Almost sure exponential stability of semi-Euler numerical scheme for nonlinear stochastic functional differential equation

Abstract

It is generally known that explicit solution can rarely be obtained for stochastic differential equations (SDEs), not to mention the nonlinear stochastic functional differential equation (SFDEs). Therefore, this paper proposes a numerical scheme called semi-Euler numerical scheme for general SFDEs, which adapts better to the connotation of the Itô differential equations. Under a generalized polynomial growth condition, the scheme admits high nonlinearity of the system, the almost sure exponential stability of the continuous model and the numerical scheme is investigated by contrast. It is confirmed that the numerical scheme preserves the stability property of the continuous model with no restriction to the step size. Compared to the literature, the polynomial growth condition in [L. Liu, F. Deng, and T. Hou, Almost sure exponential stability of implicit numerical solution for stochastic functional differential equation with extended polynomial growth condition, Applied Mathematics and Computation. 330 (2018), pp. 201–212.], [W. Fei, L. Hu, X. Mao, and M. Shen, Delay dependent stability of highly nonlinear hybrid stochastic systems, Automatica, 82 (2017), pp. 165–170.], [M. Shen, W. Fei, X. Mao, and Y. Liang, Stability of highly nonlinear neutral stochastic differential delay equations, Systems & Control Letters, 115 (2018), pp. 1–8.], etc. is generalized to that described with Lyapunov function. Besides, to provide an approach for the solvability of the implicit scheme, the generalized monotonicity of the vectorial functions is introduced. An application is given for the scheme and the stability conclusion, by virtue of the generalized monotonicity condition, the scheme is solvable also without restriction to the step size. At the end of the paper, a high order example is proposed to illustrate the theory of this paper, and a further research direction from this work is pointed out.