An Asymptotic Exit Problem for Multidimensional Markov Chains

A.A. Borovkov

1997, v.3, Issue 4, 547-564

ABSTRACT

We consider Markov chains partially homogeneous in space (see the definitions in Section 1 or in [A.A. Borovkov, Ergodicity and Stability of Stochastic Processes, J.Wiley], defined in the positive $d$-dimensional orthant $(R^+)^d$ with initial position $x$, $|x|\to\infty$. The limit distribution (as $|x|\to\infty$) of the first passage time in the $N$-neighborhood of the boundary of $(R^+)^d$ and the limit distribution of the corresponding position are found in the case when the passage occurs with probability 1. Alternatively, asymptotics of the probability $q(x)$ that the trajectory of the Markov chain never reaches the boundary of $(R^+)^d$ is considered if $q(x)\to 0$ as $\min x_i \to \infty$.

Keywords: multidimensional Markov chain,first passage problem,exit problem,multidimensional random walk,hitting time,hitting position,limit distribution,boundaryvalue problems