A Lรฉvy process is a continuous-time analogue of a random walk, and as such, is at the cradle of modern theories of stochastic processes. Martingales, Markov processes, and diffusions are extensions and generalizations of these processes. In the past, representatives of the Lรฉvy 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 Lรฉvy processes. Researchers and practitioners in fields as diverse as physics, meteorology, statistics, insurance, and finance have rediscovered the simplicity of Lรฉvy processes and their enormous flexibility in modeling tails, dependence and path behavior. This volume, with an excellent introductory preface, describes the state-of-the-art of this rapidly evolving subject with special emphasis on the non-Brownian world. Leading experts present surveys of recent developments, or focus on some most promising applications. Despite its special character, every topic is aimed at the non- specialist, keen on learning about the new exciting face of a rather aged class of processes. An extensive bibliography at the end of each article makes this an invaluable comprehensive reference text. For the researcher and graduate student, every article contains open problems and points out directions for futurearch. The accessible nature of the work makes this an ideal introductory text for graduate seminars in applied probability, stochastic processes, physics, finance, and telecommunications, and a unique guide to the world of Lรฉvy processes.
I. A Tutorial on Levy Processes -- Basic Results on Lรฉvy Processes -- II. Distributional, Pathwise, and Structural Results -- Exponential Functionals of Lรฉvy Processes -- Fluctuation Theory for Lรฉvy Processes -- Gaussian Processes and Local Times of Symmetric Lรฉvy Processes -- Temporal Change in Distributional Properties of Lรฉvy Processes -- III. Extensions and Generalizations of Lรฉvy Processes -- Lรฉvy Processes in Stochastic Differential Geometry -- Lรฉvy-Type Processes and Pseudodifferential Operators -- Semistable Distributions -- IV. Applications in Physics -- Analytic and Probabilistic Aspects of Lรฉvy Processes and Fields in Quantum Theory -- Lรฉvy Processes and Continuous Quantum Measurements -- Lรฉvy Processes in the Physical Sciences -- Some Properties of Burgers Turbulence with White or Stable Noise Initial Data -- V. Applications in Finance -- Modelling by Lรฉvy Processess for Financial Econometrics -- Application of Generalized Hyperbolic Lรฉvy 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 Lรฉvy Processes from the Perspective of Point Processes