Introductory Lectures on Fluctuations of Lévy Processes with Applications

Beschreibung

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 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.

This textbook 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.

Fills a gap in the market; namely a text which is focused on one of the fundamental corner stones of the theory of Lévy processes and does so in a pedagogical way and at graduate level

First book within the field which gives a comprehensive set of exercises with fully worked out solutions

Complementary addition to existing book on Lévy processes - stresses fluctuation theory in the context of classical applied probability models

Lévy processes are the natural continuous-time analogue of random walks; they form a rich class of stochastic processes around which a robust mathematical theory exists. Their mathematical significance is evident in 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 book aims to be mathematically rigourous 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 includes a comprehensive set of exercises with complete solutions.

Autor*in

Andreas E. Kyprianou

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Branching process Lévy process Lévy processes Maximum Random Walk Stochastic processes applied probability differential equation fluctuation theory potential analysis random walks stochastic process

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