Coterminal families and the strong Markov property

Abstract:

Let ${E_\Delta }$ be a compact metric space and assume that a strong Markov process X is defined on ${E_\Delta }$. Under the assumption that X has right continuous paths with left limits, it is shown that a version of the strong Markov property extends to coterminal families, a class of random times which can be visualized as last exit times before t from a fixed subset of ${E_\Delta }$. Since the random times are not Markov times, the conditioning $\sigma$-field and the new conditional probabilities must be defined. If X is also assumed to be nearly quasileft continuous, i.e. branching points are permitted, two different conditionings are possible—one on the “past” of the random time and one on the “past plus present"—and two different conditional probabilities must be defined.