Nonparametric estimation for Levy processes with a view towards mathematical finance

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