Parameter estimation of Lévy processes

Abstract

In chapter 1 we provide an overall introduction related to the subject of statistical inference of Lévy processes. In particular, we briefly discuss some of the main problems which plague this area, we provide an overview of this thesis and briefly discuss how the methods of estimation developed throughout this Ph.D. research project deal with some of the above-mentioned problems. Thus providing estimators that are comparable, and in certain cases better that other estimators found in literature. In chapter 2 we provide a literature review related to the statistical inference of Lévy processes for the parametric and nonparametric categories. In chapter 3 a number of parametric techniques are discussed. This chapter is primarily divided in two main sections: Estimation through Characteristic Functions and Estimation through the Stochastic Process Setup. As the name implies, in the former we propose a number of estimation techniques which make use of the characteristic function and its empirical counterpart. The first method of estimation was inspired by the properties of the inverse Fourier transform while the second and third techniques use the stochastic programming framework to reduce the problems caused by oscillatory nature of the empirical characteristic function. The properties of the estimators obtained are discussed and their performance is compared to other estimators present in literature. In the second part of chapter 3 we exploit some properties that are present in the gamma process to obtain improved estimates for the shape parameter associated to the distribution of the increments of this process. This estimator guarantees better statistical performance when compared to the maximum likelihood estimator. In chapter 4 we primarily concentrate on the nonparametric estimation of the distribution associated with the measure present in the L´evy Khintchine canonical representation. In particular, we provide two improved versions of the Rubin and Tucker estimator. The first improvement is only valid for Brownian motion, however, the second improvement is valid for other, more general, L´evy processes. The properties of these estimators are considered, and a number of simulations are also conducted with the aim of comparing the Rubin and Tucker estimator with the proposed improved versions. Chapter 5 contains a number of concluding remarks and suggestions for future research.