Applied Modeling Techniques and Data Analysis 2

11Quantization of Transformed Lévy Measures

In this chapter, we find an optimal approximation of the measure associated with a transformed version of the Lévy-Khintchine canonical representation via a convex combination of a finite number P of Dirac masses. The quality of such an approximation is measured in terms of the Monge-Kantorovich, known also as the Wasserstein metric. In essence, this procedure is equivalent to the quantization of measures. This method requires prior knowledge of the functional form of the measure. However, since this is in general not known, then we shall have to estimate it. It will be shown that the objective function used to estimate the position of the Dirac masses and their associated weights (or masses) can be expressed as a stochastic program. The properties of the estimator provided are discussed. Also, a number of simulations for different types of Lévy processes are performed and the results are discussed.

11.1. Introduction

Recently, there has been a sharp rise of interest in the study of Lévy processes. This is because their applications are far-reaching. These processes have been applied in various fields of research which include telecommunications, quantum theory, extreme value theory, insurance and finance. Lévy processes can be defined as stochastic processes that are stochastically continuous, with increments that are independent and stationary. Moreover, it is possible to find a version of such a process that is almost surely right ...