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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

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Motivated by the many and long-standing contributions of H. Gerber and E. Shiu, this book gives a modern perspective on the problem of ruin for the classical Cramér–Lundberg model and the surplus of an insurance company. The book studies martingales and path decompositions, which are the main tools used in analysing the distribution of the time of ruin, the wealth prior to ruin and the deficit at ruin. Recent developments in exotic ruin theory are also considered. In particular, by making dividend or tax payments out of the surplus process, the effect on ruin is explored. Gerber-Shiu Risk Theory can be used as lecture notes and is suitable for a graduate course. Each chapter corresponds to approximately two hours of lectures.

Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Mathematics. --- Actuarial science. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Actuarial Sciences. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Statistics --- Insurance --- Math --- Science --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities

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Motivated by the many and long-standing contributions of H. Gerber and E. Shiu, this book gives a modern perspective on the problem of ruin for the classical Cramér-Lundberg model and the surplus of an insurance company. The book studies martingales and path decompositions, which are the main tools used in analysing the distribution of the time of ruin, the wealth prior to ruin and the deficit at ruin. Recent developments in exotic ruin theory are also considered. In particular, by making dividend or tax payments out of the surplus process, the effect on ruin is explored. Gerber-Shiu Risk Theory can be used as lecture notes and is suitable for a graduate course. Each chapter corresponds to approximately two hours of lectures.

Operational research. Game theory --- Probability theory --- Mathematics --- waarschijnlijkheidstheorie --- stochastische analyse --- wiskunde --- kansrekening --- Distribution (Probability theory) --- Mathématiques --- Distribution (Théorie des probabilités) --- EPUB-LIV-FT LIVMATHE LIVSTATI SPRINGER-B

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Random walks (Mathematics) --- Additive process (Probability theory) --- Random walk process (Mathematics) --- Walks, Random (Mathematics) --- Stochastic processes --- Rutes aleatòries (Matemàtica) --- Processos de Lévy --- Passeigs aleatoris (Matemàtica) --- Processos additius (Teoria de la probabilitat) --- Processos de trajectòries aleatòries (Matemàtica) --- Recorreguts aleatoris (Matemàtica) --- Trajectòries aleatòries (Matemàtica) --- Anàlisi matemàtica --- Anàlisi numèrica --- Física matemàtica --- Processos estocàstics

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Since around the turn of the millennium there has been a general acceptance that one of the more practical improvements one may make in the light of the shortfalls of the classical Black-Scholes model is to replace the underlying source of randomness, a Brownian motion, by a Lévy process. Working with Lévy processes allows one to capture desirable distributional characteristics in the stock returns. In addition, recent work on Lévy processes has led to the understanding of many probabilistic and analytical properties, which make the processes attractive as mathematical tools. At the same time, exotic derivatives are gaining increasing importance as financial instruments and are traded nowadays in large quantities in OTC markets. The current volume is a compendium of chapters, each of which consists of discursive review and recent research on the topic of exotic option pricing and advanced Lévy markets, written by leading scientists in this field. In recent years, Lévy processes have leapt to the fore as a tractable mechanism for modeling asset returns. Exotic option values are especially sensitive to an accurate portrayal of these dynamics. This comprehensive volume provides a valuable service for financial researchers everywhere by assembling key contributions from the world's leading researchers in the field. Peter Carr, Head of Quantitative Finance, Bloomberg LP. This book provides a front-row seat to the hottest new field in modern finance: options pricing in turbulent markets. The old models have failed, as many a professional investor can sadly attest. So many of the brightest minds in mathematical finance across the globe are now in search of new, more accurate models. Here, in one volume, is a comprehensive selection of this cutting-edge research.

Stochastic processes --- Options (Finance) --- Lévy processes --- Options (Finances) --- Lévy, Processus de --- Prices --- Mathematical models --- Prix --- Modèles mathématiques --- -Lévy processes --- -332.632283 --- Random walks (Mathematics) --- Call options --- Calls (Finance) --- Listed options --- Options exchange --- Options market --- Options trading --- Put and call transactions --- Put options --- Puts (Finance) --- Derivative securities --- Investments --- -Mathematical models --- -Electronic information resources --- Electronic information resources --- E-books --- Lévy processes. --- Mathematical models. --- Lévy processes --- Lévy, Processus de --- Modèles mathématiques