Conditioning an additive functional of a markov chain to stay nonnegative. II, Hitting a high level

Abstract

Let (X-t)(t >= 0) be a continuous-time irreducible Markov chain on a finite state space E, let v: E -> R \ {0}, and let (phi(t))(t >= 0) be defined by phi(t) = integral(0)(t) v(X-s) ds. We consider the case in which the process (phi(t))(t >= 0) is oscillating and that in which (phi(t))(t >= 0) has a negative drift. In each of these cases, we condition the process (X-t, phi(t))(t >= 0) on the event that (phi(t))(t >= 0) hits level y before hitting 0 and prove weak convergence of the conditioned process as y -> infinity. In addition, we show the relationship between the conditioning of the process (phi(t))(t >= 0) with a negative drift to oscillate and the conditioning of it to stay nonnegative for a long time, and the relationship between the conditioning of (phi(t))(t >= 0) with a negative drift to drift to infinity and the conditioning of it to hit large levels before hitting 0