Exponential Mean-Square Stability of Numerical Solutions to Stochastic Differential Equations
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.
Additional Information
© 2003, Desmond J. Higham, Xuerong Mao and Andrew M. Stuart. Received 6 June 2003, revised 1 September 2003; published 28 November 2003. The first author was supported by a Research Fellowship from the Leverhulme Trust. The third author was supported by the Engineering and Physical Sciences Research Council of the UK under grant GR/N00340.Additional details
- Eprint ID
- 78115
- DOI
- 10.1112/S1461157000000462
- Resolver ID
- CaltechAUTHORS:20170612-133950435
- Leverhulme Trust
- GR/N00340
- Engineering and Physical Sciences Research Council (EPSRC)
- Created
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2017-06-12Created from EPrint's datestamp field
- Updated
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2021-11-15Created from EPrint's last_modified field
- Other Numbering System Name
- Andrew Stuart
- Other Numbering System Identifier
- J58