A random time $R$ is called a regular birth time for a Markov Process if (i) the $R$-past and $R$-future are conditionally independent with respect to $X(R)$ and (ii) the post-$R$ process evolves as a Markov process, perhaps with different probability laws. In this paper we characterize each regular birth time in terms of an earlier, coterminal time $L$. It is shown (Theorem 4.2) that to the post-$L$ process $R$ appears as an optional time, perhaps with dependency on pre-$L$ information and on a certain invariant set.
"Regular Birth Times for Markov Processes." Ann. Probab. 9 (5) 769 - 780, October, 1981. https://doi.org/10.1214/aop/1176994307