Levy Processes, Phase-type Distributions, and Martingales

Soren Asmussen
Aarhus University

Wednesday, January 11, 2017
4:15 - 5:15 PM
Location: Spilker 232


Levy processes are defined as processes with stationary independent increments and have become increasing popular as models in queueing, finance etc.; apart from Brownian motion and compound Poisson processes, some popular examples are stable processes, variance Gamma processes, CGMY Levy processes (tempered stable processes), NIG (normal inverse Gaussian) Levy processes, hyperbolic Levy processes. We consider here a dense class of Levy processes, compound Poisson processes with phase-type jumps in both directions and an added Brownian component. Within this class, we survey how to explicitly compute a number of quantities that are traditionally studied in the area of Levy processes, in particular two-sided exit probabilities and associated Laplace transforms, the closely related scale function, one-sided exit probabilities and associated Laplace transforms coming up in queueing problems, and similar quantities for a Levy process with reflection at 0. The solutions are in terms of roots to polynomials, and the basic equations are derived by purely probabilistic arguments using martingale optional stopping; a particularly useful martingale is the so-called Kella-Whitt martingale. Also, the relation to fluid models with a Brownian component is discussed.

Operations Research Colloquia: http://or.stanford.edu/oras_seminars.html