Two Operational Characterizations of Cooptional Times

Abstract

Consider a random time $\tau$ determined by the evolution of a Markov chain $X$ in discrete time and with discrete state space. Assuming that the pre-$\tau$ and post-$\tau$ processes are conditionally independent given $X_{\tau-1}$ and $0 < \tau < \infty$, it is shown that: (i) the pre-$\tau$ process reversed is Markov and in natural duality to $X$ if and only if $\tau$ is almost surely equal to a modified cooptional time; (ii) the pre-$\tau$ process itself is Markov and an $h$-transform of $X$ if and only if $\tau$ is almost surely equal to a cooptional time with, in general, the possible starts for the pre-$\tau$ process restricted. Also, a result is presented characterizing those $\tau$ for which the reversed pre-$\tau$ process is Markov in natural duality to $X$, without the assumption of conditional independence.