Strong convergence of Euler-type methods for nonlinear stochastic differential equations

D.J. Higham, X. Mao, A.M. Stuart

Research output: Contribution to journalArticle

321 Citations (Scopus)

Abstract

Traditional finite-time convergence theory for numerical methods applied to stochastic differential equations (SDEs) requires a global Lipschitz assumption on the drift and diffusion coefficients. In practice, many important SDE models satisfy only a local Lipschitz property and, since Brownian paths can make arbitrarily large excursions, the global Lipschitz-based theory is not directly relevant. In this work we prove strong convergence results under less restrictive conditions. First, we give a convergence result for Euler--Maruyama requiring only that the SDE is locally Lipschitz and that the pth moments of the exact and numerical solution are bounded for some p >2. As an application of this general theory we show that an implicit variant of Euler--Maruyama converges if the diffusion coefficient is globally Lipschitz, but the drift coefficient satisfies only a one-sided Lipschitz condition; this is achieved by showing that the implicit method has bounded moments and may be viewed as an Euler--Maruyama approximation to a perturbed SDE of the same form. Second, we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial.
LanguageEnglish
Pages1041-1063
Number of pages22
JournalSIAM Journal on Numerical Analysis
Volume40
Issue number3
DOIs
Publication statusPublished - 2002

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Strong Convergence
Stochastic Equations
Lipschitz
Euler
Differential equations
Differential equation
Convergence Results
Diffusion Coefficient
Lipschitz Property
Moment
Optimal Rate of Convergence
Convergence Theory
Excursion
Lipschitz condition
Local Properties
Implicit Method
Coefficient
Numerical methods
Exact Solution
Numerical Methods

Keywords

  • finite-time convergence
  • nonlinearity
  • computer science
  • applied mathematics

Cite this

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Strong convergence of Euler-type methods for nonlinear stochastic differential equations. / Higham, D.J.; Mao, X.; Stuart, A.M.

In: SIAM Journal on Numerical Analysis, Vol. 40, No. 3, 2002, p. 1041-1063.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Strong convergence of Euler-type methods for nonlinear stochastic differential equations

AU - Higham, D.J.

AU - Mao, X.

AU - Stuart, A.M.

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AB - Traditional finite-time convergence theory for numerical methods applied to stochastic differential equations (SDEs) requires a global Lipschitz assumption on the drift and diffusion coefficients. In practice, many important SDE models satisfy only a local Lipschitz property and, since Brownian paths can make arbitrarily large excursions, the global Lipschitz-based theory is not directly relevant. In this work we prove strong convergence results under less restrictive conditions. First, we give a convergence result for Euler--Maruyama requiring only that the SDE is locally Lipschitz and that the pth moments of the exact and numerical solution are bounded for some p >2. As an application of this general theory we show that an implicit variant of Euler--Maruyama converges if the diffusion coefficient is globally Lipschitz, but the drift coefficient satisfies only a one-sided Lipschitz condition; this is achieved by showing that the implicit method has bounded moments and may be viewed as an Euler--Maruyama approximation to a perturbed SDE of the same form. Second, we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial.

KW - finite-time convergence

KW - nonlinearity

KW - computer science

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