IDT processes and associated Lévy processes with explicit constructions†

Abstract

This article deals with IDT processes, i.e. processes which are infinitely divisible with respect to time. Given an IDT process (X t , t ≥ 0), there exists a unique (in law) Lévy process (L t ;t ≥ 0) which has the same one-dimensional marginal distributions, i.e for any t ≥ 0 fixed, we have . Such processes are said to be associated. The main objective of this work is to exhibit numerous examples of IDT processes and their associated Lévy processes. To this end, we take up ideas from Hirsch, Profeta, Roynette and Yor's monograph Peacocks and associated martingales with explicit construction (Lévy, Sato and Gaussian sheet methods) and apply them in the framework of IDT processes. This gives a new and interesting outlook on the study of processes with specified one-dimensional (1D) marginals. Also, we give an integrated weak Itô type formula for IDT processes (in the same spirit as the one for Gaussian processes) and some links between IDT processes and self-decomposability. The last sections are devoted to the study of some extensions of the notion of IDT processes in the weak sense as well as in the multi-parameter sense. In particular, a new approach for multi-parameter IDT processes is introduced and studied. The main examples of this kind of processes are the -parameter Lévy process and Lévy's R M -parameter Brownian motion. These results give some better understanding of IDT processes, and may be seen as a continuation of the works of Es-Sebaiy and Ouknine [How rich is the class of processes which are infinitely divisible with respect to time?] and Mansuy [On processes which are infinitely divisible with respect to time].