The truncated Euler–Maruyama method for stochastic differential equations

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Abstract

Influenced by Higham et al. (2003), 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. In this paper we will develop a new explicit method, called the truncated EM method, for the nonlinear SDE dx(t)=f(x(t))dt+g(x(t))dB(t)dx(t)=f(x(t))dt+g(x(t))dB(t) and establish the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition View the MathML sourcexTf(x)+p−12∣g(x)∣2≤K(1+∣x∣2). The type of convergence specifically addressed in this paper is strong-LqLq convergence for 2≤q<p2≤q<p, and pp is a parameter in the Khasminskii-type condition.
Original languageEnglish
Pages (from-to)370–384
Number of pages15
JournalJournal of Computational and Applied Mathematics
Volume290
Early online date10 Jun 2015
DOIs
Publication statusPublished - 15 Dec 2015

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Strong Convergence
Stochastic Equations
Numerical methods
Differential equations
Lipschitz condition
Differential equation
Numerical Methods
Convergence Theory
Explicit Methods
Numerical Solution

Keywords

  • stochastic differential equation
  • local Lipschitz condition
  • khasminskii-type condition
  • Truncated Euler–Maruyama method
  • strong convergence

Cite this

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abstract = "Influenced by Higham et al. (2003), 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. In this paper we will develop a new explicit method, called the truncated EM method, for the nonlinear SDE dx(t)=f(x(t))dt+g(x(t))dB(t)dx(t)=f(x(t))dt+g(x(t))dB(t) and establish the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition View the MathML sourcexTf(x)+p−12∣g(x)∣2≤K(1+∣x∣2). The type of convergence specifically addressed in this paper is strong-LqLq convergence for 2≤q",
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The truncated Euler–Maruyama method for stochastic differential equations. / Mao, Xuerong.

In: Journal of Computational and Applied Mathematics, Vol. 290, 15.12.2015, p. 370–384.

Research output: Contribution to journalArticle

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AU - Mao, Xuerong

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Y1 - 2015/12/15

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AB - Influenced by Higham et al. (2003), 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. In this paper we will develop a new explicit method, called the truncated EM method, for the nonlinear SDE dx(t)=f(x(t))dt+g(x(t))dB(t)dx(t)=f(x(t))dt+g(x(t))dB(t) and establish the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition View the MathML sourcexTf(x)+p−12∣g(x)∣2≤K(1+∣x∣2). The type of convergence specifically addressed in this paper is strong-LqLq convergence for 2≤q

KW - stochastic differential equation

KW - local Lipschitz condition

KW - khasminskii-type condition

KW - Truncated Euler–Maruyama method

KW - strong convergence

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