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Lévy processes, i.e. processes in continuous time with stationary and independent increments, are named after Paul Lévy, who made the connection with infinitely divisible distributions and described their structure. They form a flexible class of models, which have been applied to the study of storage processes, insurance risk, queues, turbulence, laser cooling, ... and of course finance, where the feature that they include examples having "heavy tails" is particularly important. Their sample path behaviour poses a variety of difficult and fascinating problems. Such problems, and also some related distributional problems, are addressed in detail in these notes that reflect the content of the course given by R. Doney in St. Flour in 2005.

Lévy processes. --- Lévy, Processus de --- Electronic books. -- local. --- Lévy processes. --- Random walks (Mathematics). --- Lâevy processes --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Theory --- Mathematical Statistics --- Random walks (Mathematics) --- Lévy, Processus de --- EPUB-LIV-FT SPRINGER-B --- Additive process (Probability theory) --- Random walk process (Mathematics) --- Walks, Random (Mathematics) --- Mathematics. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Math --- Science --- Stochastic processes --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities

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Lévy processes are the natural continuous-time analogue of random walks and form a rich class of stochastic processes around which a robust mathematical theory exists. Their mathematical significance is justified by their application in many areas of classical and modern stochastic models including storage models, renewal processes, insurance risk models, optimal stopping problems, mathematical finance and continuous-state branching processes. This text book forms the basis of a graduate course on the theory and applications of Lévy processes, from the perspective of their path fluctuations. Central to the presentation are decompositions of the paths of Lévy processes in terms of their local maxima and an understanding of their short- and long-term behaviour. The book aims to be mathematically rigorous while still providing an intuitive feel for underlying principles. The results and applications often focus on the case of Lévy processes with jumps in only one direction, for which recent theoretical advances have yielded a higher degree of mathematical transparency and explicitness. Each chapter has a comprehensive set of exercises with complete solutions.

Stochastic processes --- Lévy processes --- Lévy, Processus de --- Processus stochastiques --- Electronic books. -- local. --- Lévy processes. --- Stochastic processes. --- Lâevy processes --- Mathematical Statistics --- Mathematics --- Physical Sciences & Mathematics --- 519.23 --- Random processes --- Probabilities --- Random walks (Mathematics) --- Lévy processes --- Lévy, Processus de --- EPUB-LIV-FT LIVMATHE SPRINGER-B --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Economics, Mathematical. --- Probabilities. --- Analysis. --- Probability Theory and Stochastic Processes. --- Quantitative Finance. --- Global analysis (Mathematics). --- Distribution (Probability theory. --- Finance. --- Funding --- Funds --- Economics --- Currency question --- Distribution functions --- Frequency distribution --- Characteristic functions --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Economics, Mathematical . --- Mathematical economics --- Econometrics --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- 517.1 Mathematical analysis --- Mathematical analysis --- Methodology