Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-16T10:42:23.168Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Analysis of Explicit One-Step Methods for Stochastic Delay Differential Equations

Published online by Cambridge University Press:  01 February 2010

Christopher T. H. Baker  and
Evelyn Buckwar
Christopher T. H. Baker
Affiliation:
Department of Mathematics, The Victoria University of Manchester, Manchester M13 9PL, cthbaker@maths.man.ac.uk
Evelyn Buckwar
Affiliation:
Department of Mathematics, The Victoria University of Manchester, Manchester M13 9PL, ebuckwar@maths.man.ac.uk

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.

We consider the problem of strong approximations of the solution of stochastic differential equations of Itô form with a constant lag in the argument. We indicate the nature of the equations of interest, and give a convergence proof in full detail for explicit one-step methods. We provide some illustrative numerical examples, using the Euler–Maruyama scheme.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 3 , 2000 , pp. 315 - 335
Copyright
Copyright © London Mathematical Society 2000

References

1. Arnold, L., Stochastic differential equations: theory and applications (Wiley-Interscience, New York, 1974).Google Scholar
2. Averina, T. A. and Artemiev, S. S., Numerical analysis of systems of ordinary and stochastic differential equations (VSP, Utrecht, 1997).Google Scholar
3. Baker, C. T. H., ‘Retarded differential equations’, J. Comput. Appl. Math., in press.Google Scholar
4. Baker, C. T. H., Bocharov, G. A., Paul, C. A. H. and Rihan, F. A., ‘Modelling and analysis of time-lags in some basic patterns of cell proliferation’, J. Math. Biol. 37 (1998)341–371.Google Scholar
5. Beuter, A. and Bélair, J., ‘Feedback and delays in neurological diseases: a modelling study using dynamical systems’, Bull. Math. Biol. 55 (1993) 525–541.Google Scholar
6. Burrage, K. and Platen, E., ‘Runge-Kutta methods for stochastic differential equations’, Ann. Numer. Math. 1 (1994) 63–78.Google Scholar
7. Clark, J. M. C., ‘The discretization of stochastic differential equations: a primer’, Road Vehicle Systems and Related Mathematics, Proc. 2nd DMV-GAMM Workshop, Torino, 1987 (ed. Neunzert, H., Teubner, Stuttgart, and Kluwer Academic Publishers, Amsterdam, 1988) 163–179.Google Scholar
8. Crauel, H. and Gundlach, M. (eds), Stochastic dynamics, Papers from the Conference on Random Dynamical Systems, Bremen, April 28-May 2 1997 (Springer, New York, 1999).Google Scholar
9. Driver, R. D., Ordinary and delay differential equations, Appl. Math. Sci. 20 (Springer, New York, 1977).Google Scholar
10. Eurich, C. W. and Milton, J. G., ‘Noise-induced transitions in human postural sway’, Phys. Rev. E 54 (1996) 6681‘6684 1996).Google Scholar
11. Gaines, J. G. and Lyons, T. L., ‘Variable step size control in the numerical solution of stochastic differential equations’, SIAM J. Appl. Math. 57 (1997) 1455–1484.CrossRefGoogle Scholar
12. Gard, T. C., Introduction to stochastic differential equations (Marcel-Dekker, New York, Basel, 1988).Google Scholar
13. Hofmann, N., Müller-Gronbach, T. and Ritter, K., ‘Optimal approximation of stochastic differential equations by adaptive step-size control’, Math. Comp. 69 (2000) 1017–1034.CrossRefGoogle Scholar
14. Karatzas, I. and Shreve, S. E., Brownian motion and stochastic calculus (Springer, New York, 1991).Google Scholar
15. Kloeden, P. E. and Platen, E., Numerical solution of stochastic differential equations (Springer, Berlin, 1992).CrossRefGoogle Scholar
16. Kolmanovskiĭ, V. and Myshkis, A., Applied theory of functional-differential equations (Kluwer Academic Publishers Group, Dordrecht, 1992).Google Scholar
17. Küchler, U. and Platen, E., ‘Strong discrete time approximation of stochastic differential equations with time delay’, Discussion paper, Interdisciplinary Research Project 373 (ISSN 1436–1086) 25, Humboldt University, Berlin (1999).Google Scholar
18. Mackey, M. C., Longtin, A., Milton, J. G. and Bos, J. E., ‘Noise and critical behaviour of the pupil light reflex at oscillation onset’, Phys. Rev. A 41 (1990) 6992–7005.Google Scholar
19. Mao, X., Stochastic differential equations and their applications (Horwood Publishing, Chichester, 1997).Google Scholar
20. Mauthner, S., ‘Step size control in the numerical solution of stochastic differential equations’, J. Comput. Appl. Math. 100 (1998) 93–109.CrossRefGoogle Scholar
21. Milstein, G. N., Numerical integration of stochastic differential equations (translated and revised from the 1988 Russian original) (Kluwer, Dordrecht, 1995).CrossRefGoogle Scholar
22. Mizel, V. J. and Trutzer, V., ‘Stochastic hereditary equations: existence and asymptotic stability’, J. Integral Equations 7 (1984) 1–72.Google Scholar
23. Mohammed, S. E. A., Stochastic functional differential equations (Pitman Advanced Publishing Program, Boston, MA, 1984).Google Scholar
24. Mohammed, S. E. A., ‘Stochastic differential systems with memory. Theory, examples and applications’, Stochastic analysis and related topics VI, The Geilo Work shop, 1996, Progress in Probability (ed. Decreusefond, L., Gjerde, J., Oksendal, B. and Ustunel, A. S., Birkhauser, 1998) 1–77.Google Scholar
25. Platen, E., ‘An introduction to numerical methods for stochastic differential equations’, Acta Numerica 8 (1999) 195–244.CrossRefGoogle Scholar
26. Saito, Y. and Mitsui, T., ‘Stability analysis of numerical schemes for stochastic differential equations’, SIAM J. Numer. Anal. 33 (1996) 2254–2267.CrossRefGoogle Scholar
27. Talay, D., ‘Simulation of stochastic differential systems’, Probabilistic methods in applied physics, Lecture Notes in Phys. 451 (ed. Kree, P. and Wedig, W., Springer, Berlin, 1995) 54–96.Google Scholar
28. Tudor, M., ‘Approximation schemes for stochastic equations with hereditary argument’ (in Romanian), Stud. Cere. Mat. 44 (1992) 73–85.Google Scholar
29. Williams, D., Probability with martingales (Cambridge University Press, Cambridge, 1991).CrossRefGoogle Scholar
30. Zennaro, M., ‘Delay differential equations: theory and numerics’, Theory and numerics of ordinary and partial differential equations, Leicester, 1994 (ed. Ainsworth, M., Levesley, J., Light, W. A. and Marletta, M., Oxford University Press, New York, 1995) 291–333.Google Scholar