Markov Processes for Stochastic Modeling, 2nd Edition

11

Levy Processes

11.1 Introduction

Both the Poisson process and the Brownian motion have stationary and independent increments. However, they have different sample paths: the Brownian motion has continuous sample paths, while the Poisson process has discontinuities (or jumps) of size 1. At a high level, Levy processes are stochastic processes with both stationary and independent increments. They constitute a wide class of stochastic processes whose sample paths can be continuous, continuous with occasional discontinuities, and purely discontinuous. In this chapter, we discuss Levy process as well as power-law and stable distributions, which will enable us to study the Levy distribution. Finally, we discuss jump-diffusion processes that are used ...