Simulation and estimation of Lévy driven stochastic processes

Abstract

We carry on an exploration of Lévy processes, focusing on instrumental definitions that ease our way towards their simulation. Beginning with the basic Brownian motion and Poisson processes, we then explore algorithms to simulate Lévy processes, do maximum likelihood estimations, and follow this exploratory road studying a maximum likelihood methodology to estimate the parameters of a one dimensional stationary process of Ornstein Uhlenbeck type that is constructed via a self-decomposable distribution D. Finally, we also present the maximum empirical likelihood method specifically for Lévy processes as an alternative to the classical maximum likelihood estimation methodology, when the density function is unknown.