Nonparametric estimation of Levy processes with a view towards mathematical finance

Abstract

Model selection methods and nonparametric estimation of Levy densities are presented. The estimation relies on the properties of Levy processes for small time spans, on the nature of the jumps of the process, and on methods of estimation for spatial Poisson processes. Given a linear space S of possible Levy densities, an asymptotically unbiased estimator for the orthogonal projection of the Levy density onto S is found. It is proved that the expected standard error of the proposed estimator realizes the smallest possible distance between the true Levy density and the linear space S as the frequency of the data increases and as the sampling time period gets longer. Also, we develop data-driven methods to select a model among a collection of models. The method is designed to approximately realize the best trade-off between the error of estimation within the model and the distance between the model and the unknown Levy density. As a result of this approach and of concentration inequalities for Poisson functionals, we obtain Oracles inequalities that guarantee us to reach the best expected error (using projection estimators) up to a constant. Numerical results are presented for the case of histogram estimators and variance Gamma processes. To calibrate parametric models,a nonparametric estimation method with least-squares errors is studied. Comparison with maximum likelihood estimation is provided. On a separate problem, we review the theoretical properties of temepered stable processes, a class of processes with potential great use in Mathematical Finance.