Lévy Processes

In the past, representatives of the L,vy class were considered most useful for applications to either Brownian motion or the Poisson process. Nowadays, the need for modeling jumps, bursts, extremes and other irregular behavior of phenomena in nature and society has led to a renaissance of the theory of general L,vy processes. Researchers and practitioners in physics, meteorology, statistics, insurance and finance have rediscovered the simplicity of L,vy processes and their enormous flexibility in modeling tails, dependence and path behavior. This volume describes the state-of-the-art of this rapidly evolving subject with special emphasis on the non-Brownian world.

"This volume presents a useful summary of some of the recent scientific developments concerning Lévy processes. Both introductory and more advanced articles are included. The interested researcher will get a good overview of 'where the action is' whereas students will find numerous interesting research topics to work on . . . I am convinced that the text will contribute further to making stochastic models based on general Lévy processes even more popular. I, therefore, take pleasure in recommending this volume to all interested readers." ISI Short Book Reviews

Autorentext
In the past, representatives of the L,vy class were considered most
useful for applications to either Brownian motion or the Poisson
process. Nowadays, the need for modeling jumps, bursts, extremes and
other irregular behavior of phenomena in nature and society has led to
a renaissance of the theory of general L,vy processes. Researchers and
practitioners in physics, meteorology, statistics, insurance and
finance have rediscovered the simplicity of L,vy processes and their
enormous flexibility in modeling tails, dependence and path behavior.
This volume describes the state-of-the-art of this rapidly
evolving subject with special emphasis on the non-Brownian world.

Inhalt
I. A Tutorial on Levy Processes.- Basic Results on Lévy Processes.- II. Distributional, Pathwise, and Structural Results.- Exponential Functionals of Lévy Processes.- Fluctuation Theory for Lévy Processes.- Gaussian Processes and Local Times of Symmetric Lévy Processes.- Temporal Change in Distributional Properties of Lévy Processes.- III. Extensions and Generalizations of Lévy Processes.- Lévy Processes in Stochastic Differential Geometry.- Lévy-Type Processes and Pseudodifferential Operators.- Semistable Distributions.- IV. Applications in Physics.- Analytic and Probabilistic Aspects of Lévy Processes and Fields in Quantum Theory.- Lévy Processes and Continuous Quantum Measurements.- Lévy Processes in the Physical Sciences.- Some Properties of Burgers Turbulence with White or Stable Noise Initial Data.- V. Applications in Finance.- Modelling by Lévy Processess for Financial Econometrics.- Application of Generalized Hyperbolic Lévy Motions to Finance.- Explicit Form and Path Regularity of Martingale Representations.- Interpretations in Terms of Brownian and Bessel Meanders of the Distribution of a Subordinated Perpetuity.- VI. Numerical and Statistical Aspects.- Maximum Likelihood Estimation and Diagnostics for Stable Distributions.- Series Representations of Lévy Processes from the Perspective of Point Processes.