IDT processes and associated L\'evy processes

TL;DR

This paper explores IDT processes, their associated Lévy processes, and extends the concept to multiparameter cases, providing new examples, formulas, and links to selfdecomposability for better understanding of these stochastic processes.

Contribution

It introduces new examples and frameworks for IDT processes, including multiparameter extensions and links to selfdecomposability, expanding the theoretical understanding of these processes.

Findings

Established connections between IDT and Lévy processes.

Developed an integrated weak Itô formula for IDT processes.

Introduced multiparameter IDT processes and studied their properties.

Abstract

This article deals with IDT processes, i.e. processes which are infinitely divisible with respect to time. Given an IDT process $(X_{t},\,t\geq0)$, there exists a unique (in law) L\'evy process $(L_{t}; t\geq0)$ which has the same one-dimensional marginals distributions, i.e for any $t\geq0$ fixed, we have $$X_{t}\stackrel{(law)}{=}L_{t}.$$ Such processes are said to be associated. The main objective of this work is to exhibit numerous examples of IDT processes and their associated L\'evy processes. To this end, we take up ideas of the monograph \textit{Peacocks and associated martingales} from F. Hirsch, C. Profeta, B. Roynette and M. Yor (L\'evy, Sato and Gaussian sheet methods) and apply them in the framework of IDT processes. This gives a new interesting outlook to the study of processes whose only one-dimensional marginals are known. Also, we give an integrated weak It\^o type formula for IDT processes (in the same spirit as the one for Gaussian processes) and some links between IDT processes and selfdecomposability. 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 multiparameter sense. In particular, a new approach for multiparameter IDT processes is introduced and studied. Main examples of this kind of processes are the $\mathbb{R}_{+}^{N}$-parameter L\'{e}vy process and the L\'{e}vy's $\mathbb{R}^{M}$-parameter Brownian motion. These results give some better understanding of IDT processes, and may be seen as some continuation of the works of K. Es-Sebaiy and Y. Ouknine [\textit{How rich is the class of processes which are infinitely divisible with respect to time ?}] and R. Mansuy [\textit{On processes which are infinitely divisible with respect to time}].