Doney R.A. Fluctuation theory for Lévy processes: École d’Été de Probabilités de Saint-Flour XXXV - 2005

Springer, 2007. — 152 p. — ISBN: 3-540-48510-4, 978-3-540-48510-0.
Series: Lecture Notes in Mathematics / École d'Été de Probabilités de Saint-Flour (Book 1897).Lévy processes, i.e. processes in continuous time with stationary and independent increments, are named after Paul Lévy, who made the connection with infinitely divisible distributions and described their structure. They form a flexible class of models, which have been applied to the study of storage processes, insurance risk, queues, turbulence, laser cooling, . and of course finance, where the feature that they include examples having "heavy tails" is particularly important. Their sample path behaviour poses a variety of difficult and fascinating problems. Such problems, and also some related distributional problems, are addressed in detail in these notes that reflect the content of the course given by R. Doney in St. Flour in 2005.Introduction to Lévy Processes.
Notation, Poisson Point Processes, The Lévy–Itô Decomposition, Lévy Processes as Markov Processes.
Subordinators.
Basics, The Renewal Measure, Passage Across a Level, Arc-Sine Laws for Subordinators,
Rates of Growth, Killed Subordinators.
Local Times and Excursions.
Local Time of a Markov Process; The Regular, Instantaneous Case;
The Excursion Process, The Case of Holding and Irregular Points.
Ladder Processes and the Wiener–Hopf Factorisation.
The Random Walk Case, The Reflected and Ladder Processes, Applications, A Stochastic Bound.
Further Wiener–Hopf Developments.
Extensions of a Result due to Baxter, Les Équations Amicales of Vigon, A First Passage Quintuple Identity.
Creeping and Related Questions.
Notation and Preliminary Results, The Mean Ladder Height Problem, Creeping,
Limit Points of the Supremum Process, Regularity of the Half-Line, Summary: Four Integral Tests.
Spitzer’s Condition.
Proofs, Further Results, Tailpiece.
Lévy Processes Conditioned to Stay Positive.
Notation and Preliminaries, Definition and Path Decomposition, The Convergence Result,
Pathwise Constructions of (X, P^↑): Tanaka’s Construction, Bertoin’s Construction.
Spectrally Negative Lévy Processes.
Basics, The Random Walk Case, The Scale Function, Further Developments,
Exit Problems for the Reflected Process, Addendum.
Small-Time Behaviour.
Notation and Preliminary Results, Convergence in Probability, Almost Sure Results,
Summary of Asymptotic Results: Laws of Large Numbers, Central Limit Theorems, Exit from a Symmetric Interval.