Levy Processes and Change of Probability Measure
Document ID: 2009-1262 (previously 2009/01)
Published on: 5th January 2009
Author: Graeme Lawson
The aim of this technical note is to describe the Levy Process, examples of which include the Wiener, Poisson, Compound Poisson, Gamma process amongst many others.
With numerous examples throughout the technical note, we use an extended version of Ito’s formulae, and Girsanov Theorem to illustrate the mathematical properties and definitions of the Levy process as a modelling tool in quantitative finance. We place emphasis on the derivation of exponential martingales, which play a key role in modelling of asset price processes, and the construction of equivalent martingale measures.
We concentrate specifically on the Weiner, Process, and Compound Poisson processes, as theses are modelling tools used within the Barrie & Hibbert Economic Scenario Generator.
In the sections that follow, we will always assume for simplicity that we are always working with a scalar random variable (as opposed to vector random variables). The notions can easily be extended to vector random variables
