Some characterizations for Markov processes at first passage

Abstract

Suppose $X$ is a Markov process on the real line (or some interval). Do the distributions of its first passage times downwards (fptd) determine its law? In this paper we treat some special cases of this question. We prove that if the fptd process has the law of a subordinator, then necessarily $X$ is a L\'evy process with no negative jumps; specifying the law of the subordinator determines the law of $X$ uniquely. We further show that, likewise, the classes of continuous-state branching processes and of self-similar processes without negative jumps are also respectively characterised by a certain structure of their fptd distributions; and each member of these classes separately is determined uniquely by the precise family of its fptd laws. The road to these results is paved by (i) the identification of Markov processes without negative jumps in terms of the nature of their fptd laws, and (ii) some general results concerning the identification of the fptd distributions for such processes.