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

## 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 |
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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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