Stability With General Decay Rates Of Regime-switching Jump-diffusion Systems

Regime-switching jump-diffusion systems(RSJDSs)are two-element Markov processes,which are suitable for describing phenomena such as random factor disturbance,regime-switching,and system jumping.RSJDSs have a wide range of applications in many fields such as financial mathematics,signal transmission,biological communities,and automation control.During the past decades,the stability problem of RSJDSs has received a lot of attention.This thesis is concentrated on the stability problem with general decay rates of RSJDSs and RSJDSs with time-varying time delays,where the Lévy measure is allowed to be infinite.The details are as follows:1.In this study,we examine the p-th moment and almost sure stability with general decay rates of RSJDSs,and investigate their connections.Firstly,by restricting the Lyapunov function,the p-th moment and almost sure stability with general decay rates are proved by using the Gronwall inequality and the exponential martingale inequality,respectively.Then,under the condition that the infinitesimal operator of the Lyapunov function is negative definite,we prove the p-th moment stability with general decay rates.It is shown that the p-th moment stability with general decay rates leads to almost sure stability with general decay rates.In case p(?)(0,2]and case p≥2,this result is proved by using the martingale convergence theorem,the Burkholder-Davis-Gundy inequality,and the Kunita first inequality,respectively.On the other hand,if the infinitesimal operator of the Lyapunov function is integrable on_+,the p-th moment stability with general decay rates is proved.And we can also prove the almost sure stability with general decay rates,by using the non-negative semi-martingale convergence theorem.Finally,two variance-gamma examples are presented to verify our work.2.The stability problem of RSJDSs with time-varying delays is studied.Firstly,we prove the existence and uniqueness of the global solution of the system by the Skorokhod representation of the switching process under the global Lipschitz condition and the linear growth condition,and we also prove this result by the Khasminskii method under the local conditions.Then,the p-th moment stability and almost sure stability with general decay rates are investigated by using the Gronwall inequality,the exponential martingale inequality,and the theory of stochastic analysis.Finally,two variance-gamma examples are presented to verify our work.