Pure -Jump Levy Processes and Self-decomposability in Financial Modeling

Abstract

In this study, we review the connections between Lévy processes with jumps and self-decomposable laws. Self-decomposable laws constitute a subclass of infinitely divisible laws. Lévy processes additive processes and independent increments can be related using self-similarity property. Sato (1991) defined additive processes as a generalization of Lévy processes. In this way, additive processes are those processes with inhomogeneous (in general) and independent increments and Lévy processes correspond with the particular case in which the increments are time homogeneous. Hence Lévy processes are considerable as a particular type. Self-decomposable distributions occur as limit law an Ornstein-Uhlenbeck type process associated with a background driving Lévy process. Finally as an application, asset returns are representing by a normal inverse Gaussian process. Then to test applicability of this representation, we use the nonparametric threshold estimator of the quadratic variation, proposed by Cont and Mancini (2007).