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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. .

Stochastic processes --- Lévy processes. --- Lévy processes --- 519.282 --- Random walks (Mathematics) --- Probabilities. --- Applied mathematics. --- Engineering mathematics. --- Operations research. --- Management science. --- Probability Theory and Stochastic Processes. --- Applications of Mathematics. --- Operations Research, Management Science. --- Quantitative business analysis --- Management --- Problem solving --- Operations research --- Statistical decision --- Operational analysis --- Operational research --- Industrial engineering --- Management science --- Research --- System theory --- Engineering --- Engineering analysis --- Mathematical analysis --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk

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European hare --- Mammal populations --- Illumø (Denmark)

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This is the second of two volumes containing the revised and completed notes of lectures given at the school "Quantum Independent Increment Processes: Structure and Applications to Physics". This school was held at the Alfried-Krupp-Wissenschaftskolleg in Greifswald in March, 2003, and supported by the Volkswagen Foundation. The school gave an introduction to current research on quantum independent increment processes aimed at graduate students and non-specialists working in classical and quantum probability, operator algebras, and mathematical physics. The present second volume contains the following lectures: "Random Walks on Finite Quantum Groups" by Uwe Franz and Rolf Gohm, "Quantum Markov Processes and Applications in Physics" by Burkhard Kümmerer, Classical and Free Infinite Divisibility and Lévy Processes" by Ole E. Barndorff-Nielsen, Steen Thorbjornsen, and "Lévy Processes on Quantum Groups and Dual Groups" by Uwe Franz.

Mathematics. --- Distribution (Probability theory). --- Mathematical physics. --- Probability Theory and Stochastic Processes. --- Applications of Mathematics. --- Mathematical and Computational Physics. --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Mathematical Theory --- Physical mathematics --- Physics --- Distribution functions --- Frequency distribution --- Math --- Applied mathematics. --- Engineering mathematics. --- Probabilities. --- Physics. --- Theoretical, Mathematical and Computational Physics. --- Science --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Engineering --- Engineering analysis --- Mathematical analysis --- Distribution (Probability theory. --- Characteristic functions --- Probabilities --- Stochastic analysis. --- Probabilistic number theory. --- Lévy processes. --- Random walks (Mathematics) --- Number theory --- Analysis, Stochastic --- Stochastic processes --- Distribution (Probability theory) --- Probability Theory.

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Operational research. Game theory --- Mathematics --- Mathematical physics --- toegepaste wiskunde --- stochastische analyse --- wiskunde --- fysica --- kansrekening

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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.

Lâevy processes --- Branching processes --- Trees (Graph theory) --- Options (Finance) --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Prices --- Mathematical models --- Branching processes. --- Lévy processes. --- Lévy, Paul, --- Processes, Branching --- Mathematics. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Math --- Science --- Graph theory --- Random walks (Mathematics) --- Stochastic processes --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities