Three key aims of the proposal have been achieved. The research outcomes have formed 7 research papers listed below. The main findings of these papers are described as follows:
In papers (1) and (2), we establish some useful criteria on the exponential stability for neutral-type stochastic differential delay equations. These criteria no longer require the coefficients of the underlying equations to obey the linear growth condition nor the time delay to be a constant. Moreover, the key condition on the diffusion operator associated with the underlying equations takes a much more general form. Our new stability criteria not only cover many highly non-linear neutral-type stochastic differential delay equations with variable time delays but they can also be verified much more easily than the known criteria.
In papers (3), (6) and (7) we develop the techniques used in papers (1) and (2) to deal with more general neutral stochastic functional differential equations with Markovian switching. Our new existence-and-uniqueness theorem enables us to consider a wider class of nonlinear equations. We establish a powerful LaSalle-type theorem on the limit sets of the solutions. We also establish several new criteria on almost surely asymptotic stability, in particular, exponential and polynomial stability.
In papers (4) and (5), we develop a numerical scheme to approximate the solutions of stochastic functional differential equations with Markovian switching. We show the strong convergence of the approximate solutions to the true solutions under the very week local Lipschitz condition. Moreover, based on the scheme developed in this paper we have also carried out certain amount of numerical simulation to examine whether numerical methods can help to reveal long-time behaviours of neutral-type equations. In particular, we have found that the numerical method can be used to study the exponential stability of linear neutral-type equations and we are currently writing a new paper on it.
(1) Luo, Q., Mao, X. and Shen, Y., New criteria on exponential stability of neutral stochastic differential delay equations, Systems and Control Letters 55 (2006), 826-834.
(2) Shen, Y. and Mao, X., Asymptotic behaviours of stochastic differential delay equations, Asian Journal of Control 8(1) (2006), 21-27.
(3) Mao, X., Shen, Y. and Yuan, C., Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching, Stochastic Processes and their Applications 118 (2008), 1385--1406.
(4) Li, X., Mao, X. and Shen, Y., Approximate solutions of stochastic differential delay equations with Markovian switching, Journal of Difference Equations and Applications 16(2-3) (2010), 195-207.
(5) Mao, X., Shen, Y. and Gray, A., Almost sure exponential stability of backward Euler-Maruyama discretizations for hybrid stochastic differential equations, Journal of Computational and Applied Mathematics 235 (2011), 1213-1226.
(6) Luo, Q., Mao, X. and Shen, Y., Generalised theory on asymptotic stability and boundedness of stochastic functional differential equations, Automatica 47 (2011), 2075--2081.
(7) Hu, L., Mao, X. and Shen, Y., Stability and boundedness of nonlinear hybrid stochastic differential delay equations, Systems \& Control Letters 62 (2013), 178-187.