Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations

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Abstract

Influenced by Higham, Mao and Stuart [9], several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations (SDEs) under the local Lipschitz condition. These numerical methods include the tamed Euler–Maruyama (EM) method, the tamed Milstein method, the stopped EM, the backward EM, the backward forward EM, etc. Recently, we developed a new explicit method in [23], called the truncated EM method, for the nonlinear SDE dx(t) = f (x(t))dt + g(x(t))dB(t) and established the strong convergence theory under the local Lip- schitz condition plus the Khasminskii-type condition xT f (x) + p−1 |g(x)|2 ≤ K(1 + |x|2). However, due to the page limit there, we did not study the convergence rates for the method, which is the aim of this paper. We will, under some additional conditions, discuss the rates of Lq -convergence of the truncated EM method for 2 ≤ q < p and show that the order of Lq -convergence can be arbitrarily close to q/2.
LanguageEnglish
Pages362-375
Number of pages14
JournalJournal of Computational and Applied Mathematics
Volume296
Early online date13 Oct 2015
Publication statusPublished - Apr 2016

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Euler-Maruyama Method
Stochastic Equations
Euler
Numerical methods
Rate of Convergence
Differential equations
Differential equation
Strong Convergence
Numerical Methods
Convergence Theory
Lipschitz condition
Explicit Methods
Order of Convergence
Lipschitz
Convergence Rate
Numerical Solution

Keywords

  • stochastic differential equation
  • local Lipschitz condition
  • Khasminskii-type condition
  • truncated Euler-Maruyama method
  • convergence rate

Cite this

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title = "Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations",
abstract = "Influenced by Higham, Mao and Stuart [9], several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations (SDEs) under the local Lipschitz condition. These numerical methods include the tamed Euler–Maruyama (EM) method, the tamed Milstein method, the stopped EM, the backward EM, the backward forward EM, etc. Recently, we developed a new explicit method in [23], called the truncated EM method, for the nonlinear SDE dx(t) = f (x(t))dt + g(x(t))dB(t) and established the strong convergence theory under the local Lip- schitz condition plus the Khasminskii-type condition xT f (x) + p−1 |g(x)|2 ≤ K(1 + |x|2). However, due to the page limit there, we did not study the convergence rates for the method, which is the aim of this paper. We will, under some additional conditions, discuss the rates of Lq -convergence of the truncated EM method for 2 ≤ q < p and show that the order of Lq -convergence can be arbitrarily close to q/2.",
keywords = "stochastic differential equation, local Lipschitz condition, Khasminskii-type condition, truncated Euler-Maruyama method, convergence rate",
author = "Xuerong Mao",
year = "2016",
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language = "English",
volume = "296",
pages = "362--375",
journal = "Journal of Computational and Applied Mathematics",
issn = "0377-0427",

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T1 - Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations

AU - Mao, Xuerong

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AB - Influenced by Higham, Mao and Stuart [9], several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations (SDEs) under the local Lipschitz condition. These numerical methods include the tamed Euler–Maruyama (EM) method, the tamed Milstein method, the stopped EM, the backward EM, the backward forward EM, etc. Recently, we developed a new explicit method in [23], called the truncated EM method, for the nonlinear SDE dx(t) = f (x(t))dt + g(x(t))dB(t) and established the strong convergence theory under the local Lip- schitz condition plus the Khasminskii-type condition xT f (x) + p−1 |g(x)|2 ≤ K(1 + |x|2). However, due to the page limit there, we did not study the convergence rates for the method, which is the aim of this paper. We will, under some additional conditions, discuss the rates of Lq -convergence of the truncated EM method for 2 ≤ q < p and show that the order of Lq -convergence can be arbitrarily close to q/2.

KW - stochastic differential equation

KW - local Lipschitz condition

KW - Khasminskii-type condition

KW - truncated Euler-Maruyama method

KW - convergence rate

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M3 - Article

VL - 296

SP - 362

EP - 375

JO - Journal of Computational and Applied Mathematics

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JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

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