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Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach by Hong Xie, Chaoqun Ma, Guojun Gan

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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) For almost every ω, the path tXt(ω) is right-continuous with left limit and X0(ω) = 0.
(b) For every t and h in [0, ∞), the increment Xt+hXt is independent of t and has the same distribution as Xh.

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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