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Exponential mean square stability of numerical solutions to stochastic differential equations

Higham, D.J. and Mao, X. and Stuart, A.M. (2003) Exponential mean square stability of numerical solutions to stochastic differential equations. LMS Journal of Computation and Mathematics, 6. pp. 297-313. ISSN 1461-1570

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Abstract

Positive results are proved here about the ability of numerical simulations to reproduce the exponential mean-square stability of stochastic differential equations (SDEs). The first set of results applies under finite-time convergence conditions on the numerical method. Under these conditions, the exponential mean-square stability of the SDE and that of the method (for sufficiently small step sizes) are shown to be equivalent, and the corresponding second-moment Lyapunov exponent bounds can be taken to be arbitrarily close. The required finite-time convergence conditions hold for the class of stochastic theta methods on globally Lipschitz problems. It is then shown that exponential mean-square stability for non-globally Lipschitz SDEs is not inherited, in general, by numerical methods. However, for a class of SDEs that satisfy a one-sided Lipschitz condition, positive results are obtained for two implicit methods. These results highlight the fact that for long-time simulation on nonlinear SDEs, the choice of numerical method can be crucial.

Item type: Article
ID code: 166
Keywords: stochastic differential equations, numerical simulations, Lipschitz problems, mean-square stability, SDEs, Mathematics, Mathematics(all), Computational Theory and Mathematics
Subjects: Science > Mathematics
Department: Faculty of Science > Mathematics and Statistics
Related URLs:
Depositing user: Ms Sarah Scott
Date Deposited: 22 Feb 2006
Last modified: 04 Sep 2014 12:06
URI: http://strathprints.strath.ac.uk/id/eprint/166

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