Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-13T05:32:41.190Z Has data issue: false hasContentIssue false
  • English
  • Français

Simulations of Two-Step Maruyama Methods for Nonlinear Stochastic Delay Differential Equations

Published online by Cambridge University Press:  03 June 2015

Wanrong Cao  and
Zhongqiang Zhang
Wanrong Cao*
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, China Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
Zhongqiang Zhang*
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
*
Corresponding author. Email: wrcao@seu.edu.cn
Email: zhongqiang_zhang@brown.edu
Get access

Abstract

In this paper, we investigate the numerical performance of a family of P-stable two-step Maruyama schemes in mean-square sense for stochastic differential equations with time delay proposed in for a certain class of nonlinear stochastic delay differential equations with multiplicative white noises. We also test the convergence of one of the schemes for a time-delayed Burgers’ equation with an additive white noise. Numerical results show that this family of two-step Maruyama methods exhibit similar stability for nonlinear equations as that for linear equations.

Keywords

65C2065C3065Q20P-stability in mean-square sensetwo-step Maruyama methodsnonlinear stochastic delay differential systemBurgers’ equation
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 6 , December 2012 , pp. 821 - 832
Copyright
Copyright © Global-Science Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Barwell, V. K., Special stability problems for functional equations, BIT, 15 (1975), pp. 130–135.Google Scholar
[2]Bellen, A. and Zennaro, M., Numerical Methods for Delay Differential Equations, Oxford, Clarendon Press, 2003.Google Scholar
[3]Baker, C. T. H. and Buckwar, E., Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations, J. Comput. Appl. Math., 184 (2005), pp. 404–427.CrossRefGoogle Scholar
[4]Beretta, E., Kolmanovskii, V. B. and Shaikhet, L, Stability of epidemic model with time delays influenced by stochastic perturbations, Math. Comput. Simul., 45 (1998), pp. 269–277.CrossRefGoogle Scholar
[5]Beuter, A. and Vasilakos, K., Effects of noise on a delayed visual feedback system, J. Theor. Bio., 165 (1993), pp. 389–407.Google Scholar
[6]Beuter, A. and Belair, J., Feedback and delays in neurological diseases: a modeling study using dynamical systems, Bull Math. Bio., 55 (1993), pp. 525–541.Google Scholar
[7]Brugnano, L., Burrage, K. and Burrage, P. M., Adams-type methods for the numerical solution of stochastic ordinary differential equations, BIT, 40 (2000), pp. 451–470.Google Scholar
[8]Buckwar, E. and Winkler, R., Multi-step Maruyama methods for stochastic delay differential equations, Stoch. Anal. Appl., 25 (2007), pp. 933–959.Google Scholar
[9]Buckwar, E. and Winkler, R., Multi-step methods for SDEs and their application to problems with small noise, SIAM J. Numer. Anal., 44 (2006), pp. 779–803.Google Scholar
[10]Cao, W. and Zhang, Z., On mean-square stability of two-step Maruyama methods for stochastic delay differential equations, J. Comput. Appl. Math., Submitted.Google Scholar
[11]Denk, G. and Schaüffler, S., Adams methods for the efficient solution of stochastic differential equations with additive noise, Computing, 59 (1996), pp. 153–161.Google Scholar
[12]Ewald, B. D. and Tèmam, R., Numerical analysis of stochastic schemes in geophysics, SIAM J. Numer. Anal., 42 (2005), pp. 2257–2276.Google Scholar
[13]Fan, Z., Waveform relaxation method for stochastic differential equations with constant delay, Appl. Numer. Math., 61 (2011), pp. 229–240.Google Scholar
[14]Grigoriu, M., Control of time delay linear systems with Gaussian white noise, Prob. Eng. Mech., 12 (1997), pp. 89–96.CrossRefGoogle Scholar
[15]Hobson, D. G. and Rogers, L. C. G., Complete models with stochastic balatility, Math. Fina., 8 (1998), pp. 27–48.Google Scholar
[16]Kloeden, P. E. and Platen, E., Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992.CrossRefGoogle Scholar
[17]Li, R., Convergence and stability of numerical solutions to SDDEs with Markovian switching, Appl. Math. Comput., 175 (2006), pp. 1080–1091.Google Scholar
[18]Mao, X., Stochastic Differential Equations and Applications, Horwood, 1997.Google Scholar
[19]Mao, X., Razumikhin-Type theorems on exponential stability of stochastic functional differential equations, Stochastic Proc. Appl., 65 (1996), pp. 233–250.Google Scholar
[20]Mao, X., Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions, Appl. Math. Comput., 217 (2011), pp. 5512–5524.Google Scholar
[21]Mao, X., Exponential Stability of Stochastic Differential Equations, Marcel Dekker, Inc., New York, 1994.Google Scholar
[22]Milstein, G. N., Numerical Integration of Stochastic Differential Equations, Kluwer Academic Publishers Group, Dordrecht, 1995.Google Scholar
[23]Mohammed, S. E. A., Stochastic Functional Differential Equations, Research Notes in Mathematics, Pitman, London, 99 (1984).Google Scholar
[24]Paola, M. D. and Pirrotta, A., Time delay induced effects on control of linear systems under random excitation, Prob. Eng. Mech., 16 (2001), pp. 43–51.Google Scholar
[25]Tian, H. J. and Kuang, J. X., The numerical stability of linear multistep methods for delay differential equations with many delays, SIAM J. Numer. Anal., 33 (1996), pp. 883–889.Google Scholar
[26]Tsimring, L. S. and Pikovsky, A., Noise-induced dynamics in bistable systems with delay, Phys. Rev. Lett., 87 (2001), pp. 250602-1–25062-4.Google Scholar
[27]Zhou, S. and Wu, F., Convergence of numerical solutions to neutral stochastic delay differential equations with Markovian switching, J. Comput. Appl. Math., 229 (2009), pp. 85–96.Google Scholar
[28]Liu, W., Asymptotic behavior of solutions of time-delayed Burgers’ equation, Dis. Cont. Dyn. Syst. Series B, 2 (2002), pp. 47–56.Google Scholar
[29]Zhang, Z., Rozovskii, B. and Karniadakis, G. E., Stochastic collocation methods for stochastic differential equations driven by white noise, Submitted to SISC.Google Scholar