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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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This monograph highlights the connection between the theory of neutron transport and the theory of non-local branching processes. By detailing this frequently overlooked relationship, the authors provide readers an entry point into several active areas, particularly applications related to general radiation transport. Cutting-edge research published in recent years is collected here for convenient reference. Organized into two parts, the first offers a modern perspective on the relationship between the neutron branching process (NBP) and the neutron transport equation (NTE), as well as some of the core results concerning the growth and spread of mass of the NBP. The second part generalizes some of the theory put forward in the first, offering proofs in a broader context in order to show why NBPs are as malleable as they appear to be. Stochastic Neutron Transport will be a valuable resource for probabilists, and may also be of interest to numerical analysts and engineers in the field of nuclear research.

Probabilities. --- Stochastic processes. --- Markov processes. --- Applied Probability. --- Probability Theory. --- Stochastic Processes. --- Markov Process. --- Probabilitats --- Processos estocàstics