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Markov Processes for Stochastic Modeling, 2nd Edition by Oliver Ibe

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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 ...

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