Δ^{p / 2 - 1} \int_{t_{k}}^{t} {| g_{Δ} ({\bar{x}}_{Δ} (s)) |}^{p} d s) \leq c_{p} Δ^{\frac{p}{2}} (h {(Δ))}^{p} \end{array}$ Mathematical equation

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

$E {| x (t) - {\bar{x}}_{Δ} (t) |}^{\bar{p}} \leq {(E {| x (t) - {\bar{x}}_{Δ} (t) |}^{p})}^{\frac{\bar{p}}{p}} \leq {(c_{p} Δ^{\frac{p}{2}} (h {(Δ))}^{p})}^{\frac{\bar{p}}{p}} = c_{p} Δ^{\frac{\bar{p}}{2}} (h {(Δ))}^{\bar{p}} .$ Mathematical equation

This proof is completed.

Let $R > 0$ Mathematical equation be an arbitrary number and $x_{Δ} (t)$ be the continuous-time truncated EM solution defined by Eq. (13), define

$ρ_{R} = i n f {t \geq 0 : | x_{Δ} (t) | \geq R}$ Mathematical equation (16)

Lemma 4 Let Assumption 2 hold, $α \geq 2 α_{i} \lor 2 (i = 1, \dots, n)$ Mathematical equation and $a > \sum_{i = 1}^{n} a_{i} + a_{0} / 2$ . Then for $Δ \in (0, Δ^{*})$ and $p > 0$ , there exists a positive constant $c_{p}$ such that

$\underset{0 < Δ < Δ^{*}}{s u p} \underset{0 \leq t \leq T}{s u p} E {| x_{Δ} (t) |}^{p} \leq c_{p}, \forall T > 0 .$ Mathematical equation

Moreover,

$P {ρ_{R} \leq T} \leq \frac{c_{p}}{R^{p}} .$ Mathematical equation

Proof We prove the results are true for $p \geq 2$ Mathematical equation first. For any $s > 0$ , there exists a unique nonnegative integer $k$ such that $s \in [t_{k}, t_{k + 1})$ . For $s \leq ρ_{R}$ , we see that $| x_{Δ} (s) | \leq R$ by definition of $ρ_{R}$ ; for $s \leq ρ_{R}$ , we deduce that $| {\bar{x}}_{Δ} (s) | \leq R$ as well due to ${\bar{x}}_{Δ} (s) \equiv {\bar{x}}_{Δ} (t_{k}) = x_{Δ} (t_{k})$ Mathematical equation for $s \in [t_{k}, t_{k + 1})$ . The Itô formula then gives

$\begin{array}{l} E {| x_{Δ} (t \land ρ_{R}) |}^{p} \leq E {| x_{0} |}^{p} + E \int_{0}^{t \land ρ_{R}} p {| x_{Δ} (s) |}^{p - 2} (x_{Δ} {(s)}^{T} f_{Δ} ({\bar{x}}_{Δ} (s)) + \frac{p - 1}{2} {| g_{Δ} ({\bar{x}}_{Δ} (s)) |}^{2}) d s \\ \leq E {| x_{0} |}^{2} + \int_{0}^{t \land ρ_{R}} p {| x_{Δ} (s) |}^{p - 2} ({\bar{x}}_{Δ} {(s)}^{T} f_{Δ} ({\bar{x}}_{Δ} (s)) + \frac{p - 1}{2} {| g_{Δ} ({\bar{x}}_{Δ} (s)) |}^{2}) d s + \int_{0}^{t \land ρ_{R}} p {| x_{Δ} (s) |}^{p - 2} {(x_{Δ} (s) - {\bar{x}}_{Δ} (s))}^{T} f_{Δ} ({\bar{x}}_{Δ} (s)) d s . \end{array}$ Mathematical equation

Depending on whether $| {\bar{x}}_{Δ} (s) | \leq μ^{- 1} (h (Δ))$ Mathematical equation , the rest of the proof falls into two cases:

Case 1: For any ${\bar{x}}_{Δ} (s) \in R^{\bar{n}}$ Mathematical equation with $| {\bar{x}}_{Δ} (s) | \leq μ^{- 1} (h (Δ))$ , by Lemma 2, we have

$\begin{array}{l} E {| x_{Δ} (t \land ρ_{R}) |}^{p} \leq E {| x_{0} |}^{p} + E \int_{0}^{t \land ρ_{R}} p {| x_{Δ} (s) |}^{p - 2} (a_{0} {| {\bar{x}}_{Δ} (s) |}^{2} + \sum_{i = 1}^{n} a_{i} {| {\bar{x}}_{Δ} (s) |}^{α_{i} + 1} - a {| {\bar{x}}_{Δ} (s) |}^{α + 2}) d s \\ + E \int_{0}^{t \land ρ_{R}} p {| x_{Δ} (s) |}^{p - 2} {(x_{Δ} (s) - {\bar{x}}_{Δ} (s))}^{T} f_{Δ} ({\bar{x}}_{Δ} (s)) d s \\ \leq E {| x_{0} |}^{p} + 2 p a_{0} E \int_{0}^{t \land ρ_{R}} {| x_{Δ} (s) |}^{p - 2} {| {\bar{x}}_{Δ} (s) |}^{2} d s \\ + E \int_{0}^{t \land ρ_{R}} p {| x_{Δ} (s) |}^{p - 2} (- a_{0} {| {\bar{x}}_{Δ} (s) |}^{2} + \sum_{i = 1}^{n} a_{i} {| {\bar{x}}_{Δ} (s) |}^{α_{i} + 1} - a {| {\bar{x}}_{Δ} (s) |}^{α + 2}) d s \\ + E \int_{0}^{t \land ρ_{R}} p {| x_{Δ} (s) |}^{p - 2} {(x_{Δ} (s) - {\bar{x}}_{Δ} (s))}^{T} f_{Δ} ({\bar{x}}_{Δ} (s)) d s . \end{array}$ Mathematical equation

According to Lemma $1, α \geq 2 α_{i}$ Mathematical equation and $a > \sum_{i = 1}^{n} a_{i}$ , there is a constant $c_{0}$ such that

$a_{0} {| {\bar{x}}_{Δ} (s) |}^{2} - \sum_{i = 1}^{n} a_{i} {| {\bar{x}}_{Δ} (s) |}^{α_{i} + 1} + a {| {\bar{x}}_{Δ} (s) |}^{α + 2} \geq c_{0} {| {\bar{x}}_{Δ} (s) |}^{2} .$ Mathematical equation

This, together with the Young's inequality $a^{p - 2} b \leq \frac{p - 2}{p} a^{p} + \frac{2}{p} b^{p / 2}$ Mathematical equation for $a, b \geq 0$ , implies

$\begin{array}{l} E {| x_{Δ} (t \land ρ_{R}) |}^{p} \leq E {| x_{0} |}^{p} + 2 a_{0} E \int_{0}^{t \land ρ_{R}} [(p - 2) {| x_{Δ} (s) |}^{p} + 2 {| {\bar{x}}_{Δ} (s) |}^{p}] d s + (p - 2) E \int_{0}^{t \land ρ_{R}} {| x_{Δ} (s) |}^{p} d s \\ + 2 E \int_{0}^{t \land ρ_{R}} {| x_{Δ} (s) - {\bar{x}}_{Δ} (s) |}^{p / 2} {| f_{Δ} ({\bar{x}}_{Δ} (s)) |}^{p / 2} d s \\ \leq E {| x_{0} |}^{2} + (a_{0} + 1) (p - 2) E \int_{0}^{t \land ρ_{R}} {| x_{Δ} (s) |}^{p} d s + 4 a_{0} E \int_{0}^{t \land ρ_{R}} {| {\bar{x}}_{Δ} (s) |}^{p} d s + 2 h {(Δ)}^{p / 2} E \int_{0}^{t \land ρ_{R}} {| x_{Δ} (s) - {\bar{x}}_{Δ} (s) |}^{p / 2} d s . \end{array}$ Mathematical equation

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

$E \int_{0}^{t \land ρ_{R}} {| x_{Δ} (s) - {\bar{x}}_{Δ} (s) |}^{p / 2} d s \leq \int_{0}^{t} {(E {| x_{Δ} (s \land ρ_{R}) - {\bar{x}}_{Δ} (s \land ρ_{R}) |}^{p})}^{1 / 2} d s \leq c_{p} h {(Δ)}^{p / 2} Δ^{p / 4} .$ Mathematical equation

Noticing that $\underset{0 \leq t \leq T}{s u p} E {| {\bar{x}}_{Δ} (s) |}^{p} \leq \underset{0 \leq t \leq T}{s u p} E {| x_{Δ} (s) |}^{p}$ Mathematical equation ,

$\underset{0 \leq t \leq T}{s u p} E {| x_{Δ} (t \land ρ_{R}) |}^{p} \leq E {| x_{0} |}^{2} + [2 (a_{0} + 1) p - 2] \int_{0}^{T} \underset{0 \leq v \leq s}{s u p} E {| x_{Δ} (v \land ρ_{R}) |}^{p} d s + c_{p} h {(Δ)}^{p} Δ^{p / 4} .$ Mathematical equation

Using the Gronwall inequality , let $R \to \infty$ Mathematical equation , the Fatou lemma gives

$\underset{0 \leq Δ \leq Δ^{*}}{l i m} \underset{0 \leq t \leq T}{s u p} E {| x_{Δ} (t) |}^{p} \leq (E {| x_{0} |}^{2} + c_{p} h {(Δ)}^{p} Δ^{p / 4}) e^{(2 (a_{0} + 1) p - 2) T} \equiv c_{p} .$ Mathematical equation

Case 2: For any ${\bar{x}}_{Δ} (s) \in R^{\bar{n}}$ Mathematical equation with $| {\bar{x}}_{Δ} (s) | > μ^{- 1} (h (Δ))$ , we obtain by Lemma 2 :

$\begin{array}{l} E {| x_{Δ} (t \land ρ_{R}) |}^{p} \leq E {| x_{0} |}^{p} + p \tilde{a} E \int_{0}^{t \land ρ_{R}} {| x_{Δ} (s) |}^{p - 2} {| {\bar{x}}_{Δ} (s) |}^{2} d s \\ + E \int_{0}^{t \land ρ_{R}} p {| x_{Δ} (s) |}^{p - 2} [\frac{a_{0}}{2} {| π ({\bar{x}}_{Δ} (s)) |}^{4} - a {| π ({\bar{x}}_{Δ} (s)) |}^{α + 2} + \sum_{i = 1}^{n} \frac{a_{i}}{2} {| π ({\bar{x}}_{Δ} (s)) |}^{2 α_{i} + 2}] d s \\ + E \int_{0}^{t \land ρ_{R}} p {| x_{Δ} (s) |}^{p - 2} {(x_{Δ} (s) - {\bar{x}}_{Δ} (s))}^{T} f_{Δ} ({\bar{x}}_{Δ} (s)) d s . \end{array}$ Mathematical equation

Note that ${| π ({\bar{x}}_{Δ} (s)) |}^{2} \leq | {\bar{x}}_{Δ} (s) |$ Mathematical equation , then

$\begin{array}{l} E {| x_{Δ} (t \land ρ_{R}) |}^{p} \leq E {| x_{0} |}^{p} + E \int_{0}^{t \land ρ_{R}} p {| x_{Δ} (s) |}^{p - 2} {(x_{Δ} (s) - {\bar{x}}_{Δ} (s))}^{T} f_{Δ} ({\bar{x}}_{Δ} (s)) d s - \frac{a_{0}}{2} {| π ({\bar{x}}_{Δ} (s)) |}^{4} \\ - E \int_{0}^{t \land ρ_{R}} p {| x_{Δ} (s) |}^{p - 2} [a_{0} {| π ({\bar{x}}_{Δ} (s)) |}^{2} + a {| π ({\bar{x}}_{Δ} (s)) |}^{α + 2} - \sum_{i = 1}^{n} \frac{a_{i}}{2} {| π ({\bar{x}}_{Δ} (s)) |}^{2 α_{i} + 2}] d s \\ + p a_{0} E \int_{0}^{t \land ρ_{R}} {| x_{Δ} (s) |}^{p - 2} {| π ({\bar{x}}_{Δ} (s)) |}^{2} d s + p \tilde{a} E \int_{0}^{t \land ρ_{R}} {| x_{Δ} (s) |}^{p - 2} {| {\bar{x}}_{Δ} (s) |}^{2} d s . \end{array}$ Mathematical equation

Recalling that $α \geq 2 α_{i} \lor 2$ Mathematical equation and $a > \sum_{i = 1}^{n} a_{i} + a_{0} / 2$ , by Lemma 1, there is a constant $c_{0}$ such that

$a_{0} {| π ({\bar{x}}_{Δ} (s)) |}^{2} + a {| π ({\bar{x}}_{Δ} (s)) |}^{α + 2} - \frac{a_{0}}{2} {| π ({\bar{x}}_{Δ} (s)) |}^{4} - \sum_{i = 1}^{n} \frac{a_{i}}{2} {| π ({\bar{x}}_{Δ} (s)) |}^{2 α_{i} + 2} \geq c_{0} {| π ({\bar{x}}_{Δ} (s)) |}^{2} .$ Mathematical equation

Therefore,

$E {| x_{Δ} (t \land ρ_{R}) |}^{p} = E {| x_{0} |}^{p} + 2 (a_{0} + \tilde{a}) E \int_{0}^{t \land ρ_{R}} ({| {\bar{x}}_{Δ} (s) |}^{p} + \frac{p - 2}{2} {| x_{Δ} (s) |}^{p}) d s + (p - 2) E \int_{0}^{t \land ρ_{R}} {| x_{Δ} (s) |}^{p} d s + c_{p} h {(Δ)}^{p} Δ^{p / 4} .$ Mathematical equation

Noticing that $\underset{0 \leq t \leq T}{s u p} E {| {\bar{x}}_{Δ} (s) |}^{p} \leq \underset{0 \leq t \leq T}{s u p} E {| x_{Δ} (s) |}^{p}$ Mathematical equation , we obtain

$\underset{0 \leq t \leq T}{s u p} E {| x_{Δ} (t \land ρ_{R}) |}^{p} \leq E {| x_{0} |}^{p} + c_{p} \int_{0}^{T} [\underset{0 \leq u \leq s}{s u p} E {| x_{Δ} (u \land ρ_{R}) |}^{p}] d s + c_{p} h {(Δ)}^{p} Δ^{p / 4}$ Mathematical equation

The Gronwall inequality and Fatou lemma imply

$\underset{0 \leq Δ \leq Δ^{*}}{l i m} \underset{0 \leq t \leq T}{s u p} E {| x_{Δ} (t) |}^{2} \leq (E {| x_{0} |}^{2} + c_{p} h {(Δ)}^{p} Δ^{p / 4}) e^{c_{p} T} \equiv c_{p}$ Mathematical equation

For $0 < \bar{p} < 2$ Mathematical equation , the Lyapunov inequality gives the desired result. The second part of this lemma easily follows.

Lemma 5 Let Assumptions 1 and 2 hold, $p > 0, α \geq 2 α_{i} (i = 1, \dots, n)$ Mathematical equation and $a > \sum_{i = 1}^{n} a_{i}$ . Then for any real number $R < μ^{- 1} (h (Δ))$ and $Δ \in (0, Δ^{*})$ , there exists a positive constant $c_{p}$ such that

$E {| x_{Δ} (t \land θ_{R}) - x (t \land θ_{R}) |}^{p} \leq c_{p} Δ^{\frac{p}{2}} h {(Δ)}^{p}, 0 \leq t \leq T,$ Mathematical equation

where $θ_{R} = ρ_{R} \land σ_{R}$ Mathematical equation .

Proof Denote $e (t) = x (t) - x_{Δ} (t)$ Mathematical equation . Assume $p \geq 2$ first. For given $Δ \in (0, Δ^{*})$ and any real number $R < μ^{- 1} (h (Δ)), 0 \leq s \leq t \land θ_{R}$ implies that $| {\bar{x}}_{Δ} (s) | \lor | x (s) | \lor | x_{Δ} (s) | \leq R < μ^{- 1} (h (Δ)))$ and

$f_{Δ} ({\bar{x}}_{Δ} (s)) = f ({\bar{x}}_{Δ} (s)), g_{Δ} ({\bar{x}}_{Δ} (s)) = g ({\bar{x}}_{Δ} (s)) .$ Mathematical equation

The Itô formula and Assumption 1 give

$\begin{array}{l} E {| e (t \land θ_{R}) |}^{p} \leq E \int_{0}^{t \land θ_{R}} p | e (s) |^{p - 2} [e {(s)}^{T} (f (x (s)) - f ({\bar{x}}_{Δ} (s)) + \frac{p - 1}{2} {| g (x (s)) - g ({\bar{x}}_{Δ} (s)) |}^{2}] d s \\ \leq k_{R} E \int_{0}^{t \land θ_{R}} (p | e (s) |^{p - 1} | x (s) - {\bar{x}}_{Δ} (s) | + \frac{p (p - 1)}{2} | e (s) |^{p - 2} {| x (s) - {\bar{x}}_{Δ} (s) |}^{2}) d s . \end{array}$ Mathematical equation

The Young inequality implies

$\begin{array}{l} E {| e (t \land θ_{R}) |}^{p} \leq c_{p} E \int_{0}^{t \land θ_{R}} (| e (s) |^{p} + {| x_{Δ} (s) - {\bar{x}}_{Δ} (s) |}^{p}) d s \leq c_{p} E \int_{0}^{t \land θ_{R}} (| e (s) |^{p} + 2^{p} {| x (s) - x_{Δ} (s) |}^{p} + 2^{p} {| x_{Δ} (s) - {\bar{x}}_{Δ} (s) |}^{p}) d s \\ \leq c_{p} E \int_{0}^{t} {| e (s \land θ_{R}) |}^{p} d s + c_{p} Δ^{\frac{p}{2}} h {(Δ)}^{p} . \end{array}$ Mathematical equation

Applying the Gronwall inequality to the above inequality, we achieve the desired result.

For $0 < \bar{p} < 2$ Mathematical equation , picking a $p > 2$ , we have

$E {| e (t \land θ_{R}) |}^{\bar{p}} \leq {[E {| e (t \land θ_{R}) |}^{p}]}^{\frac{\bar{p}}{p}} \leq {[c_{p} Δ^{\frac{p}{2}} h {(Δ)}^{p}]}^{\frac{\bar{p}}{p}} \leq c_{p} Δ^{\frac{\bar{p}}{2}} h {(Δ)}^{\bar{p}}$ Mathematical equation

by the Lyapunov inequality.

Theorem 1 Let Assumptions 1 and 2 hold with $α \geq 2 α_{i} \lor 2 (i = 1, \dots, n)$ Mathematical equation and $a > \sum_{i = 1}^{n} a_{i} + a_{0} / 2$ , $Δ \in (0, Δ^{*})$ with $h (Δ) \geq μ ({(Δ^{\frac{q}{2}} h {(Δ)}^{q})}^{- \frac{1}{p - q}}), p > 2$ , and $q \in [2, p)$ . Then there exists a positive constant $c_{p}$ such that

$E {| x (T) - x_{Δ} (T) |}^{q} \leq c_{p} Δ^{q / 2} h {(Δ)}^{q}$ Mathematical equation

Proof For $p \geq 2$ Mathematical equation , by the Young's inequality $x^{q} y \leq \frac{δ q}{p} x^{p} + \frac{p - q}{p δ^{\frac{q}{p - q}}} y^{\frac{q}{p - q}}, \forall x, y, δ > 0$ , we have

$E | e (T) |^{q} = E [| e (T) |^{q} I_{{θ_{R} > T}}] + E [| e (T) |^{q} I_{{θ_{R} \leq T}}] \leq E [| e (T) |^{q} I_{{θ_{R} > T}}] + \frac{q δ}{p} E [| e (T) |^{p}] + \frac{p - q}{p δ^{\frac{q}{p - q}}} P (θ_{R} \leq T)$ Mathematical equation (17)

The Hölder inequality and Lemma 5 imply that

$E [{| e_{Δ} (T) |}^{q} I_{{θ_{R} > T}}] = E [{| e_{Δ} (T \land θ_{R}) |}^{q}] \leq c_{p} Δ^{q / 2} h {(Δ)}^{q}$ Mathematical equation (18)

By Lemma 4, we have

$\begin{array}{l} E | e (T) |^{p} \leq E | x (T) |^{p} + E {| x_{Δ} (T) |}^{p} \leq c_{p}, \\ P {θ_{R} \leq T} \leq P {σ_{R} \land ρ_{R} \leq T} = P {σ_{R} \leq T} + P {ρ_{R} \leq T} \leq \frac{c_{p}}{R^{p}} . \end{array}$ Mathematical equation

This, together with Lemma 5, yields $E {| e_{Δ} (T) |}^{p} \leq c_{p} Δ^{q / 2} h {(Δ)}^{q} + \frac{c_{p} q δ}{p} + \frac{c_{p} (p - q)}{p R^{p} δ^{\frac{q}{p - q}}} .$ Mathematical equation

Choose $δ = Δ^{\frac{q}{2}} h {(Δ)}^{q}, R = {(Δ^{\frac{q}{2}} h {(Δ)}^{q})}^{- \frac{1}{p - q}}$ Mathematical equation , then $E {| x (T) - x_{Δ} (T) |}^{q} \leq c_{p} Δ^{q / 2} h {(Δ)}^{q} .$

Example 1Consider a scalar $S D E$ Mathematical equation

$d x (t) = (- x (t) + x^{2} (t) - 3 x^{5} (t)) d t + | x (t) |^{3 / 2} d w (t)$ Mathematical equation (19)

Denote $f (x) = - x + x^{2} - 3 x^{5}, g (x) = | x |^{3 / 2}$ Mathematical equation . Then

$x^{T} f (x) + \frac{p - 1}{2} | g (x) |^{2} = - x^{2} + x^{3} - 3 x^{6} + \frac{p - 1}{2} | x |^{3} .$ Mathematical equation

Now, we design functions $μ (x), h (x)$ Mathematical equation and choose $Δ^{*}$ so that condition (10) holds. First, it is easy to see that

$\underset{| x | \leq u}{s u p} | f (x) | \lor | g (x) | \leq 5 u^{5}, u \geq 1 .$ Mathematical equation

Pick $μ (u) = 5 u^{5}, u \geq 1$ Mathematical equation , then $μ^{- 1} (u) = {(\frac{u}{5})}^{\frac{1}{5}}, u \geq 0$ . Let $h (Δ) = Δ^{- ϵ}, ϵ \in [0, \frac{1}{4}], Δ > 0$ . Choose $Δ^{*}$ satisfying $Δ^{*} \leq 5^{- ϵ^{- 1}}$ , then $h (Δ^{*}) \geq μ (1), h (Δ) Δ^{\frac{1}{4}} = Δ^{- ϵ + \frac{1}{4}} < 1$ . It is easy to see that Assumption 2 holds, by Theorem 1, for any $Δ \in (0, Δ^{*}), E {| x (T) - x_{Δ} (T) |}^{p} \leq c_{p} Δ^{\frac{p}{2}} h {(Δ)}^{p}$ Mathematical equation .

In Fig. 1, we plot the truncated EM approximation (11) of Eq. (19) with $T = 2$ Mathematical equation for initial value $x (0) = 1$ . The figure illustrates that the numerical solution has convergence property.

Fig. 1

Numerical simulation of the path $x (t)$ Mathematical equation with $T = 2$ for Eq. (19)

2 Strong Convergence Rate over a Finite Time Interval

Section 1 has established the strong convergence rate of the truncated EM solution at a fixed time $T > 0$ Mathematical equation . In this section, we consider the path-dependent strong convergence rate over a finite time interval $[0, T]$ , which requires a stronger assumption on the diffusion coefficient.

Assumption 3 (linear growth condition) For $x \in R^{\bar{n}}$ Mathematical equation , there is a positive constant $k_{g}$ such that

$| g (x) |^{2} \leq k_{g} (1 + | x |^{2})$ Mathematical equation (20)

Since $g$ Mathematical equation is linearly growing, it is not necessary to truncate it in this section. Consequently, $g_{Δ} (\cdot)$ shall be replaced by $g (\cdot)$ in Equations (11) to (13), i.e., in the definition of the discrete-time and continuous-time truncated EM solutions.

Lemma 6 Let Assumptions 2 and 3 hold with $α \geq 2 α_{i} (i = 1,2, \dots, n)$ Mathematical equation and $a > \sum_{i = 1}^{n} a_{i}$ . Then for any $p \geq 2$ andgiven $T > 0$ , there exists a positive constant $c_{p}$ such that $E [\underset{0 \leq s \leq T}{s u p} | x (s) |^{p}] \leq c_{p}$ .

Proof The Itô formula gives

$| x (t) |^{p} \leq | x (0) |^{p} + \int_{0}^{t} p (a_{0} | x (s) |^{p} + \sum_{i = 1}^{n} a_{i} | x (s) |^{α_{i} + p - 1} - a | x (s) |^{α + p}) d s + \int_{0}^{t} p | x (s) |^{p - 2} x {(s)}^{T} g (x (s)) d w (s)$ Mathematical equation (21)

By Lemma 1 , for any $x \in R^{\bar{n}}$ Mathematical equation and $α \geq 2 α_{i} (i = 1,2, \dots, n)$ and $a > \sum_{i = 1}^{n} a_{i}$ , there exists a constant $c_{0}$ such that

$a_{0} | x (s) |^{p} - \sum_{i = 1}^{n} a_{i} | x (s) |^{α_{i} + p - 1} + a | x (s) |^{α + p} \geq c_{0} | x (s) |^{p} .$ Mathematical equation

Inequality (21) then becomes

$| x (t) |^{p} \leq {| x_{0} |}^{p} + 2 p a_{0} \int_{0}^{t} | x (s) |^{p} d s + \int_{0}^{t} p | x (s) |^{p - 2} x {(s)}^{T} g (x (s)) d w (s) .$ Mathematical equation

It follows that

$E [\underset{0 \leq s \leq T}{s u p} {| x (s \land σ_{R}) |}^{p}] \leq E {| x_{0} |}^{p} + 2 p a_{0} \int_{0}^{T} E [\underset{0 \leq u \leq s}{s u p} {| x (u \land σ_{R}) |}^{p}] d s + E [\underset{0 \leq s \leq T}{s u p} \int_{0}^{t \land σ_{R}} p | x (s) |^{p - 2} x {(s)}^{T} g (x (s)) d w (s)] .$ Mathematical equation

By the Burkhölder-Davis-Gundy inequality, we obtain

$\begin{array}{l} E [\underset{0 \leq s \leq t}{s u p} \int_{0}^{t \land σ_{R}} p | x (s) |^{p - 2} x {(s)}^{T} g (x (s)) d w (s)] \leq 4 \sqrt[]{2} p E {[\int_{0}^{t \land σ_{R}} | x (s) |^{2 p - 2} | g (x (s)) |^{2} d s]}^{1 / 2} \\ \leq \frac{1}{2} E [\underset{0 \leq s \leq T}{s u p} {| x (s \land σ_{R}) |}^{p}] + 16 p^{2} E [\int_{0}^{t \land σ_{R}} | x (s) |^{p - 2} | g (x (s)) |^{2} d s] . \end{array}$ Mathematical equation

The above estimate, together with Assumption 3, yields

$E [\underset{0 \leq s \leq T}{s u p} {| x (s \land σ_{R}) |}^{p}] \leq c_{p} + c_{p} \int_{0}^{T} E [\underset{0 \leq u \leq s}{s u p} {| x (u \land σ_{R}) |}^{p}] d s$ Mathematical equation (22)

Finally, the desired result is obtained by the Gronwall inequality and the Fatou Lemma.

From the procedure of the proof in Lemma 6, we can see that Assumption 3 plays an important role in establishing the estimate (22), which makes the Gronwall inequality applicable to establish the moment boundedness.

Lemma 7 Let Assumption 3 hold. Then for $p > 0$ Mathematical equation and any $Δ \in (0, Δ^{*})$ with $h (Δ) \geq 1$ , there exists a positive constant $c_{p}$ such that $E {| x_{Δ} (t) - {\bar{x}}_{Δ} (t) |}^{p} \leq c_{p} Δ^{\frac{p}{2}} (h {(Δ))}^{p} .$

Proof We assume $p \geq 2$ Mathematical equation first. For any $t \geq 0$ , there exists a unique nonnegative integer $k$ such that $t \in [t_{k}, t_{k + 1})$ .

$\begin{array}{l} E {| x (t \land ρ_{R}) - {\bar{x}}_{Δ} (t \land ρ_{R}) |}^{p} \leq E {| \int_{t_{k}}^{t \land ρ_{R}} f_{Δ} ({\bar{x}}_{Δ} (s)) d s + \int_{t_{k}}^{t \land ρ_{R}} g ({\bar{x}}_{Δ} (s)) d w (s) |}^{p} \\ \leq 2^{p - 1} (Δ^{p - 1} E \int_{t_{k}}^{t \land ρ_{R}} {| f_{Δ} ({\bar{x}}_{Δ} (s)) |}^{p} d s + Δ^{p / 2 - 1} \int_{t_{k}}^{t \land ρ_{R}} {| g ({\bar{x}}_{Δ} (s)) |}^{p} d s) \leq 2^{p - 1} Δ^{p} (h {(Δ))}^{p} + 2^{p - 1} Δ^{p / 2 - 1} c_{p} E \int_{t_{k}}^{t \land ρ_{R}} (1 + {| {\bar{x}}_{Δ} (s) |}^{p}) d s . \end{array}$ Mathematical equation

By the linear growth condition (3) and the Doob martingale inequality, it is easy to see that

$\underset{0 \leq s \leq t}{s u p} E {| {\bar{x}}_{Δ} (s) |}^{p} \leq \underset{0 \leq s \leq t}{s u p} E {| x_{Δ} (s) |}^{p} \leq c_{p} (1 + h {(Δ)}^{p}) .$ Mathematical equation

This, together with the Fatou Lemma, implies that

$E {| x (t) - {\bar{x}}_{Δ} (t) |}^{p} \leq c_{p} Δ^{p} (h {(Δ))}^{p} + c_{p} Δ^{p / 2} (1 + h {(Δ)}^{p}) \leq c_{p} Δ^{\frac{p}{2}} (h {(Δ))}^{p} .$ Mathematical equation

For $0 < \bar{p} < 2$ Mathematical equation , the Lyapunov inequality gives

$E {| x (t) - {\bar{x}}_{Δ} (t) |}^{\bar{p}} \leq {(E {| x (t) - {\bar{x}}_{Δ} (t) |}^{p})}^{\frac{\bar{p}}{p}} \leq {(c_{p} Δ^{\frac{p}{2}} (h {(Δ))}^{p})}^{\frac{\bar{p}}{p}} \leq c_{p} Δ^{\frac{\bar{p}}{2}} (h {(Δ))}^{\bar{p}} .$ Mathematical equation

The proof of the above lemma is different from that of Lemma 3 in that: $| g_{Δ} | (\leq h (Δ))$ Mathematical equation is bounded in Lemma 3, but $g$ here may be unbounded because we do not truncate $g$ . Thus, the stopping time is necessary to apply the Doob martingale inequality.

Lemma 8 Let Assumptions 2 and 3 hold with $α \geq 2 α_{i} (i = 1,2, \dots, n)$ Mathematical equation and $a > \sum_{i = 1}^{n} a_{i}$ . Then for any $p > 0$ , there exists a positive constant $c_{p}$ such that $E [\underset{0 \leq s \leq t}{s u p} {| x_{Δ} (s) |}^{p}] \leq c_{p} .$

Proof Let us first assume $p \geq 2$ Mathematical equation . Repeating the same process as in the proof of Lemma 4, we obtain ${| x_{Δ} (t) |}^{p} \leq c_{p} + c_{p} \int_{0}^{t} ({| x_{Δ} (s) |}^{p} + {{\bar{x}}_{Δ} (s) |}^{p}) d s + 2 h {(Δ)}^{p} Δ^{p / 4} + \int_{0}^{t} p {| x_{Δ} (s) |}^{p - 1} x_{Δ} {(s)}^{T} g ({\bar{x}}_{Δ} (s)) d w (s)$ Mathematical equation . Therefore,

$E [\underset{0 \leq s \leq t}{s u p} {| x_{Δ} (s \land ρ_{R}) |}^{p}] \leq c_{p} + c_{p} E [\underset{0 \leq s \leq t}{s u p} \int_{0}^{s \land ρ_{R}} {| x_{Δ} (v) |}^{p} d v] + 2 h {(Δ)}^{p} Δ^{p / 4} + E [\underset{0 \leq s \leq t}{s u p} \int_{0}^{s \land ρ_{R}} p {| x_{Δ} (v) |}^{p - 2} x_{Δ} {(v)}^{T} g ({\bar{x}}_{Δ} (v)) d w (v)] .$ Mathematical equation

By the Burkhölder-Davis-Gundy inequality, the Hölder inequality and Lemma 7, we may compute

$\begin{array}{l} E [\underset{0 \leq s \leq t}{s u p} | \int_{0}^{s \land ρ_{R}} p {| x_{Δ} (v) |}^{p - 2} x_{Δ} {(v)}^{T} g ({\bar{x}}_{Δ} (v)) d w (v) |] \leq c_{p} E {[\int_{0}^{t \land ρ_{R}} {| x_{Δ} (v) |}^{2 p - 2} {| g ({\bar{x}}_{Δ} (v)) |}^{2} d s]}^{1 / 2} \\ \leq \frac{1}{2} E [\underset{0 \leq s \leq t}{s u p} {| x_{Δ} (s \land ρ_{R}) |}^{p}] + c_{p} E [\int_{0}^{t \land ρ_{R}} {| x_{Δ} (s) |}^{p - 2} {| g ({\bar{x}}_{Δ} (s)) |}^{2} d s] \\ \leq \frac{1}{2} E [\underset{0 \leq s \leq t}{s u p} {| x_{Δ} (s \land ρ_{R}) |}^{p}] + c_{p} E \int_{0}^{t \land ρ_{R}} (1 + {| x_{Δ} (s) |}^{p} + {| {\bar{x}}_{Δ} (s) |}^{p}) d s . \end{array}$ Mathematical equation

Observing that $E [s u p_{0 \leq v \leq s} {| {\bar{x}}_{Δ} (v) |}^{p}] \leq E [s u p_{0 \leq v \leq s} {| x_{Δ} (v) |}^{p}]$ Mathematical equation , we arrive at

$E [\underset{0 \leq s \leq t}{s u p} {| x_{Δ} (s \land ρ_{R}) |}^{p}] \leq c_{p} + c_{p} h {(Δ)}^{p} Δ^{p / 4} + c_{p} \int_{0}^{t} E [\underset{0 \leq v \leq s}{s u p} {| x_{Δ} (v \land ρ_{R}) |}^{p}] d s .$ Mathematical equation

The Gronwall inequality yields $E [\underset{0 \leq s \leq t}{s u p} {| x_{Δ} (s \land ρ_{R}) |}^{p}] \leq (c_{p} + c_{p} h {(Δ)}^{p} Δ^{p / 4}) e^{c_{p} T} \equiv c_{p}$ Mathematical equation .

Let $R \to \infty$ Mathematical equation , the Fatou lemma gives the desired result.

For $0 < \bar{p} < 2$ Mathematical equation , the Lyapunov inequality shall ensure the result.

Lemma 9 Fix $T > 0$ Mathematical equation and $p \geq 2$ . Let Assumptions 1 to 3 hold with $α \geq 2 α_{i} (i = 1,2, \dots, n)$ and $a > \sum_{i = 1}^{n} a_{i}$ . Then for $| x_{0} | < R$ and $Δ \in (0, Δ^{*})$ sufficiently small such that $μ^{- 1} (h (Δ)) > R$ , there exists a positive constant $c_{p}$ Mathematical equation such that

$E [\underset{0 \leq s \leq T}{s u p} {| x (s \land θ_{R}) - x_{Δ} (s \land θ_{R}) |}^{p}] \leq c_{p} Δ^{\frac{p}{2}} h {(Δ)}^{p} .$ Mathematical equation

Proof Denote $e (s) = x (s) - x_{Δ} (s)$ Mathematical equation . For $s \leq t \land θ_{R}$ , we have $| x_{Δ} (s) | \lor | {\bar{x}}_{Δ} (s) | \lor | x (s) | \leq R \leq$ $μ^{- 1} (h (Δ)), f_{Δ} ({\bar{x}}_{Δ} (s)) = f ({\bar{x}}_{Δ} (s))$ , and $g_{Δ} ({\bar{x}}_{Δ} (s)) = g ({\bar{x}}_{Δ} (s))$ . The Itô formula gives

${| e (t \land θ_{R}) |}^{2} \leq \int_{0}^{t \land θ_{R}} [2 e {(s)}^{T} (f (x (s)) - f ({\bar{x}}_{Δ} (s))) + {| g (x (s)) - g ({\bar{x}}_{Δ} (s)) |}^{2}] d s + \int_{0}^{t \land θ_{R}} 2 e {(s)}^{T} (g (x (s)) - g ({\bar{x}}_{Δ} (s))) d w (s) \equiv J_{1} + J_{2} + J_{3} .$ Mathematical equation

Here $J_{1}, J_{2}$ Mathematical equation and $J_{3}$ are the three integrals inside the expression. For $J_{1}$ , we apply the Hölder's inequality to get

$\begin{array}{l} E [\underset{0 \leq s \leq t}{s u p} {| J_{1} |}^{\frac{p}{2}}] \leq {(T^{p - 1} E \int_{0}^{t \land θ_{R}} 2^{p} | e (s) |^{p} {| f (x (s)) - f ({\bar{x}}_{Δ} (s)) |}^{p} d s)}^{1 / 2} \\ \leq 2^{- 2} 3^{- \frac{p}{2} + 1} E [\underset{0 \leq s \leq t}{s u p} {| e (s \land θ_{R}) |}^{p}] + 3^{\frac{p}{2} - 1} T^{p - 1} 2^{p} E \int_{0}^{t \land θ_{R}} {| f (x (s)) - f ({\bar{x}}_{Δ} (s)) |}^{p} d s . \end{array}$ Mathematical equation

This, together with Assumption 1, implies

$E [\underset{0 \leq s \leq t}{s u p} {| J_{1} |}^{\frac{p}{2}}] \leq 2^{- 2} 3^{- \frac{p}{2} + 1} E [\underset{0 \leq s \leq t}{s u p} {| e (t \land θ_{R}) |}^{p}] + 3^{\frac{p}{2} - 1} T^{p - 1} 2^{p} k_{R}^{p} E \int_{0}^{t \land θ_{R}} {| x (s) - {\bar{x}}_{Δ} (s) |}^{p} d s .$ Mathematical equation

For $J_{2}$ Mathematical equation , we have

$E [\underset{0 \leq s \leq t}{s u p} {| J_{2} |}^{\frac{p}{2}}] \leq k_{R}^{p} T^{\frac{p}{2} - 1} E \int_{0}^{t \land θ_{R}} {| x (s) - {\bar{x}}_{Δ} (s) |}^{p} d s$ Mathematical equation

By the Burkhölder-Davis-Gundy and Hölder inequalities, we may compute

$\begin{array}{l} E [\underset{0 \leq s \leq t}{s u p} {| J_{3} |}^{\frac{p}{2}}] \leq E [\underset{0 \leq u \leq t}{s u p} {| \int_{0}^{u \land θ_{R}} 2 e {(s)}^{T} (g (x (s)) - g ({\bar{x}}_{Δ} (s))) d w (s) |}^{\frac{p}{2}}] \leq c_{p} E {(\int_{0}^{t \land θ_{R}} 4 | e (s) |^{2} {| g (x (s)) - g ({\bar{x}}_{Δ} (s)) |}^{2} d s)}^{\frac{p}{4}} \\ \leq c_{p} E {(\int_{0}^{t \land θ_{R}} T^{p / 2 - 1} 2^{p} c_{p}^{2} {| e (s \land θ_{R}) |}^{p} {| g (x (s)) - g ({\bar{x}}_{Δ} (s)) |}^{p} d s)}^{\frac{1}{2}} \leq 2^{- 2} 3^{- \frac{p}{2} + 1} E [\underset{0 \leq s \leq t}{s u p} {| e (s \land θ_{R}) |}^{p}] + c_{p} E \int_{0}^{t \land θ_{R}} {| x (s) - {\bar{x}}_{Δ} (s) |}^{p} d s . \end{array}$ Mathematical equation

Summarizing up, we reach

$\begin{array}{l} E [\underset{0 \leq s \leq t}{s u p} {| e (t \land θ_{R}) |}^{p}] \leq \frac{1}{2} E [\underset{0 \leq s \leq t}{s u p} {| e (s \land θ_{R}) |}^{p}] + c_{p} E \int_{0}^{t \land θ_{R}} {| x (s) - {\bar{x}}_{Δ} (s) |}^{p} d s \\ \leq c_{p} E \int_{0}^{t \land θ_{R}} {| x (s) - x_{Δ} (s) |}^{p} d s + c_{p} E \int_{0}^{t \land θ_{R}} {| x_{Δ} (s) - {\bar{x}}_{Δ} (s) |}^{p} d s \end{array}$ Mathematical equation (23)

This, together with Lemma 7, implies

$E [\underset{0 \leq s \leq t}{s u p} {| e (s \land θ_{R}) |}^{p}] \leq c_{p} \int_{0}^{t} E [\underset{0 \leq u \leq s}{s u p} {| e (u \land θ_{R}) |}^{p}] d s + c_{p} Δ^{p / 2} h {(Δ)}^{p} .$ Mathematical equation

The Gronwall inequality gives $E [\underset{0 \leq s \leq t}{s u p} {| e (s \land θ_{R}) |}^{p}] \leq c_{p} Δ^{p / 2} h {(Δ)}^{p} .$ Mathematical equation

The proof is completed.

Theorem 2 Let Assumptions 1 to 3 hold with $α \geq 2 α_{i} \lor 2 (i = 1,2, \dots, n)$ Mathematical equation and $a > \sum_{i = 1}^{n} a_{i} + a_{0} / 2$ . Then for $p > 2$ and given $q \in [2, p)$ , the truncated EM scheme described by Equation (13) has the property

$E [\underset{0 \leq t \leq T}{s u p} {| x (t) - x_{Δ} (t) |}^{q}] \leq c_{p} Δ^{q / 2} h {(Δ)}^{q} .$ Mathematical equation

Proof Denote $e (t) = x (t) - x_{Δ} (t)$ Mathematical equation . For $p > 2$ , the Young's inequality $x^{q} y \leq \frac{δ q}{p} x^{p} + \frac{p - q}{p δ^{\frac{q}{p - q}}} y^{\frac{q}{p - q}}, x, y$ , $δ > 0$ , gives that

$E [\underset{0 \leq t \leq T}{s u p} | e (t) |^{q}] = E [\underset{0 \leq t \leq T}{s u p} | e (t) |^{q} I_{{θ_{R} > T}}] + \frac{q δ}{p} E [\underset{0 \leq t \leq T}{s u p} | e (t) |^{p}] + \frac{p - q}{p δ^{\frac{q}{p - q}}} P (θ_{R} \leq T)$ Mathematical equation (24)

The Hölder inequality and Lemma 9 imply that

$E [\underset{0 \leq t \leq T}{s u p} {| e_{Δ} (t) |}^{q} I_{{θ_{R} > T}}] = E [\underset{0 \leq t \leq T}{s u p} {| e_{Δ} (t \land θ_{R}) |}^{q}] \leq c_{p} Δ^{q / 2} h {(Δ)}^{q}$ Mathematical equation (25)

By Lemmas 6 and 8, we obtain

$E [\underset{0 \leq t \leq T}{s u p} | e (t) |^{p}] \leq E [\underset{0 \leq t \leq T}{s u p} | x (t) |^{p}] + E [\underset{0 \leq t \leq T}{s u p} {| x_{Δ} (t) |}^{p}] \leq c_{p} .$ Mathematical equation

Applying Lemma 4, we get

$P {θ_{R} \leq T} \leq P {σ_{R} \land ρ_{R} \leq T} = P {σ_{R} \leq T} + P {ρ_{R} \leq T} \leq \frac{c_{p}}{R^{p}} .$ Mathematical equation

Choosing $δ = Δ^{\frac{q}{2}} h {(Δ)}^{q}, R = {(Δ^{\frac{q}{2}} h {(Δ)}^{q})}^{- \frac{1}{p - q}}$ Mathematical equation , we achieve $E [\underset{0 \leq t \leq T}{s u p} {| x (t) - x_{Δ} (t) |}^{q}] \leq c_{p} Δ^{q / 2} h {(Δ)}^{q} .$

The proof is completed.

Section 1 discusses the strong convergence rate at time $T$ Mathematical equation under a very general polynomial growth condition (6). Imposing the linear growth condition (20) on the diffusion coefficient, we obtain the strong convergence rate over $[0, T]$ after the $p$ -th moment uniform boundedness is established.

Example 2 Consider a one-dimensional stochastic differential equation

$d x (t) = (x (t) + 2 x^{2} (t) - 8 x^{5} (t)) d t + 0.5 x (t) d w (t)$ Mathematical equation (26)

It is easy to see that $f (x) = x + 2 x^{2} - 8 x^{5}, g (x) = 0.5 x$ Mathematical equation satisfy Assumptions 1, 2 and 3. Clearly, for $u \geq 1, s u p_{| x | \leq u} | f (x) | \lor | g (x) | \leq 11 u^{5}$ . Define $μ (u) = 11 u^{5}$ , then $μ^{- 1} (u) = {(\frac{u}{11})}^{1 / 5}$ . Define $h (λ) = λ^{- ε}, ε \in [0,1 / 4]$ , obviously, $l i m_{Δ \to 0} h (Δ) = \infty, Δ^{\frac{1}{4}} h (Δ) < 1$ Mathematical equation . Choose $Δ^{*} \leq 11^{- ε^{- 1}}$ , then $h (Δ^{*}) \geq μ (1)$ . By Theorem 2, for any $Δ \in (0, Δ^{*}), E {| x (t) - x_{Δ} (t) |}^{p} \leq c_{p} Δ^{\frac{p}{2}} h {(Δ)}^{p}$ .

In Fig. 2, we demonstrate that the numerical solution calculated by Eq. (26) converges strongly in the sense to the numerical exact solution with an order approximately equal to 0.5. We perform 2 000 sample paths and average over all the paths.

Fig. 2

Convergence rate plot for Eq.(26)

3 Conclusion

In this paper, we discussed the strong convergence of the truncated EM methods for stochastic differential equations with super-linearly growing coefficients. The strong convergence rate at a single time $T$ Mathematical equation under a very general polynomial growth condition (6) is established in Theorem 1. Imposing the linearly growing condition (20) on the diffusion coefficient, the strong convergence rate over $[0, T]$ is established in Theorem 2. By using the Gronwall inequality, we proved the $p$ Mathematical equation -th moment uniform boundedness of both the exact and the approximate truncated EM solutions, which is the crucial result in examining the convergence rates.

References

Mao X , Szpruch L. Strong convergence rates for backward Euler-Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients[J]. Stochastics: An International Journal of Probability and Stochastic Processes, 2013, 85(1): 144-171. [CrossRef] [MathSciNet] [Google Scholar]
Zhou S B, Jin H. Numerical solution to highly nonlinear neutral-type stochastic differential equation[J]. Applied Numerical Mathematics, 2019, 140: 48-75. [CrossRef] [MathSciNet] [Google Scholar]
Zhou S B, Jin H. Implicit numerical solutions to neutral-type stochastic systems with superlinearly growing coefficients[J]. Journal of Computational and Applied Mathematics, 2019, 350: 423-441. [CrossRef] [MathSciNet] [Google Scholar]
Hutzenthaler M, Jentzen A, Kloeden P E. Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients[J]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2011, 467(2130): 1563-1576. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
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Wang X J, Gan S Q. The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients[J]. Journal of Difference Equations and Applications, 2013, 19(3): 466-490. [CrossRef] [MathSciNet] [Google Scholar]
Mao X R. The truncated Euler-Maruyama method for stochastic differential equations[J]. Journal of Computational and Applied Mathematics, 2015, 290: 370-384. [CrossRef] [MathSciNet] [Google Scholar]
Mao X R. Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations[J]. Journal of Computational and Applied Mathematics, 2016, 296: 362-375. [CrossRef] [MathSciNet] [Google Scholar]
Lan G, Xia F. Strong convergence rates of modified truncated Euler-Maruyama method for stochastic differential equations[J]. Journal of Computational and Applied Mathematics, 2018, 334: 1-17. [CrossRef] [MathSciNet] [Google Scholar]
Liu W, Mao X R, Tang J W, et al. Truncated Euler-Maruyama method for classical and time-changed non-autonomous stochastic differential equations. Applied Numerical Mathematics, 2020, 153: 66-81. [CrossRef] [MathSciNet] [Google Scholar]
Yang H, Wu F K, Kloeden P E, et al. The truncated Euler-Maruyama method for stochastic differential equations with Hölder diffusion coefficients[J]. Journal of Computational and Applied Mathematics, 2020, 366: 112379. [CrossRef] [MathSciNet] [Google Scholar]
Mao X R, Wei F Y, Wiriyakraikul T. Positivity preserving truncated Euler-Maruyama Method for stochastic Lotka-Volterra competition model[J]. Journal of Computational and Applied Mathematics, 2021, 394: 113566. [CrossRef] [Google Scholar]
Yang H F, Huang J H. Convergence and stability of modified partially truncated Euler-Maruyama method for nonlinear stochastic differential equations with Hölder continuous diffusion coefficient[J]. Journal of Computational and Applied Mathematics, 2022, 404:113895. [CrossRef] [MathSciNet] [Google Scholar]
Zhou S B. Strong convergence and stability of backward Euler-Maruyama scheme for highly nonlinear hybrid stochastic differential delay equation[J]. Calcolo, 2015, 52(4): 445-473. [CrossRef] [MathSciNet] [Google Scholar]
Zhou S B, Jin H. Strong convergence of implicit numerical methods for nonlinear stochastic functional differential equations[J]. Journal of Computational and Applied Mathematics, 2017, 324: 241-257. [CrossRef] [MathSciNet] [Google Scholar]
Zhou S B, Hu C Z. Numerical approximation of stochastic differential delay equation with coefficients of polynomial growth[J]. Calcolo, 2017, 54(1): 1-22. [CrossRef] [MathSciNet] [Google Scholar]
Guo Q, Mao X R, Yue R X. The truncated Euler-Maruyama method for stochastic differential delay equations[J]. Numerical Algorithms, 2018, 78(2): 599-624. [CrossRef] [MathSciNet] [Google Scholar]

All Figures

	Fig. 1 Numerical simulation of the path $x (t)$ with $T = 2$ for Eq. (19)
In the text

	Fig. 2 Convergence rate plot for Eq.(26)
In the text

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[R1] Mao X , Szpruch L. Strong convergence rates for backward Euler-Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients[J]. Stochastics: An International Journal of Probability and Stochastic Processes, 2013, 85(1): 144-171. [CrossRef] [MathSciNet] [Google Scholar]

[R2] Zhou S B, Jin H. Numerical solution to highly nonlinear neutral-type stochastic differential equation[J]. Applied Numerical Mathematics, 2019, 140: 48-75. [CrossRef] [MathSciNet] [Google Scholar]

[R3] Zhou S B, Jin H. Implicit numerical solutions to neutral-type stochastic systems with superlinearly growing coefficients[J]. Journal of Computational and Applied Mathematics, 2019, 350: 423-441. [CrossRef] [MathSciNet] [Google Scholar]

[R4] Hutzenthaler M, Jentzen A, Kloeden P E. Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients[J]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2011, 467(2130): 1563-1576. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

[R5] Liu W, Mao X R. Strong convergence of the stopped Euler-Maruyama method for nonlinear stochastic differential equations[J]. Applied Mathematics and Computation, 2013, 223: 389-400. [CrossRef] [MathSciNet] [Google Scholar]

[R6] Sabanis S. A note on tamed Euler approximations[J]. Electronic Communications in Probability, 2013, 18(1): 1-10. [CrossRef] [MathSciNet] [Google Scholar]

[R7] Wang X J, Gan S Q. The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients[J]. Journal of Difference Equations and Applications, 2013, 19(3): 466-490. [CrossRef] [MathSciNet] [Google Scholar]

[R8] Mao X R. The truncated Euler-Maruyama method for stochastic differential equations[J]. Journal of Computational and Applied Mathematics, 2015, 290: 370-384. [CrossRef] [MathSciNet] [Google Scholar]

[R9] Mao X R. Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations[J]. Journal of Computational and Applied Mathematics, 2016, 296: 362-375. [CrossRef] [MathSciNet] [Google Scholar]

[R10] Lan G, Xia F. Strong convergence rates of modified truncated Euler-Maruyama method for stochastic differential equations[J]. Journal of Computational and Applied Mathematics, 2018, 334: 1-17. [CrossRef] [MathSciNet] [Google Scholar]

[R11] Liu W, Mao X R, Tang J W, et al. Truncated Euler-Maruyama method for classical and time-changed non-autonomous stochastic differential equations. Applied Numerical Mathematics, 2020, 153: 66-81. [CrossRef] [MathSciNet] [Google Scholar]

[R12] Yang H, Wu F K, Kloeden P E, et al. The truncated Euler-Maruyama method for stochastic differential equations with Hölder diffusion coefficients[J]. Journal of Computational and Applied Mathematics, 2020, 366: 112379. [CrossRef] [MathSciNet] [Google Scholar]

[R13] Mao X R, Wei F Y, Wiriyakraikul T. Positivity preserving truncated Euler-Maruyama Method for stochastic Lotka-Volterra competition model[J]. Journal of Computational and Applied Mathematics, 2021, 394: 113566. [CrossRef] [Google Scholar]

[R14] Yang H F, Huang J H. Convergence and stability of modified partially truncated Euler-Maruyama method for nonlinear stochastic differential equations with Hölder continuous diffusion coefficient[J]. Journal of Computational and Applied Mathematics, 2022, 404:113895. [CrossRef] [MathSciNet] [Google Scholar]

[R15] Zhou S B. Strong convergence and stability of backward Euler-Maruyama scheme for highly nonlinear hybrid stochastic differential delay equation[J]. Calcolo, 2015, 52(4): 445-473. [CrossRef] [MathSciNet] [Google Scholar]

[R16] Zhou S B, Jin H. Strong convergence of implicit numerical methods for nonlinear stochastic functional differential equations[J]. Journal of Computational and Applied Mathematics, 2017, 324: 241-257. [CrossRef] [MathSciNet] [Google Scholar]

[R17] Zhou S B, Hu C Z. Numerical approximation of stochastic differential delay equation with coefficients of polynomial growth[J]. Calcolo, 2017, 54(1): 1-22. [CrossRef] [MathSciNet] [Google Scholar]

[R18] Guo Q, Mao X R, Yue R X. The truncated Euler-Maruyama method for stochastic differential delay equations[J]. Numerical Algorithms, 2018, 78(2): 599-624. [CrossRef] [MathSciNet] [Google Scholar]