Abstract
This paper deals with almost sure and moment exponential stability of a class of predictor-corrector methods applied to the stochastic differential equations of Itô-type. Stability criteria for this type of methods are derived. The methods are shown to maintain almost sure and moment exponential stability for all sufficiently small timesteps under appropriate conditions. A numerical experiment further testifies these theoretical results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Carletti, K. Burrage and P. M. Burrage, Numerical simulation of stochastic ordinary differential equations in biomathematical modelling, Mathematics and Computers in Simulation, 2004, 64: 271–277.
Y. Xu, L. J. S. Allena, A. S. Perelson, Stochastic model of an influenza epidemic with drug resistance, Journal of Theoretical Biology, 2007, 248: 179–193.
Y. Cao and D. Samuels, Discrete stochastic simulation methods for chemically reacting systems, Methods in Enzymology, 2009, 454: 115–140.
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.
K. Burrage and P. M. Burrage, High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations, Applied Numerical Mathematics, 1996, 22: 81–101.
K. Burrage and T. Tian, Predictor-corrector methods of Runge-Kutta type for stochastic differential equations, SIAM Journal on Numerical Analysis, 2002, 40(4): 1516–1537.
Y. Saito and T. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM Journal on Numerical Analysis, 1996, 33(6): 2254–2267.
G. N. Milstein, E. Platen, and H. Schurz, Balanced implicit methods for stiff stochastic systems, SIAM Journal on Numerical Analysis, 1998, 35(3): 1010–1019.
P. M. Burrage, Runge-Kutta methods for stochastic differential equations, Ph.D.Thesis, The University of Queensland, 1999.
D. J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM Journal on Numerical Analysis, 2000, 38(3): 753–769.
A. Bryden and D. J. Higham, On the boundedness of asymptotic stability regions for the stochastic theta method, BIT Numerical Mathematics, 2003, 43: 1–6.
D. J. Higham, X. Mao, and A. M. Stuart, Exponential mean-square stability of numerical solutions to stochastic differential equations, LMS Journal of Computation and Mathematics, 2003, 6: 297–313.
D. J. Higham, X. Mao, and C. Yuan, Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations, SIAM Journal on Numerical Analysis, 2007, 45(2): 592–609.
S. Pang, F. Deng, and X. Mao, Almost sure and moment exponential stability of Euler-Maruyama discretizations for hybrid stochastic differential equations, Journal of Computational and Applied Mathematics, 2008, 213: 127–141.
N. Bruti-Liberati and E. Platen, Strong predictor-corrector Euler methods for stochastic differential equations, Stochastics and Dynamics, 2008, 8: 561–581.
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is supported by NSFC under Grant Nos. 11171125 and 91130003, NSFH under Grant No. 2011CDB289, and the Freedom Explore Program of Central South University.
This paper was recommended for publication by Editor Ningning YAN.
Rights and permissions
About this article
Cite this article
Niu, Y., Zhang, C. Almost sure and moment exponential stability of predictor-corrector methods for stochastic differential equations. J Syst Sci Complex 25, 736–743 (2012). https://doi.org/10.1007/s11424-012-0183-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-012-0183-5