Infinite Divisibility for Stochastic Processes and Time Change

Abstract

General results concerning infinite divisibility, selfdecomposability, and the class L m property as properties of stochastic processes are presented. A new concept called temporal selfdecomposability of stochastic processes is introduced. Lévy processes, additive processes, selfsimilar processes, and stationary processes of Ornstein-Uhlenbeck type are studied in relation to these concepts. In the latter half of the paper time change of stochastic processes is studied, where chronometers (stochastic processes that serve to change time) and base processes (processes to be time-changed) are independent but do not, in general, have independent increments. Conditions for inheritance of infinite divisibility and selfdecomposability under time change are given.

How rich is the class of processes which are infinitely divisible with respect to time?

“…As in the one-parameter case [3], we characterize the multiparameter IDT Gaussian processes. Moreover, we define multiparameter temporal selfdecomposable processes similar to those introduced by Barndorff-Nielsen, Meajima and Sato [1] and we prove that multiparameter IDT processes are temporal selfdecomposable.…”

“…We will define the multiparameter temporally selfdecomposable processes which extend the one introduced in [1] and relate this notion with the IDT processes.…”

“…where (X (1) , t ≥ 0), ..., (X (n) , t ≥ 0) are independent copies of X and d = denotes equality in all finite-dimensional distributions. The notion of IDT processes has been introduced by Mansuy [3] as a generalization of Lévy processes.…”

“…Various properties of IDT processes have been already investigated in [3], related for instance to their temporal self-decomposability and the characterization of IDT Gaussian processes. Regarded as a contribution to this expending topic, it is the purpose of this paper to extend some results on Lévy processes studied in [1], [2] and [4] to the case of IDT processes. In particular, we shall prove that, IDT processes are more tractable than Lévy processes, since they could be obtained by combining the selfsimilarity and strict stability.…”

“…Time-changed Lévy processes where the chronometers are more general than subordinators arise now in many fields of application, see for instance [1] and the reference therein. We shall prove (Theorem 3.6) the inheritance of IDT under time change when base processes are Lévy processes.…”