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Positive recurrence for a class of jump diffusions

In this paper, the author study positive recurrence for a wide class of jump diffusions by using Khasminskii’s approach developed in. We show that the property of positive recurrence is independent of the choice of the bounded domain in the state space. Moreover, the author establish a sufficient condition for positive recurrence using Liapunov functions. | POSITIVE RECURRENCE FOR A CLASS OF JUMP DIFFUSIONS TRAN QUAN KY Department of Mathematics, University of Education, Hue University Email address: quankysp@gmail.com Abstract: This work is concerned with asymptotic properties of a class of diffusion processes with jumps. In particular, we show that the property of positive recurrence is independent of the choice of the bounded domain in the state space. A sufficient condition for positive recurrence using Liapunov functions is derived. Keywords: jump diffusion, Liapunov function, positive recurrence 1 INTRODUCTION This work focuses on positive recurrence for a class of jump diffusion processes. Our motivation stems from the study of a family of Markov processes in which both continuous dynamics and jump discontinuity coexist. Such systems have drawn new as well as resurgent attention because of the urgent needs of systems modeling, analysis, and optimization in a wide variety of applications. Asymptotic properties of diffusion processes and associated partial differential equations are well known in the literature. We refer to [2, 4] and references therein. Results for switching diffusion processes can be found in [7]. One of the important problems in stochastic systems is their longtime behavior. In the literature, criteria for certain types of weak stability including Harris recurrence and positive Harris recurrence for continuous time Markovian processes based on Foster-Liapunov inequalities were developed in [5]. Using results in that paper, some sufficient conditions for ergodicity of Lévy type operators in dimension one are presented in [6] under the assumption of Lebesgue-irreducibility. In a recent work [1], the authors discuss positive recurrence for jump processes with no diffusion part. Compared to the case of diffusion processes, even though the classical approaches such as Liapunov function methods Journal of Science and Education, College of Education, Hue University ISSN 1859-1612, No. 01(45)/2018: