Conditioning an Additive Functional of a Markov Chain to Stay Non-negative Ii: Hitting a High Level

Let (Xt)t≥0 be a continuous-time irreducible Markov chain on a finite statespace E, let v : E → R\{0} and let (φt)t≥0 be defined by φt = ∫ t 0 v(Xs)ds. We consider the cases where the process (φt)t≥0 is oscillating and where (φt)t≥0 has a negative drift. In each of the cases we condition the process (Xt, φt)t≥0 on the event that (φt)t≥0 hits level y before hitting zero and prove weak convergence of the conditioned process as y → ∞. In addition, we show the relation between conditioning the process (φt)t≥0 with a negative drift to oscillate and conditioning it to stay non-negative until large time, and relation between conditioning (φt)t≥0 with a negative drift to drift to drift to +∞ and conditioning it to hit large levels before hitting zero.