(Georgia Institute of Technology, 2004-11)
Figueroa-Lopez, Enrique; Houdré, Christian
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.