Almost sure exponential stability of implicit numerical solution for stochastic functional differential equation with extended polynomial growth condition

Stability of numerical solutions to stochastic differential equations (SDEs) has received an increasing attention from researchers in applied mathematics and engineering areas, but there has been so far little work on the stability analysis of numerical solutions to highly nonlinear stochastic functional differential equations (SFDEs). The aim of this paper is to study the almost sure exponential stability of the backward Euler scheme for highly nonlinear SFDEs. Firstly, a stability criterion is established for the almost sure exponential stability of the underlying nonlinear SFDE under an extended polynomial growth condition, the global existence of the solutions of the underlying equation is simultaneously shown with a new technique, the Fatou's Lemma. Secondly, the almost sure exponential stability of the backward Euler scheme is investigated by contrast. The stability criterion shows that the numerical scheme preserves the almost sure exponential stability of the analytic solution under the same growth condition, which improves some related result in literature as a corollary by way. To describe the scheme conveniently, a concept, the interpolation segment, is formally proposed in the paper, which provides a way for constructing a discretized approximation to a continuous function applied in the scheme. To achieve the required results, the classical nonnegative semi-martingale convergence theorem is reformed to a practical version, namely the so called practical nonnegative semi-martingale boundedness lemma in the paper. At the end of the paper, a numerical example is proposed to illustrate the adaption of the growth condition assumed in the paper.