Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-19T08:02:03.217Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  01 February 2010

Desmond J. Higham
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow G1 1XHdjh@maths.strath.ac.uk, http://www.maths.strath.ac.uk/~aas96106/
Xuerong Mao
Affiliation:
Department of Statistics and Modelling Science, University of Strathclyde, Glasgow Gl 1XH, xuerong@stams.strath.ac.uk, http://www.stams.strath.ac.uk/people/staff/bios/XuerongMao/index.php
Andrew M. Stuart
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7ALstuart@maths.warwick.ac.uk, http://www.maths.warwick.ac.uk/staff/stuart.html

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
Copyright
Copyright © London Mathematical Society 2003

References

1Arnold, L., Stochastic differential equations: theory and applications (Wiley, NewYork, 1972).Google Scholar
2Baker, C.T.H. and Buckwar, E., ‘Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, to stochastic delay differential equations’, Numerical Analysis Report 390, University of Manchester, 2001.Google Scholar
3Dekker, K. and Verwer, J.G., Stability of Runge–Kutta methods for stiff nonlinear equations (North-Holland, Amsterdam, 1984).Google Scholar
4Friedman, A., Stochastic differential equations and their applications (Academic Press, New York, 1976).Google Scholar
5Hairer, E., Nørsett, S.P. and Wanner, G., Solving ordinary differential equations I: nonstiff problems, 2nd edn (Springer, Berlin, 1993).Google Scholar
6Higham, D.J., ‘Mean-square and asymptotic stability of the stochastic theta method‘, SIAM J. Numer. Anal. 38 (2000) 753769.CrossRefGoogle Scholar
7Higham, D.J., Mao, X. and Stuart, A.M., ‘Strong convergence of numerical methods for nonlinear stochastic differential equations’, SIAM J. Numer. Anal. 40 (2002) 10411063.CrossRefGoogle Scholar
8Mattingly, J., Stuart, A.M. and Higham, D.J., ‘Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise’, Stock. Process. Appl. 101 (2002) 185232.CrossRefGoogle Scholar
9Khasminskii, R.Z., Stochastic stability of differential equations (Sijthoff and Noord-hoff, Alphen aan den Rijn, 1981).Google Scholar
10Mao, X., Stability of stochastic differential equations with respect to semimartingales (Longman Scientific and Technical, London, 1991).Google Scholar
11Mao, X., Exponential stability of stochastic differential equations (Marcel Dekker, New York, 1994).Google Scholar
12Mao, X., Stochastic differential equations and applications (Horwood, Chichester, 1997).Google Scholar
13Roberts, G.O. and Tweedie, R.L., ‘Exponential convergence of Langevin diffusions and their discrete approximations’, Bernoulli 2 (1996) 341363.CrossRefGoogle Scholar
14Saito, Y. and Mitsui, T., ‘Stability analysis of numerical schemes for stochastic differential equations’, SIAM J. Numer. Anal. 33 (1996) 22542267.CrossRefGoogle Scholar
15Schurz, H., Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications, PhD Thesis, Humboldt University (Logos Verlag, Berlin, 1997).Google Scholar
16Smart, D.R., Fixed point theorems (Cambridge University Press, 1974).Google Scholar
17Stuart, A.M. and Humphries, A.R., Dynamical systems and numerical analysis (Cambridge University Press, 1996).Google Scholar
18Talay, D., Approximation of the invariant probability measure of stochastic Hamiltonian dissipative systems with non globally Lipschitz co-efficients, Progress in Stochastic Structural Dynamics 152 (ed. Bouc, R. and Soize, C., L.M.A.-CNRS, 1999).Google Scholar