Abstract

Lévy processes (LP) are gaining popularity in actuarial and financial modeling. The Lévy measure is a key factor in the versatility of LP applications. The estimation of the Lévy measure from data is shown to be useful in analyzing the aggregate claims processes in Risk Theory. Starting with infinitely divisible distributions (IDD), some nice constructions are obtained for finite sums of Lévy processes. The Lévy properties of compound Poisson processes are extensively used in the thesis. Examples illustrate the close relationship between IDDs and Lévy processes. The Poisson random measure associated with jumps of a Lévy process exceeding a given threshold is discussed and a new derivation is obtained. The relation with subordinators (increasing Lévy processes) is explored. Intuitive ideas and results are obtained for the jump function G appearing in Lévy's characterization of the Lévy-Khinchine formula. A non-parametric estimator of G is discussed. A detailed relation between G and p, the Lévy measure, is derived, yielding an estimator of p. The latter gives an estimator of the Poisson rate n f and the claim size distribution F f for claims larger than the threshold f. Extensive numerical simulations illustrate the paths of gamma, inverse Gaussian and {460}-stable claim subordinators and their corresponding estimates for n f and F f