Expression of analytical solution and almost sure exponential stability for linear regime-switching jump diffusion systems

Abstract

In this paper, we consider linear regime-switching jump diffusion systems. First, for the homogeneous and nonhomogeneous linear regime-switching jump diffusion systems, we respectively present the expressions of the analytical solutions, which depend on the fundamental matrix of the homogeneous systems. Then, for several classes of the linear homogeneous regime-switching jump diffusion systems, the expression of solutions are given explicitly by showing the expression of the fundamental matrices. For multi-dimensional linear homogeneous regime-switching jump diffusion systems, we obtain that system is exponentially stable almost surely if and only if the sample Lyapunov exponent of the fundamental matrix is less than zero. Furthermore, through the expressions of the explicit solutions, the sufficient conditions dependent directly on coefficients are developed for the almost sure exponential stability of multi-dimensional linear homogeneous regime-switching jump diffusion systems. Especially, for a class of multi-dimensional linear homogeneous regime-switching jump diffusion systems, the sufficient and necessary condition of almost sure exponential stability is obtained. Finally, two examples, whose almost sure exponential stability cannot be analyzed by the existing related work, are given and their numerical simulations are presented to illustrate the obtained results about almost sure exponential stability.