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 andMancini (2007).
Highlights
The usual models of modern finance are based on the assumption of normality for asset returns
The laws of self-decomposable distributions class are be constitute as a limit laws of Levy – driven Ornstein-Uhlenbeck process
We focus on Levy process and it’s increments laws
Summary
The usual models of modern finance are based on the assumption of normality for asset returns. A remarkable number of empirical studies have shown that the assumption of normally distributed observations is a poor approximation for the real data This is because the returns have features such as jumps, semi-heavy tails and asymmetry. When price process model includes the jumps, the perfect hedging is imposable In this case, market participants can not hedge risks by using only underlying assets. The increments of additive process provide us more flexible models These processes were studied by (Madan, Carr and Chang(1998)), (Carr et al(2007)) being obtained from self-decomposable distributions. Aim of these paper is to review pure- jump Levy process arising from self-decomposable distributions in financial modeling and to test the presence of a Brownian motion component and discriminating between finite or infinite variation jumps.
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