Explicit numerical approximations for stochastic differential equations in finite and infinite horizons: truncation methods, convergence in pth moment, and stability

Xiaoyue Li, Xuerong Mao, George Yin

Research output: Contribution to journalArticle

Abstract

Solving stochastic differential equations (SDEs) numerically, explicit Euler-Maruyama (EM) schemes are used most frequently under global Lipschitz conditions for both drift and diffusion coefficients. In contrast, without imposing the global Lipschitz conditions, implicit schemes are often used for SDEs but require additional computational effort; along another line, tamed EM schemes and truncated EM schemes have been developed recently. Taking advantages of being explicit and easily implementable, truncated EM schemes are proposed in this paper. Convergence of the numerical algorithms is studied, and pth moment boundedness is obtained. Furthermore, asymptotic properties of the numerical solutions such as the exponential stability in pth moment and stability in distribution are examined. Several examples are given to illustrate our findings.
LanguageEnglish
Number of pages46
JournalIMA Journal of Numerical Analysis
Early online date9 Apr 2018
DOIs
StateE-pub ahead of print - 9 Apr 2018

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Finite Horizon
Infinite Horizon
Numerical Approximation
Truncation
Stochastic Equations
Differential equations
Differential equation
Moment
Euler
Lipschitz condition
Asymptotic stability
Euler line
Implicit Scheme
Exponential Stability
Numerical Algorithms
Diffusion Coefficient
Asymptotic Properties
Boundedness
Numerical Solution

Keywords

  • local Lipschitz condition
  • explicit EM scheme
  • finite horizon
  • infinite horizon
  • pth moment convergence
  • moment bound
  • stability
  • invariant measure

Cite this

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title = "Explicit numerical approximations for stochastic differential equations in finite and infinite horizons: truncation methods, convergence in pth moment, and stability",
abstract = "Solving stochastic differential equations (SDEs) numerically, explicit Euler-Maruyama (EM) schemes are used most frequently under global Lipschitz conditions for both drift and diffusion coefficients. In contrast, without imposing the global Lipschitz conditions, implicit schemes are often used for SDEs but require additional computational effort; along another line, tamed EM schemes and truncated EM schemes have been developed recently. Taking advantages of being explicit and easily implementable, truncated EM schemes are proposed in this paper. Convergence of the numerical algorithms is studied, and pth moment boundedness is obtained. Furthermore, asymptotic properties of the numerical solutions such as the exponential stability in pth moment and stability in distribution are examined. Several examples are given to illustrate our findings.",
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AB - Solving stochastic differential equations (SDEs) numerically, explicit Euler-Maruyama (EM) schemes are used most frequently under global Lipschitz conditions for both drift and diffusion coefficients. In contrast, without imposing the global Lipschitz conditions, implicit schemes are often used for SDEs but require additional computational effort; along another line, tamed EM schemes and truncated EM schemes have been developed recently. Taking advantages of being explicit and easily implementable, truncated EM schemes are proposed in this paper. Convergence of the numerical algorithms is studied, and pth moment boundedness is obtained. Furthermore, asymptotic properties of the numerical solutions such as the exponential stability in pth moment and stability in distribution are examined. Several examples are given to illustrate our findings.

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KW - infinite horizon

KW - pth moment convergence

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KW - stability

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