Ebook: Lévy Processes: Theory and Applications

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

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Content:
Front Matter....Pages i-xi
Front Matter....Pages 1-1
Basic Results on L?vy Processes....Pages 3-37
Front Matter....Pages 39-39
Exponential Functionals of L?vy Processes....Pages 41-55
Fluctuation Theory for L?vy Processes....Pages 57-66
Gaussian Processes and Local Times of Symmetric L?vy Processes....Pages 67-88
Temporal Change in Distributional Properties of L?vy Processes....Pages 89-107
Front Matter....Pages 109-109
L?vy Processes in Stochastic Differential Geometry....Pages 111-137
L?vy-Type Processes and Pseudodifferential Operators....Pages 139-168
Semistable Distributions....Pages 169-183
Front Matter....Pages 185-185
Analytic and Probabilistic Aspects of L?vy Processes and Fields in Quantum Theory....Pages 187-224
L?vy Processes and Continuous Quantum Measurements....Pages 225-239
L?vy Processes in the Physical Sciences....Pages 241-266
Some Properties of Burgers Turbulence with White or Stable Noise Initial Data....Pages 267-279
Front Matter....Pages 281-281
Modelling by L?vy Processess for Financial Econometrics....Pages 283-318
Application of Generalized Hyperbolic L?vy Motions to Finance....Pages 319-336
Explicit Form and Path Regularity of Martingale Representations....Pages 337-360
Interpretations in Terms of Brownian and Bessel Meanders of the Distribution of a Subordinated Perpetuity....Pages 361-375
Front Matter....Pages 377-377
Maximum Likelihood Estimation and Diagnostics for Stable Distributions....Pages 379-400
Series Representations of L?vy Processes from the Perspective of Point Processes....Pages 401-415
Back Matter....Pages 417-418

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Content:
Front Matter....Pages i-xi
Front Matter....Pages 1-1
Basic Results on L?vy Processes....Pages 3-37
Front Matter....Pages 39-39
Exponential Functionals of L?vy Processes....Pages 41-55
Fluctuation Theory for L?vy Processes....Pages 57-66
Gaussian Processes and Local Times of Symmetric L?vy Processes....Pages 67-88
Temporal Change in Distributional Properties of L?vy Processes....Pages 89-107
Front Matter....Pages 109-109
L?vy Processes in Stochastic Differential Geometry....Pages 111-137
L?vy-Type Processes and Pseudodifferential Operators....Pages 139-168
Semistable Distributions....Pages 169-183
Front Matter....Pages 185-185
Analytic and Probabilistic Aspects of L?vy Processes and Fields in Quantum Theory....Pages 187-224
L?vy Processes and Continuous Quantum Measurements....Pages 225-239
L?vy Processes in the Physical Sciences....Pages 241-266
Some Properties of Burgers Turbulence with White or Stable Noise Initial Data....Pages 267-279
Front Matter....Pages 281-281
Modelling by L?vy Processess for Financial Econometrics....Pages 283-318
Application of Generalized Hyperbolic L?vy Motions to Finance....Pages 319-336
Explicit Form and Path Regularity of Martingale Representations....Pages 337-360
Interpretations in Terms of Brownian and Bessel Meanders of the Distribution of a Subordinated Perpetuity....Pages 361-375
Front Matter....Pages 377-377
Maximum Likelihood Estimation and Diagnostics for Stable Distributions....Pages 379-400
Series Representations of L?vy Processes from the Perspective of Point Processes....Pages 401-415
Back Matter....Pages 417-418
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