Convergence and almost sure exponential stability of implicit numerical methods for a class of highly nonlinear neutral stochastic differential equations with constant delay

作者：

Highlights：

• This manuscript represents the continuation of the previous work of the author, related to a class of highly nonlinear neutral stochastic differential equations with time-dependent delay.

• Main results of this manuscript are convergence in probability and almost sure exponential stability of the backward Euler approximate solution for a class of stochastic differential equations with constant delay.

• Conditions under which both, the exact and approximate solutions share the property of the almost sure exponential stability are revealed.

• This paper also illustrates that, in some cases, stochastic differential equations with constant delay should be studied separately from those with time-dependent delay. In that sense, it should be stressed that results of this paper are obtained under weaker conditions comparing to those which would be obtained by replacing the time-dependent delay by the constant delay.

摘要

•This manuscript represents the continuation of the previous work of the author, related to a class of highly nonlinear neutral stochastic differential equations with time-dependent delay.•Main results of this manuscript are convergence in probability and almost sure exponential stability of the backward Euler approximate solution for a class of stochastic differential equations with constant delay.•Conditions under which both, the exact and approximate solutions share the property of the almost sure exponential stability are revealed.•This paper also illustrates that, in some cases, stochastic differential equations with constant delay should be studied separately from those with time-dependent delay. In that sense, it should be stressed that results of this paper are obtained under weaker conditions comparing to those which would be obtained by replacing the time-dependent delay by the constant delay.

论文关键词：60H10,Neutral stochastic differential delay equations,Nonlinear growth conditions,One-sided Lipschitz condition,Backward Euler method,Global a.s. asymptotic exponential stability

论文评审过程：Received 5 February 2014, Revised 1 December 2014, Available online 10 December 2014.

论文官网地址：https://doi.org/10.1016/j.cam.2014.12.002