Lévy Matters I: Recent Progress in Theory and Applications: Foundations, Trees and Numerical Issues in Finance

ISBN-13: 9783642140068 / Angielski / Miękka / 2010 / 206 str.

Thomas Duquesne; Oleg Reichmann; Ken-Iti Sato

Opis

Over the past 10-15 years, we have seen a revival of general Levy processes theory as well as a burst of new applications. In the past, Brownian motion or the Poisson process have been considered as appropriate models for most applications. Nowadays, the need for more realistic modelling of irregular behaviour of phen- ena in nature and society like jumps, bursts, and extremeshas led to a renaissance of the theory of general Levy processes. Theoretical and applied researchers in elds asdiverseas quantumtheory, statistical physics, meteorology, seismology, statistics, insurance, nance, and telecommunication have realised the enormous exibility of Lev y models in modelling jumps, tails, dependence and sample path behaviour. L evy processes or Levy driven processes feature slow or rapid structural breaks, extremal behaviour, clustering, and clumping of points. Toolsandtechniquesfromrelatedbut disctinct mathematical elds, such as point processes, stochastic integration, probability theory in abstract spaces, and differ- tial geometry, have contributed to a better understanding of Le vy jump processes. As in many other elds, the enormous power of modern computers has also changed the view of Levy processes. Simulation methods for paths of Levy p- cesses and realisations of their functionals have been developed. Monte Carlo simulation makes it possible to determine the distribution of functionals of sample paths of Levy processes to a high level of accuracy.

Spis treści

Fractional Integrals and Extensions of Selfdecomposability.- Packing and Hausdorff Measures of Stable Trees.- Numerical Analysis of Additive, Lévy and Feller Processes with Applications to Option Pricing.

Blurb

This is the first volume of a subseries of the Lecture Notes in Mathematics which will appear randomly over the next years. Each volume will describe some important topic in the theory or applications of Lévy processes and pay tribute to the state of the art of this rapidly evolving subject with special emphasis on the non-Brownian world. The three expository articles of this first volume have been chosen to reflect the breadth of the area of Lévy processes. The first article by Ken-iti Sato characterizes extensions of the class of selfdecomposable distributions on R^d. The second article by Thomas Duquesne discusses Hausdorff and packing measures of stable trees. The third article by Oleg Reichmann and Christoph Schwab presents numerical solutions to Kolmogoroff equations, which arise for instance in financial engineering, when Lévy or additive processes model the dynamics of the risky assets.