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Almost sure exponential stability of the θ-method for stochastic differential equations

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  • Chen, Lin
  • Wu, Fuke

Abstract

Our previous work shows that the backward Euler–Maruyama (BEM) method may reproduce the almost sure stability of stochastic differential equations (SDEs) without the linear growth condition for the drift coefficient (see Wu et al. (2010)) but the Euler–Maruyama (EM) method cannot. It is well known that the θ-method is more general and may be specialized as the BEM and EM by choosing θ=1 and θ=0. Then it is very interesting to examine the interval in which the θ-method holds the same stability property as the BEM method. This paper shows that when θ∈(1/2,1], the θ-method may reproduce the almost sure stability of the exact solution of SDEs. Finally, a two-dimensional example is presented to illustrate this result.

Suggested Citation

  • Chen, Lin & Wu, Fuke, 2012. "Almost sure exponential stability of the θ-method for stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 82(9), pages 1669-1676.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:9:p:1669-1676
    DOI: 10.1016/j.spl.2012.05.004
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    Cited by:

    1. Lan, Guangqiang & Xia, Fang & Zhao, Mei, 2020. "pth moment (p∈(0,1)) and almost sure exponential stability of the exact solutions and modified truncated EM method for stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 160(C).
    2. Haoyi Mo & Xueyan Zhao & Feiqi Deng, 2017. "Exponential mean-square stability of the θ-method for neutral stochastic delay differential equations with jumps," International Journal of Systems Science, Taylor & Francis Journals, vol. 48(3), pages 462-470, February.

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