On Diffusion Approximation of Discrete Markov Dynamical Systems

Abstract

The paper is devoted to stochastic analysis of finite dimensional difference equation with dependent on ergodic Markov chain increments, which are proportional to small parameter e. A point-form solution of this difference equation may be represented as vertexes of a time-dependent continuous broken line given on the segment [0,1] with e-dependent scaling of intervals between vertexes. Tending e to zero one may apply stochastic averaging and diffusion approximation procedures and construct continuous approximation of the initial stochastic iterations as an ordinary or stochastic Ito differential equation. The paper proves that for sufficiently small e these equations may be successfully applied not only to approximate finite number of iterations but also for asymptotic analysis of iterations, when number of iterations tends to infinity. Keywords—Markov dynamical system, diffusion approximation, equilibrium stochastic stability.