The goal of this sequel is to provide the foundations of the mathematics of Lévy processes for the readers with undergraduate knowledge of stochastic processes as simple as possible. The simplicity is a key because, for the beginners such as finance majors without the experience in stochastic processes, some available books on Lévy processes are not accessible. Lévy processes constitute a wide class of stochastic processes whose sample paths can be continuous, continuous with occasionally discontinuous, and purely discontinuous. Traditional examples of Lévy processes include a Brownian motion with drift (i.e. the only continuous Lévy processes), a Poisson process, a compound Poisson process, a jump diffusion process, and a Cauchy process. All of these are well studied and well applied stochastic processes. We define and characterize Lévy processes using theorems such as the Lévy-Itô decomposition and the Lévy-Khinchin representation and in terms of their infinite divisibilities and the Lévy measures􀁁. In the last decade and in the field of quantitative finance, there was an explosion of literatures modeling the log asset prices using purely non-Gaussian Lévy processes which are pure jump Lévy processes with infinite activity. To raise a few examples of purely non-Gaussian Lévy processes used in finance, variance gamma processes, tempered stable processes, and generalized hyperbolic Lévy motions. We cover these purely non-Gaussian Lévy processes in the next sequel with a finance application. This is because we like to keep this sequal as simple as possible with the pourpose of providing the introductory foundations of the mathematics of Lévy processes.
Introduction to the Mathematics of Lévy Processes.pdf
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