Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach

CHAPTER 30

LÉVY PROCESSES

Lévy processes are stochastic processes that have independent and stationary increments, The basic theory of Lévy processes was established in the 1930s. Recently, Lévy processes have been used as asset price models in mathematical finance. In this chapter, we present Lévy processes and their main properties.

30.1 Basic Concepts and Facts

Definition 30.1 (Lévy Process). Let (Ω, , P) be a probability space and a filtration on the probability space. A stochastic process X = {X_t : t ≥ 0} is called a Lévy process in R^d with respect to if it is adapted to the filtration and

(a) For almost every ω, the path t → X_t(ω) is right-continuous with left limit and X₀(ω) = 0.

(b) For every t and h in [0, ∞), the increment X_t+h − X_t is independent of _t and has the same distribution as X_h.

A process {Y_t : t ≥ 0} is considered a Lévy process, without mentioning a filtration, if it is a Lévy process with respect to the filtration generated by itself.

Definition 30.2 (Jump). Let {X_t : t ≥ 0} be a Lévy process. For every ω Ω and every t [0, ∞), let

Here X_t−(ω) = 0 for t = 0. ...