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

  • Desmond J. Higham (a1), Xuerong Mao (a2) and Andrew M. Stuart (a3)
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.

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References
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1Arnold, L., Stochastic differential equations: theory and applications (Wiley, NewYork, 1972).
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.
3Dekker, K. and Verwer, J.G., Stability of Runge–Kutta methods for stiff nonlinear equations (North-Holland, Amsterdam, 1984).
4Friedman, A., Stochastic differential equations and their applications (Academic Press, New York, 1976).
5Hairer, E., Nørsett, S.P. and Wanner, G., Solving ordinary differential equations I: nonstiff problems, 2nd edn (Springer, Berlin, 1993).
6Higham, D.J., ‘Mean-square and asymptotic stability of the stochastic theta method‘, SIAM J. Numer. Anal. 38 (2000) 753769.
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.
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.
9Khasminskii, R.Z., Stochastic stability of differential equations (Sijthoff and Noord-hoff, Alphen aan den Rijn, 1981).
10Mao, X., Stability of stochastic differential equations with respect to semimartingales (Longman Scientific and Technical, London, 1991).
11Mao, X., Exponential stability of stochastic differential equations (Marcel Dekker, New York, 1994).
12Mao, X., Stochastic differential equations and applications (Horwood, Chichester, 1997).
13Roberts, G.O. and Tweedie, R.L., ‘Exponential convergence of Langevin diffusions and their discrete approximations’, Bernoulli 2 (1996) 341363.
14Saito, Y. and Mitsui, T., ‘Stability analysis of numerical schemes for stochastic differential equations’, SIAM J. Numer. Anal. 33 (1996) 22542267.
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).
16Smart, D.R., Fixed point theorems (Cambridge University Press, 1974).
17Stuart, A.M. and Humphries, A.R., Dynamical systems and numerical analysis (Cambridge University Press, 1996).
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).
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LMS Journal of Computation and Mathematics
  • ISSN: -
  • EISSN: 1461-1570
  • URL: /core/journals/lms-journal-of-computation-and-mathematics
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