13. Positive Self-similar Markov Processes

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 Lévy 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 Lévy 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 Lévy 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.

Whilst our exposition of general positive self-similar Markov processes will, in the beginning, insist that their initial value lies in (0,∞), we will also look at the more complicated case that the point of issue is the origin. This discussion leads us to the concept of recurrent extensions of positive self-similar Markov processes. With this theory in hand, we will conclude the chapter by looking at elements of fluctuation theory for positive self-similar Markov processes associated with spectrally negative Lévy processes.