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Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations

Higham, D.J. and Mao, X. and Yuan, C. (2007) Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations. SIAM Journal on Numerical Analysis, 45 (2). pp. 592-609. ISSN 0036-1429

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Abstract

Relatively little is known about the ability of numerical methods for stochastic differential equations (SDEs) to reproduce almost sure and small-moment stability. Here, we focus on these stability properties in the limit as the timestep tends to zero. Our analysis is motivated by an example of an exponentially almost surely stable nonlinear SDE for which the Euler-Maruyama (EM)method fails to reproduce this behavior for any nonzero timestep. We begin by showing that EM correctly reproduces almost sure and small-moment exponential stability for sufficiently small timesteps on scalar linear SDEs. We then generalize our results to multidimensional nonlinear SDEs. We show that when the SDE obeys a linear growth condition, EM recovers almost surely exponential stability very well. Under the less restrictive condition that the drift coefficient of the SDE obeys a one-sided Lipschitz condition, where EM may break down, we show that the backward Euler method maintains almost surely exponential stability.

Item type: Article
ID code: 4547
Keywords: backward Euler, Euler-Maruyama, implicit, one-sided Lipschitz condition, linear growth condition, Lyapunov exponent, stochastic theta method, numerical mathematics, Mathematics, Numerical Analysis
Subjects: Science > Mathematics
Department: Faculty of Science > Mathematics and Statistics
Depositing user: Strathprints Administrator
Date Deposited: 01 Nov 2007
Last modified: 24 Jul 2015 01:34
URI: http://strathprints.strath.ac.uk/id/eprint/4547

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