Chapter 15: Levy Processes
A Levy process is a cadlag stochastic process that is infinitely divisible into independent and identically distributed random variables. Levy processes can be subdivided into finite and infinite activity Levy processes. Finite activity Levy processes include the Merton and the Kou jump-diffusion models. In Merton and Kou, the Levy formalism provides an elegant structure to append a jump component to a Brownian diffusion model. An infinite activity Levy process, such as the alpha-stable distribution, can have an infinite number of jumps in a finite time period. Infinite activity processes can be also be created from a Brownian motion process with the time change represented as a gamma process to produce the variance gamma process. Similarly, Brownian motion subordinated with an inverse Gaussian time change component creates the normal inverse Gaussian (NIG) process.
15.2 Levy-Khintchine Formula
The tenets of a stochastic Levy process are that X has stationary and independent increments, with probability 1, and is stochastically continuous (Applebaum 2004). When is infinitely divisible, its characteristic function is broadly described by ...