Given a Markov chain with stationary transition probabilities, we study random times $\tau$ determined by the evolution of the Markov chain for which either the pre-$\tau$ or post-$\tau$ process is Markovian with stationary transition probabilities. A complete description is given of all such random times which admit a conditional independence property analogous to the strong Markov property at a stopping time.
"Birth, Death and Conditioning of Markov Chains." Ann. Probab. 5 (3) 430 - 450, June, 1977. https://doi.org/10.1214/aop/1176995803