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Exponential Mean-Square Stability of Numerical Solutions to Stochastic Differential Equations

Published online by Cambridge University Press:  01 February 2010

Desmond J. Higham ,
Xuerong Mao  and
Andrew M. Stuart
Desmond J. Higham
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow G1 [email protected], http://www.maths.strath.ac.uk/~aas96106/
Xuerong Mao
Affiliation:
Department of Statistics and Modelling Science, University of Strathclyde, Glasgow Gl 1XH, [email protected], http://www.stams.strath.ac.uk/people/staff/bios/XuerongMao/index.php
Andrew M. Stuart
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 [email protected], http://www.maths.warwick.ac.uk/staff/stuart.html

Abstract

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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.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 6 , 2003 , pp. 297 - 313
Copyright
Copyright © London Mathematical Society 2003

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