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 = {Xt : t ≥ 0} is called a Lévy process in Rd with respect to if it is adapted to the filtration and
A process {Yt : 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 {Xt : t ≥ 0} be a Lévy process. For every ω Ω and every t [0, ∞), let
Here Xt−(ω) = 0 for t = 0. ...
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