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 application appears in the theory of
many areas of classical and modern stochastic processes including
storage models, renewal processes, insurance risk models, optimal
stopping problems, mathematical finance, continuous-state branching
processes and positive self-similar Markov processes.
This textbook is based on a series of graduate courses concerning the
theory and application of Lévy processes from the perspective of their
path fluctuations. Central to the presentation is the decomposition of
paths in terms of excursions from the running maximum as well as an
understanding of 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 tractability.
The second edition additionally addresses recent developments in the
potential analysis of subordinators, Wiener-Hopf theory, the theory of
scale functions and their application to ruin theory, as well as
including an extensive overview of the classical and modern theory of
positive self-similar Markov processes. Each chapter has a comprehensive
set of exercises.
