Lévy Processes: Theory and Applications

ISBN-13: 9781461266570 / Angielski / Miękka / 2012 / 418 str.

OLE E. Barndorff-Nielsen; Thomas Mikosch; Sidney I. Resnick

Opis

A Levy 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 Levy 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 Levy processes. Researchers and practitioners in fields as diverse as physics, meteorology, statistics, insurance, and finance have rediscovered the simplicity of Levy 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 Levy processes.

Recenzje

"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

Spis treści

Preface I. A tutorial on Lévy processes Sato, K.: Basic results on Lévy processes II. Distributional, pathwise and structural results Carmona, P. / Petit, F. / Yor, M.: Exponentials functionals of Levy processes Doney, R.: Fluctuation theory for Levy processes Marcus, M.B. / Rosen, J.: Gaussian processes and the local times of symmetric Lévy processes Watanabe, T.: Temporal change in distributional properties of Lévy processes III: Extensions and generalisations of Lévy processes Applebaum, D.: Lévy processes in stochastic differential geometry Jac. / Schilling, R.L.: Lévy-type processes and pseudo-differential operators Maejima, M.: Semi-stable distributions IV. Applications in physics Albeverio, S. / Rüdiger, B. / Wu, J-L.: Analytic and probabilistic aspects of Lévy processes and fields in quantum theory Holevo, A.S.: Lévy processes and continuous quantum measurements Woyczynski, W.A.: Lévy processes in the physical sciences Bertoin, J.: Some properties of Burgers turbulence with white or stable noise V. Applications in finance Barndorff-Nielsen, O.E / Shepard, N.: Modelling by Lévy processes for financial econometrics Eberlein, E.: Application of generalized hyperbolic Lévy motions to finance Ma, J. / Protter, P. / Zhang, J: Explicit form and path regularity of martingale representation Yor, M.: Interpretations in terms of Brownian and Bessel meanders of the distribution of a subordinated perpetuity VI. Numerical and statistical aspects Nolan, J.P.: Maximum likelihood estimation and diagnostics for stable distributions Rosinski, J.: Series representations of Lévy processes from the perspective of point processes

O autorze

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.