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Asymptotic properties of jump-diffusion processes with state-dependent switching. (English) Zbl 1191.60091
The author deals with a class of jump-diffusion processes with state-dependent switching. A jump-diffusion process with state-dependent switching is a combination of a diffusion process with both random jumping and state-dependent switching, it is often called a hybrid jump-diffusion process or a jump-diffusion process with regime switching. First, the existence and uniqueness of the solution of a system of stochastic integro-differential equations are obtained with the aid of successive construction methods. Next, the non-explosiveness is proved by truncation arguments. Then, the Feller continuity is established by means of introducing some auxiliary processes and by making use of the Radon-Nikodym derivatives. Furthermore, the strong Feller continuity is proved by virtue of the relation between the transition probabilities of jump-diffusion processes and the corresponding diffusion processes. Finally, on the basis of the above results, the exponential ergodicity is obtained under the Foster-Lyapunov drift conditions. Two examples are also provided for illustration.

60J60 Diffusion processes
60J27 Continuous-time Markov processes on discrete state spaces
34D23 Global stability of solutions to ordinary differential equations
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