The truncated Euler–Maruyama method for stochastic differential delay equations

Abstract

The numerical solutions of stochastic differential delay equations (SDDEs) under the generalized Khasminskii-type condition were discussed by Mao [15], and the theory there showed that the Euler-Maruyama (EM) numerical solutions converge to the true solutions in probability. However, there is so far no result on the strong convergence (namely in L p ) of the numerical solutions for the SDDEs under this generalized condition. In this paper, we will use the truncated EM method developed by Mao [16] to study the strong convergence of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition. e.g., [3,7,10,13,18,19]). The numerical solutions of SDDEs under the generalized Khasminskiitype condition were discussed by Mao [15], and the theory there showed that the Euler-Maruyama (EM) numerical solutions converge to the true solutions in probability. However, there is so far no result on the strong convergence (namely in L p ) of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition.Recently, Mao [16] develops a new explicit numerical method, called the truncated EM method, for SDEs under the Khasminskii-type condition plus the local Lipschitz condition and establishes the strong convergence theory. In this paper, we will use this new truncated EM method to study the strong convergence of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition.This paper is organized as follows: We will introduce necessary notion, state the generalized Khasminskii-type condition and define the truncated EM numerical solutions for SDDEs in Section 2. We will establish the strong convergence theory for the truncated EM numerical solutions in Sections 3 and 4 and discuss the convergence rates in Section 5. In each of these three sections we will illustrate our theory by examples. We will see from these examples that the truncated EM numerical method can be applied to approximate the solutions of many highly nonlinear SDDEs. We will finally conclude our paper in Section 6. The Truncated Euler-Maruyama MethodThroughout this paper, unless otherwise specified, we use the following notation. Let | · | be the Euclidean norm in R 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| = trace(A T A). Let R + = [0, ∞) and τ > 0. Denote by C([−τ, 0]; R n ) the family of continuous functions from [−τ, 0] to R n with the norm ϕ = sup −τ ≤θ≤0 |ϕ(θ)|. Let (Ω, F , {F t } t≥0 , P) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F 0 contains all P-null sets). Let B(t) = (B 1 (t), · · · , B m (t)) T be an m-dimensional Brownian motion defined on the probability space. Moreover, for two real numbers a and b, we use a ∨ b = max(a, b) and a ∧ b = min(a, b). If G is a set, its indicator function is denoted by I G , namely I G (x) = 1 if x ∈ G and 0 otherwise. If a is a real number, we denote by ⌊a⌋ the largest inte...

The Partially Truncated Euler–Maruyama Method for Highly Nonlinear Stochastic Delay Differential Equations with Markovian Switching

“…The following Lemma show that the truncated functions F ∆ and G ∆ preserve the generalized Khasminskii-type condition for any ∆ ∈ (0, ∆ * ] as shown Lemma 4.2 in [5] and we state it here as a Lemma for the use of this paper.…”

Advances in the truncated Euler–Maruyama method for stochastic differential delay equations

“…In Section 3, we will establish the strong convergence theory without this restrictive condition. In Section 4, we will compare our new result with the one in [8] to highlight our significant contribution in this paper. In Section 5, we will establish the stronger convergence theory for the solutions over a finite time interval and this was not discussed in [8].…”

“…The generalised Khasminskii-type theorems established by [16,20] replace the linear growth condition with the generalised Khasminskii-type condition in terms of Lyapunov functions. The numerical solutions of SDDEs under the generalised Khasminskii-type condition were discussed by Mao [18], and the theory there showed that the Euler-Maruyama (EM) numerical solutions converge to the true solutions in probability Influenced by [19], Guo et al [8] were the first to study the strong convergence of the truncated EM method for the SDDEs under the generalised Khasminskii-type condition. In this paper, we will explain, via an example, that a main condition imposed in [8] is sometimes so restrictive that the step size would be too small for the truncated EM method to be applicable.…”

“…This paper is organised as follows: In Section 2, we will introduce the necessary notation, recall the truncated EM method and review one of the main results of [8] and then point out a restrictive condition imposed in [8] via an example. In Section 3, we will establish the strong convergence theory without this restrictive condition.…”

Strong convergence rates of modified truncated EM methods for neutral stochastic differential delay equations

“…Here f (a(x, y)) ≡ f (ax, ay) for any a ∈ R, x, y ∈ R d . It is obvious that the functions f ∆ and g ∆ defined above are different from the truncated functions defined in [3].…”