Lévy processes: properties and Simulation

Specialeforsvar ved Thomas Povey

Titel: Lévy processes: properties and Simulation

 Abstract: The understanding of a Lévy process lies in the study of its jump component.A Lévy process can in a unique way be decomposed into a continuous part and a compound Poisson-like process. The main theorem of this thesis is the Lévy-Ito decomposition of a Lévy process. To get an understanding of this, we investigated point process theory, because there is a connection between the general Poisson process and Lévy process. We showed a link between Poisson integrals and compound Poisson structure, called compound Poisson representation of Poisson integrals. Furthermore, we showed the important independent property of Poisson integrals. A connection between the class of infinitely divisible distributions and Lévy processes was explored. The Lévy-Khinchine formula was imposed. This result gives a unique representation of the characteristic function of an infinitely divisible distribution determined by its characteristic triplet which includes the Lévy measure, which we explored. The Lévy-Khinchine formula was a central result in our study of Lévy processes. Finally, we proved the Lévy-Itô representation for a pure Lévy jump process and the main result concerned a general Lévy process. Different simulation aspects was studied. Simulation of Lévy processes was performed. Solutions to stochastic differential equations driven by Lévy processes was carried out. The Euler scheme was applied. Simulations result was visualized



Vejleder:  Thomas Mikosch
Censor:    Søren Asmussen, Aarhus Universitet