Abstract

In this study, we review the connections between L\'{e}vy processes with jumps and self-decomposable laws. Self-decomposable laws constitute a subclass of infinitely divisible laws. L\'{e}vy processes additive processes and independent increments can be related using self-similarity property. Sato (1991) defined additive processes as a generalization of L\'{e}vy processes. In this way, additive processes are those processes with inhomogeneous (in general) and independent increments and L\'{e}vy processes correspond with the particular case in which the increments are time homogeneous. Hence L\'{e}vy processes are considerable as a particular type. Self-decomposable distributions occur as limit law an Ornstein-Uhlenbeck type process associated with a background driving L\'{e}vy 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).

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