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

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 θ-method is more general than the BEM and EM methods, it is very interesting to examine the interval in which the stochastic θ-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 θ-method can capture the stability for θ∈(1/2, 1]. For θ∈[0, 1/2), a counterexample shows that the stochastic θ-method cannot reproduce the stability of the exact solution. Finally, two examples are given to illustrate our conclusions.