Convergence, boundedness, and ergodicity of regime-switching diffusion processes with infinite memory
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Convergence, boundedness, and ergodicity of regime-switching diffusion processes with infinite memory Jun LI1,2 , Fubao XI1,3 1 School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China 2 Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China 3 Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, China
c Higher Education Press 2020
Abstract We study a class of diffusion processes, which are determined by solutions X(t) to stochastic functional differential equation with infinite memory and random switching represented by Markov chain Λ(t). Under suitable conditions, we investigate convergence and boundedness of both the solutions X(t) and the functional solutions Xt . We show that two solutions (resp., functional solutions) from different initial data living in the same initial switching regime will be close with high probability as time variable tends to infinity, and that the solutions (resp., functional solutions) are uniformly bounded in the mean square sense. Moreover, we prove existence and uniqueness of the invariant probability measure of two-component MarkovFeller process (Xt , Λ(t)), and establish exponential bounds on the rate of convergence to the invariant probability measure under Wasserstein distance. Finally, we provide a concrete example to illustrate our main results. Keywords Regime-switching diffusion process, infinite memory, convergence, boundedness, Feller property, invariant measure, Wasserstein distance MSC2020 60J60, 60K37, 34K50, 34K34 1
Introduction
Emerging and existing applications in ecological systems, biological model, financial engineering, wired and wireless communication, and queueing networks need the mathematical modeling, analysis, and computation of hybrid systems in which continuous dynamics and discrete events coexist. In Received April 9, 2020; accepted June 15, 2020 Corresponding author: Fubao XI, E-mail: [email protected]
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Jun LI, Fubao XI
particular, taking random disturbance into consideration, the so-called regimeswitching diffusions are one of such hybrid models. A regime-switching diffusion process is a two-component process (X(t), Λ(t)), a continuous component X(t) evolving in line with one of many diffusions, with a choice of dynamics that changes at the jump times of the second component, and a discrete component Λ(t) taking values in a set consisting of isolated points. Due to a wide range of applications, such processes have drawn a great deal of attention in recent years; see [1,5,13,18,19,22,25–28,30] and references therein. For the comprehensive treatment of hybrid switching diffusions, we refer to monographs of Mao and Yuan [15] and Yin and Zhu [29]. More often than not, delays (also called memory) are ubiquitous and inevitable in the real world. To deal with more realistic situation, effort has also been devoted to the development of dynamic models that take the influence of past history into consideration; see, for instance, [10–12] for determinist
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