Conditioned stable Lévy processes and the Lamperti representation

Abstract

By killing a stable Lévy process when it leaves the positive half-line, or by conditioning it to stay positive, or by conditioning it to hit 0 continuously, we obtain three different positive self-similar Markov processes which illustrate the three classes described by Lamperti [10]. For each of these processes, we compute explicitly the infinitesimal generator from which we deduce the characteristics of the underlying Lévy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Lévy processes before their first passage time across level 0 which we describe here. As an application, we give the law of the minimum before an independent exponential time of a certain class of Lévy processes. It provides the explicit form of the spacial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of Lévy processes.

Stable Lévy processes, self-similarity and the unit ball

“…Stable processes killed on entering (−∞, 0). This first example was introduced in detail in Caballero and Chaumont (2006a); see also Kyprianou et al (2015). To some extent, the former of these two references marks the beginning of the modern treatment of stable processes through the theory of self-similar Markov processes.…”

Continuous-State Branching Processes and Self-Similarity

“…From the previous subsection, a stable Lévy process with no negative jumps conditioned to stay positive is tantamount to a Doob-h transform of the killed process where h(x) = x. According to Caballero and Chaumont [4], both the process X and its conditioned version belong to the class of PSSMPs; that is to say, positive Markov processes satisfying property (8).…”

“…E. KYPRIANOU AND J. C. PARDO Finally, we note that ξ and ξ * are two examples of so-called Lamperti stable processes (see, for instance, [2], [4], [6], [10], and [24] for related expositions and the formal definition of a Lamperti stable process).…”

Continuous-State Branching Processes and Self-Similarity

“…From the previous subsection, a stable Lévy process with no negative jumps conditioned to stay positive is tantamount to a Doob-h transform of the killed process where h(x) = x. According to Caballero and Chaumont [4], both the process X and its conditioned version belong to the class of positive self-similar Markov processes; that is to say positive Markov processes satisfying the property (3.9). From Lamperti's work [20] it is known that the family of positive self-similar Markov processes up to its first hitting time of 0 may be expressed as the exponential of a Lévy process, time changed by the inverse of its exponential functional.…”