Choice of theta and its effects on stability in the stochastic theta-method of stochastic delay differential equations

Lin Chen, Fuke Wu

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4 Citations (Scopus)

Abstract

The second author s work [F. Wu, X. Mao, and L. Szpruch, Almost sure exponential stability of numerical solutions for stochastic delay differential equations, Numer. Math. 115 (2010), pp. 681-697] and Mao s papers [D.J. Higham, X. Mao, and C. Yuan, Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations, SIAM J. Numer. Anal. 45 (2007), pp. 592-607; X. Mao, Y. Shen, and G. Alison, Almost sure exponential stability of backward Euler-Maruyama discretizations for hybrid stochastic differential equations, J. Comput. Appl. Math. 235 (2011), pp. 1213-1226] showed that the backward Euler-Maruyama (BEM) method may reproduce the almost sure stability of stochastic differential equations (SDEs) without the linear growth condition of the drift coefficient and the counterexample shows that the Euler-Maruyama (EM) method cannot. Since the stochastic theta-method is more general than the BEM and EM methods, it is very interesting to examine the interval in which the stochastic theta method can capture the stability of exact solutions of SDEs. Without the linear growth condition of the drift term, this paper concludes that the stochastic theta-method can capture the stability for theta is an element of (1/2, 1]. For theta is an element of [0, 1/2), a counterexample shows that the stochastic theta-method cannot reproduce the stability of the exact solution. Finally, two examples are given to illustrate our conclusions
Original languageEnglish
Pages (from-to)2106 - 2122
Number of pages17
JournalInternational Journal of Computer Mathematics
Volume89
Issue number15
DOIs
Publication statusPublished - 2012
Externally publishedYes

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