Search

Focusing on asymptotic behavior of a stochastic SIR epidemic model represented by a system of stochastic differential equations with a degenerate diffusion, this paper provides suficient conditions that are very close to the necessary ones for the permanence. In addition, this paper develops ergodicity of the underlying system. It is proved that the transition probabilities converge in total variation norm to the invariant measure. Our result gives a precise characterization of the support of the invariant measure. Rates of convergence are also ascertained. It is shown that the rate is not too far from exponential in that the convergence speed is of the form of a polynomial of any degree. [ABSTRACT FROM AUTHOR]

This work is concerned with rates of convergence of Markov chain approximation methods for controlled switching diffusions. The cost function is defined on an infinite horizon with stopping times and without discount. Displaying both continuous dynamics and discrete events, the discrete events are modeled by continuous-time Markov chains to delineate a random environment and other random factors that cannot be represented by diffusion processes. This paper presents a first attempt using a probabilistic approach to treat such rates of convergence problems. In addition, in contrast to the significant developments in the literature using partial differential equation (PDE) methods for the approximation of controlled diffusions, there do not yet appear to be any PDE results to date for rates of convergence of numerical solutions for controlled switching diffusions, to the best of our knowledge. Although some of the working conditions in this paper such as the one-dimensional continuous state variable, nondegenerate diffusions, and control only on the drift may be seemingly strong, they are adequate as the starting point for using this new approach to treat the rates of convergence problems. Moreover, in the literature, to prove the convergence using Markov chain approximation methods for control problems involving cost functions with stopping (even for uncontrolled diffusion without switching), an added assumption was used to avoid the socalled tangency problem. As a by-product of our approach, by modifying the value function, it is demonstrated that the anticipated tangency problem will not arise in the sense of convergence in probability and in the sense of L1. [ABSTRACT FROM AUTHOR]

Aiming at reduction of complexity, this work is concerned with two-time-scale Markov chains and applications to quasi-birth-death queues. Asymptotic expansions of probability vectors are constructed and justified. Lumping all states of the Markov chain in each subspace into a single state, an aggregated process is shown to converge to a continuous-time Markov chain whose generator is an average with respect to the stationary measures. Then a suitably scaled sequence is shown to converge to a switching diffusion process. Extensions of the results are presented together with examples of quasi-birth-death queues. [ABSTRACT FROM AUTHOR]

This work is concerned with diffusion processes having fast and slow components. It was known that under suitable assumptions the slow component can be approximated by the Markov process with averaged characteristics. In this work, asymptotic expansions for the solutions of the Kolmogorov backward equations are constructed and justified. Certain probabilistic conclusions and examples are also provided. [ABSTRACT FROM AUTHOR]