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

Higham, D.J. and Mao, X. and Stuart, A.M. (2002) Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM Journal on Numerical Analysis, 40 (3). pp. 1041-1063. ISSN 0036-1429

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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.

Item type: Article
ID code: 172
Keywords: finite-time convergence, nonlinearity, computer science, applied mathematics, Electronic computers. Computer science, Mathematics, Numerical Analysis
Subjects: Science > Mathematics > Electronic computers. Computer science
Science > Mathematics
Department: Faculty of Science > Mathematics and Statistics
Related URLs:
Depositing user: Ms Sarah Scott
Date Deposited: 03 Mar 2006
Last modified: 04 Sep 2014 11:02
URI: http://strathprints.strath.ac.uk/id/eprint/172

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