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Introductory Lectures on Fluctuations of Lévy Processes with Applications
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.
Lévy Processes and Applications -- TheLévy–Itô Decomposition and Path Structure -- More Distributional and Path-Related Properties -- General Storage Models and Paths of Bounded Variation -- Subordinators at First Passage and Renewal Measures -- The Wiener–Hopf Factorisation -- Lévy Processes at First Passage and Insurance Risk -- Exit Problems for Spectrally Negative Processes -- Applications to Optimal Stopping Problems -- Continuous-State Branching Processes.
Probability Theory and Stochastic Processes