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.

**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 *X*_{0}(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. ...

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