Nonparametric estimation for Levy processes with a view towards mathematical finance

Abstract

Nonparametric methods for the estimation of the Levy density of a Levy process X are developed. Estimators that can be written in terms of the "jumps" of X are introduced, and so are discrete-data based approximations. A model selection approach made up of two steps is investigated. The first step consists in the selection of a good estimator from a linear model of proposed Levy densities, while the second is a data-driven selection of a linear model among a given collection of linear models. By providing lower bounds for the minimax risk of estimation over Besov Levy densities, our estimators are shown to achieve the "best" rate of convergence. A numerical study for the case of histogram estimators and for variance Gamma processes, models of key importance in risky asset price modeling driven by Levy processes, is presented.