This work focuses on regime-switching jump diffusions, which include three classes of random processes, Brownian motions, Poisson processes, and Markov chains. First, a scalar linear system is treated as a benchmark model. Then stabilization of systems with one-sided linear growth is considered. Next, nonlinear systems that have a finite explosion time are treated, in which regularization (expl...

We investigate the almost surely asymptotic stability of gene regulatory networks (GRNs) with Markovian switching. Previous research has described GRNs as coupled nonlinear stochastic systems under parametric perturbations without considering the important aspect of different time-delays in the subsystems. However, a realistic model of a GRN is that of a hybrid stochastic retarded system that r...

We give necessary and sufficient conditions for the almost sure (a.s.) relative stability of the overshoot of a random walk when it exits from a two-sided symmetric region with curved boundaries. The boundaries are of power-law type, ±rn, r > 0, n = 1, 2, · · · , where 0 ≤ b < 1, b 6= 1/2. In these cases, the a.s. stability occurs if and only if the mean step length of the random walk is finite...

We are interested in the strong convergence and almost sure stability of Euler-Maruyama (EM) type approximations to the solutions of stochastic differential equations (SDEs) with non-linear and nonLipschitzian coefficients. Motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence. In addition we examin...

This paper is concerned with the stability and numerical analysis of solution to highly nonlinear stochastic differential equations with jumps. By the Itô formula, stochastic inequality and semi-martingale convergence theorem, we study the asymptotic stability in the pth moment and almost sure exponential stability of solutions under the local Lipschitz condition and nonlinear growth condition....

We consider population-size-dependent branching processes (PSDBPs) which eventually become extinct with probability one. For these processes, we derive maximum likelihood estimators for the mean number of offspring born to individuals when current population size is z≥1. As standard in process theory, an asymptotic analysis requires us condition on non-extinction up a finite generation n and le...

A neutral stochastic diierential diierence equation

This paper is concerned with stability of a class of randomly switched systems of ordinary differential equations. The system under consideration can be viewed as a two-component process (X(t), α(t)), where the system is linear in X(t) and α(t) is a continuous-time Markov chain with a finite state space. Conditions for almost surely exponential stability and instability are obtained. The condit...