Fluctuation Theory for Lévy Processes

Lévy processes, that is, processes in continuous time with stationary and independent increments, 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 they include particularly important examples having "heavy tails." Their sample path behaviour poses a variety of challenging and fascinating problems, which are addressed in detail.

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.

Includes supplementary material: sn.pub/extras

Inhalt
to Lévy Processes.- Subordinators.- Local Times and Excursions.- Ladder Processes and the WienerHopf Factorisation.- Further WienerHopf Developments.- Creeping and Related Questions.- Spitzer's Condition.- Lévy Processes Conditioned to Stay Positive.- Spectrally Negative Lévy Processes.- Small-Time Behaviour.