Path decompositions for real Levy processes
Path-wise solutions of stochastic differential equations driven by Lévy processes.
In this paper we show that a path-wise solution to the following integral equationYt = ∫0t f(Yt) dXt, Y0 = a ∈ Rd,exists under the assumption that Xt is a Lévy process of finite p-variation for some p ≥ 1 and that f is an α-Lipschitz function for some α > p. We examine two types of solution, determined by the solution's behaviour at jump times of the process X, one we call geometric, the other forward. The geometric solution is obtained by adding fictitious time and solving an associated...
Paul Lévy et l’arithmétique des lois de probabilités
Ce court texte reprend un exposé donné le 15 Décembre 2011 au Laboratoire de Probabilités et Modèles Aléatoires, lors d’une journée en hommage à Paul Lévy. On y rappellera comment des considérations sur l’arithmétique des lois de probabilités ont conduit Lévy à étudier les processus à accroissements indépendants.
Proof(s) of the Lamperti representation of continuous-state branching processes.