Positive Self-similar Markov Processes

Abstract

In this chapter, our objective is to explore in detail the general class of so-called positive self-similar Markov processes. Emphasis will be placed on the bijection between this class and the class of Levy processes which are killed at an independent and exponentially distributed time. This bijection can be expressed through a straightforward space-time transformation and, thereby it, we are able to explore a number of specific examples of positive self-similar Markov processes, which illuminate a variety of explicit and semi-explicit fluctuation identities for Levy processes. Our first such family of examples will be positive self-similar Markov processes that are obtained when considering path transformations of stable processes and conditioned stable processes. Here, the underlying associated Levy processes are known as Lamperti-stable processes. Known properties of stable processes, when transferred through the aforementioned space-time transform, will give us explicit fluctuation identities for Lamperti-stable processes; in particular, we will obtain their Wiener–Hopf factorisation. Another family of examples we will consider is continuous-state branching processes and continuous-state branching processes with immigration, which are also self-similar.