Applications of Raw Time-Changes to Markov Processes

Abstract

The technique of raw time-change is applied to give another proof that the Knight-Pittenger procedure of deleting excursions of a strong Markov process from a set $A$ which meet a disjoint set $B$ yields a strong Markov process. A natural filtration is associated with the new process, and generalizations are given. Under natural hypotheses, the debuts of a class of nonadapted homogeneous sets are shown to be killing times of a strong Markov process. These are generalized (i.e. raw) terminal times. Let $A_t$ be an increasing nonadapted continuous process, and let $T_t$ be its right continuous inverse satisfying a hypothesis which ensures that the collection of $\sigma$-fields $\mathscr{F}_{T(t)}$ is increasing. The optional times of $\mathscr{F}_{T(t)}$ are characterized in terms of killing operators and the points of increase of $A$, and it is shown that $\mathscr{F}_{T(t)} = \mathscr{F}_{T(t+)}$.