Continuous-State Branching Processes and Self-Similarity

Abstract

In this paper we study the α-stable continuous-state branching processes (for α ∈ (1, 2]) and the latter process conditioned never to become extinct in the light of positive self-similarity. Understanding the interaction of the Lamperti transformation for continuous state branching processes and the Lamperti transformation for positive self-similar Markov processes gives access to a number of explicit results concerning the paths of α-stable continuousstate branching processes and α-stable continuous-state branching processes conditioned never to become extinct.

Implicit renewal theory for exponential functionals of Lévy processes

“…The exponential functionals of a Lévy processes have become a key object in stochastic modelling, for instance one founds these r.v. in the following areas: random algorithms [30], composition structures [27,26], generalized Ornstein-Uhlenbeck processes [43], finance [65], financial time series [36], population dynamics [23] and [45], random media [60], risk theory [22] and [25], self-similar fragmentation theory [5] and [9], self-similar Markov processes theory [12,16,37,40,19,49], self-similar branching processes [41] and [51], random networks [34], telecommunications [30], quantum mechanics [20] and many others, see e.g. [12].…”

Spine representations for non-compact models of random geometry

“…The preceding formula does not make sense for x = 0. However, [21,Theorem 2] shows that the laws P ↑ x converge weakly as x ↓ 0 to a limiting law denoted by P ↑ 0 , which is characterized by the following two properties: (i) for every t > 0, the law of Z t under P ↑ 0 is given by…”

Fluctuations of Stable Processes and Exponential Functionals of Hypergeometric Lévy Processes

“…dt, can be rather complicated. Nonetheless, this distribution is known explicitly for the case when X is either: a standard Poisson processes, Brownian motion with drift, a particular class of spectrally negative Lamperti stable (see for instance [18,28,34]), spectrally positive Lévy processes satisfying the Cramér condition (see for instance [36]). In the class of Lévy processes with double-sided jumps the distibution of the exponential functional is known in closed form only in the case of Lévy processes with hyperexponential jumps, see the recent paper by Cai and Kou [14].…”