Numerical solutions of stochastic differential equations with piecewise continuous arguments under Khasminskii-type conditions. (English) Zbl 1251.65004

Summary: We investigate the convergence of the Euler method to stochastic differential equations with piecewise continuous arguments (SEPCAs). The classical Khasminskii-type theorem gives a powerful tool to examine the global existence of solutions for stochastic differential equations (SDEs) without the linear growth condition by the use of the Lyapunov functions. However, there is no such result for SEPCAs. Firstly, this paper shows SEPCAs which have nonexplosion global solutions under local Lipschitz condition without the linear growth condition. Then the convergence in probability of numerical solutions to SEPCAs under the same conditions is established. Finally, an example is provided to illustrate our theory.


65C30 Numerical solutions to stochastic differential and integral equations
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Full Text: DOI


[1] A. Friedman, Differential Equations and Applications. Volume 1 and 2, Academic Press, New York, NY, USA, 1975. · Zbl 0323.60056
[2] R. Z. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Groningen, The Netherlands, 1980, Russian Translation Nauka, Moscow, Russia, 1969. · Zbl 1259.60058
[3] X. Mao, Stability of Stochastic Differential Equations with Respect to Semimartingales, Long-man Scientific and Technical, London, UK, 1994.
[4] X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, New York, NY, USA, 1994.
[5] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Limited, Chichester, UK, 1997. · Zbl 0892.60057
[6] X. Mao, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, UK, 2006. · Zbl 1126.60002
[7] X. Mao, “A note on the LaSalle-type theorems for stochastic differential delay equations,” Journal of Mathematical Analysis and Applications, vol. 268, no. 1, pp. 125-142, 2002. · Zbl 0996.60064
[8] K. L. Cooke and J. Wiener, “Retarded differential equations with piecewise constant delays,” Journal of Mathematical Analysis and Applications, vol. 99, no. 1, pp. 265-297, 1984. · Zbl 0557.34059
[9] J. Wiener, Generalized Solutions of Functional Differential Equations, World scientific, Singapore, 1993. · Zbl 0874.34054
[10] X. Mao, “Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5512-5524, 2011. · Zbl 1215.65015
[11] H. Y. Dai and M. Z. Liu, “Mean square stability of stochastic differential equations with piecewise continuous arguments,” Journal of Natural Science of Heilongjiang University, vol. 25, no. 5, pp. 625-629, 2008. · Zbl 1199.60205
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.