Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T10:36:34.986Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Solutions of Stochastic Functional Differential Equations

Published online by Cambridge University Press:  01 February 2010

Xuerong Mao
Xuerong Mao
Affiliation:
Department of Statistics and Modelling Science, University of Strathclyde, Glasgow G1 [email protected], http://www.stams.strath.ac.uk/~xuerong

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.

In this paper, the strong mean square convergence theory is established for the numerical solutions of stochastic functional differential equations (SFDEs) under the local Lipschitz condition and the linear growth condition. These two conditions are generally imposed to guarantee the existence and uniqueness of the true solution, so the numerical results given here were obtained under quite general conditions.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 6 , 2003 , pp. 141 - 161
Copyright
Copyright © London Mathematical Society 2003

References

1Baker, C. T. H. and Buckwar, E., ‘Numerical analysis of explicit one-step methods for stochastic delay differential equations’, LMSJ. Comput. Math. 3 (2000)315–335.CrossRefGoogle Scholar
2Buckwar, E., ‘Introduction to the numerical analysis of stochastic delay differential equations‘, J. Comput. Appl. Math 125 (2000)297–307.CrossRefGoogle Scholar
3Fleury, G. and Bernard, P., ‘Convergence of numerical schemes for stochastic differential equations’, Monte Carlo Methods Appl. 7 (2001)35–4.Google Scholar
4Gyöngy, I., ‘A note on Euler's approximations’, Potential Anal. 8 (1998)205–216.CrossRefGoogle Scholar
5Gyöngy, I. and Krylov, N., ‘‘Existence of strong solutions for Itô's stochastic equations via approximations’, Probab. Theory Related Fields 105 (1996)143–158.CrossRefGoogle Scholar
6Hale, J.K. and Lunel, V. S. M., Introduction to functional differential equations (Springer, Berlin, 1993).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)1041–1063.CrossRefGoogle Scholar
8Hu, Y., ‘Semi-implicit Euler-Maruyama scheme for stiff stochastic equations’, Stochastic analysis and related topics V: The Silvri Workshop, Progr. Probab. 38 (ed. Koerezlioglu, H., Birkhauser, Boston, MA, 1996) 183–302.CrossRefGoogle Scholar
9Hu, Y., Mohammed, S. E. A. and Yan, F., ‘Numerical solution of stochastic differential systems with memory’, Preprint, Southern Illinois University, 2001.Google Scholar
10Kloeden, P. E. and Platen, E., Numerical solutions of stochastic differential equations (Springer, Berlin, 1992).CrossRefGoogle Scholar
11Kolmanovskii, V. and Myshkis, A., Applied theory of fundamental differential equations (Kluwer Academic Publishers, Dordrecht, 1992).CrossRefGoogle Scholar
12Küchler, U. and Platen, E., ‘Strong discrete time approximation of stochastic differential equations with time delay’, Math. Comput. Simulation 54 (2000)189–205.CrossRefGoogle Scholar
13Mao, X., Approximate solutions for a class of stochastic evolution equations with variable delays—part II, Numerical Functional Analysis and Optimisation 15 (1994)65–76.CrossRefGoogle Scholar
14Mao, X., Exponential stability of stochastic differential equations (Marcel Dekker, New York, 1994).Google Scholar
15Mao, X., Stochastic differential equations and applications (Horwood, England, 1997).Google Scholar
16Mao, X. and Sabanis, S., ‘Numerical solutions of stochastic differential delay equations under local Lipschitz condition’, J. Comput. Appl. Math. 151 (2003)215–227.CrossRefGoogle Scholar
17Milstein, G.N., Numerical integration of stochastic differential equations (Kluwer Academic Publishers Group, Dordrecht, 1995).CrossRefGoogle Scholar
18Platen, E., ‘An introduction to numerical methods for stochastic differential equations’, Acta Numerica 8 (1999)195–244.CrossRefGoogle Scholar