Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-23T08:37:56.483Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak Convergence of the Euler Scheme for Stochastic Differential Delay Equations

Published online by Cambridge University Press:  01 February 2010

Evelyn Buckwar ,
Rachel Kuske ,
Salah-Eldin Mohammed  and
Tony Shardlow
Evelyn Buckwar
Affiliation:
Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom, [email protected]
Rachel Kuske
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada, [email protected]
Salah-Eldin Mohammed
Affiliation:
Department of Mathematics, Southern Illinois University at Carbondale, Carbondale, IL 62901, USA, [email protected]
Tony Shardlow
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom, [email protected]

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 study weak convergence of an Euler scheme for nonlinear stochastic delay differential equations (SDDEs) driven by multidimensional Brownian motion. The Euler scheme has weak order of convergence 1, as in the case of stochastic ordinary differential equations (SODEs) (i.e., without delay). The result holds for SDDEs with multiple finite fixed delays in the drift and diffusion terms. Although the set-up is non-anticipating, our approach uses the Malliavin calculus and the anticipating stochastic analysis techniques of Nualart and Pardoux.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 11 , 2008 , pp. 60 - 99
Copyright
Copyright © London Mathematical Society 2008

References

1.Ahmed, T. A., ‘Stochastic Functional Differential Equations with Discontinuous Initial Data’, MSc Thesis, University of Khartoum, Sudan, 1983.Google Scholar
2.Baker, C. T. H. and Buckwar, E., ‘Numerical analysis of explicit one-step methods for stochastic delay differential equations’, LMS J. Comput. Math. 3 (2000) 315–335.CrossRefGoogle Scholar
3.Baker, C. T. H. and Buckwar, E., ‘Continuous Θ-methods for the stochastic pantograph equation’, Electron. Trans. Numer. Anal. 11 (2000) 131–151.Google Scholar
4.Bell, D. and Mohammed, S.-E. A., ‘The Malliavin calculus and stochastic delay equations’, J. Fund. Analysis 99 (1991) 75–99.CrossRefGoogle Scholar
5.Beuter, A. and Vasilakos, K., ‘Effects of noise on a delayed visual feedback system’, J. Theor. Biology 165 (1993) 389–407.Google Scholar
6.Buckwar, E. and Shardlow, T., ‘Weak approximation of stochastic differential delay equations’, IMA J. Numer. Anal. 25 (2005) 57–86.CrossRefGoogle Scholar
7.Bally, V. and Talay, D., ‘The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function’, Probab. Theory Related Fields 104 (1996) 43–60.CrossRefGoogle Scholar
8.Buldú, J .M., Garcia-Ojalvo, J., Mirasso, C. R., Torrent, M. C. and Sancho, J. M., ‘Effect of external noise correlation in optical coherence resonance’, Phys. Rev. E 64 (2001) 051109-1–051109-4.Google ScholarPubMed
9.Clément, E., Kohatsu-Higa, A. and Lamberton, D., ‘A duality approach for the weak approximation of stochastic differential equations’, Ann. Appl. Probab. 16 (2006) 1124–1154.CrossRefGoogle Scholar
10.Eurich, C. W. and Milton, J. G., ‘Noise-induced transitions in human postural sway’, Phys. Rev. E 54 (1996) 6681–6684.Google ScholarPubMed
11.Fournié, E., Lasry, J.-M., Lebuchoux, J. and Lions, P.-L., ‘Applications of Malliavin calculus to Monte Carlo methods in finance’, Finance Stoch. 3 (1999) 391–412.Google Scholar
12.Fournié, E., Lasry, J.-M., Lebuchoux, J. and Lions, P.-L., ‘Applications of Malliavin calculus to Monte-Carlo methods in finance. II’, Finance Stoch. 5 (2001) 201–236.CrossRefGoogle Scholar
13.Goldoboin, D., Rosenblum, M. and Pikovsky, A., ‘Coherence of noisy oscillators with delayed feedback’, Physica A 327 (2003) 124–128.CrossRefGoogle Scholar
14.Hu, Y., ‘Strong and weak order of time discretization schemes of stochastic differential equations’, Séminaire de Probabilitiés XXX (ed. Azema, J., Meyer, P. A. and Yor, M.), Lecture Notes in Mathematics 1626 (Springer, 1996) 218–227.CrossRefGoogle Scholar
15.Hu, Y., Mohammed, S.-E. A. and Yan, F., ‘Discrete-time approximations of stochastic delay equations: The Milstein scheme’, Ann. Probab 32 (2004) 265–314.CrossRefGoogle Scholar
16.Huber, D. and Tsimring, L. S., ‘Dynamics of an ensemble of noisy bistable elements with global time delayed coupling’, Phys. Rev. Lett. 91 (2003) 260601.CrossRefGoogle ScholarPubMed
17.E. Kloeden, P. and Platen, E., Numerical Solutions of Stochastic Differential Equations (Springer, New York-Berlin, 1995).Google Scholar
18.E. Kloeden, P., Platen, E. and Schurz, H., Numerical Solutions of SDE Through Computer Experiment (Springer, New York-Berlin, 1997).Google Scholar
19.Higa, A. Kohatsu-, ‘Weak approximations. A Malliavin calculus approach’, Math. Comput. 70 (2001) 135–172.CrossRefGoogle Scholar
20.Higa, A. Kohatsu- and Pettersson, R., ‘Variance reduction methods for simulation of densities on Wiener space’, SIAM J. Numer. Anal. 40 (2002) 431–450.CrossRefGoogle Scholar
21.Küchler, U. and Platen, E., ‘Weak discrete time approximation of stochastic differential equations with time delay’, Math. Comput. Simulation 59 (2002) 497–507.CrossRefGoogle Scholar
22.Longtin, A., Milton, J. G., Bos, J. and Mackey, M. C., ‘Noise and critical behavior of the pupil light reflex at oscillation onset’, Phys. Rev. A 41 (1990) 6992–7005.CrossRefGoogle ScholarPubMed
23.Mao, X. and Sabnis, S., ‘Numerical solutions of stochastic differential delay equations under local Lipschitz condition’, J. Comput. Appl. Math. 151 (2003) 215–227.CrossRefGoogle Scholar
24.Mao, X., ‘Numerical solutions of stochastic functional differential equations’, LMS J. Comput. Math. 6 (2003) 141–161.CrossRefGoogle Scholar
25.Masoller, C., ‘Numerical investigation of noise-induced resonance in a semiconductor laser with optical feedback’, Physica D 168–169 (2002), 171–176.Google Scholar
26.Masoller, C., ‘Distribution of residence times of time-delayed bistable systems driven by noise’, Phys. Rev. Lett. 90 (2003) 020601.CrossRefGoogle ScholarPubMed
27.Milstein, G. N. and Tret'yakov, M. V., ‘Numerical methods in the weak sense for stochastic differential equations with small noise’, SIAM J. Numer. Anal. 34 (1997) 2142–2167.CrossRefGoogle Scholar
28.Milstein, G. N. and Tret'yakov, M. V., Stochastic Numerics for Mathematical Physics (Springer, Berlin, 2004).CrossRefGoogle Scholar
29.Mohammed, S.-E. A., ‘Stochastic Functional Differential Equations’, Research Notes in Mathematics 99 (Pitman, Boston, 1984).Google Scholar
30.Mohammed, S.-E. A., ‘Stochastic differential systems with memory: Theory, examples and applications’, Stochastic Analysis (ed. Decreusefond, L., Gjerde, J., Øksendal, B. and Ustunel, A. S.), Progress in Probability 42 (Birkhäuser, 1998) 1–77.Google Scholar
31.Mohammed, S.-E. A. and Scheutzow, M. K. R., ‘Lyapunov exponents of linear stochastic functional differential equations driven by semimartingales. I: The multiplicative ergodic theory‘, Ann. Inst. Henri Poincaré 32 (1996) 69–105.Google Scholar
32.Mohammed, S.-E. A. and Scheutzow, M. K. R., ‘Lyapunov exponents of linear stochastic functional differential equations. II: Examples and case studies’, Ann. Probab. 25 (1997) 1210–1240.CrossRefGoogle Scholar
33.Nualart, D. and Pardoux, E., ‘Stochastic calculus with anticipating integrands’, Probability Theory and Related Fields 78 (1988) 535–581.CrossRefGoogle Scholar
34.Nualart, D., The Malliavin Calculus and Related Topics (Springer, 1995).CrossRefGoogle Scholar
35.Nualart, D., Analysis on Wiener Space and Anticipating Stochastic Calculus, Lecture Notes in Mathematics 1690 (Springer, New York, 1998) 123–227.Google Scholar
36.Pikovsky, A. and Tsimring, L. S., ‘Noise-induced dynamics in bistable systems with delay’, Phys. Rev. Lett. 87 (2001) 250602-1–250602-4.Google Scholar
37.Tudor, C. and Tudor, M., ‘On approximation of solutions for stochastic delay equations’, Stud. Cercet. Mat. 39 (1987) 265–274.Google Scholar
38.Tudor, C. and Tudor, M., ‘Approximation of linear stochastic functional equations’, Rev. Roum. Math. Pures Appl. 35 (1990) 81–99.Google Scholar
39.Tudor, M., ‘Approximation schemes for stochastic equations with hereditary argument’, Stud. Cercet. Mat. 44 (1992) 73–85.Google Scholar