Quantization of transformed Lévy measures

Abstract

In this paper we find an optimal approximation of the measure associated to a transformed version of the Levy-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 Levy processes are performed and the results are discussed.