Levy processes

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A Lévy process is an important type of stochastic process (namely, a family of $\mathbb{R}^d$ valued random variables each indexed by a positive number $t\geq 0$). In the context of parabolic integro-differential equations they play the same role that Brownian motion and more general diffusions play in the theory of second order parabolic equations.

Informally speaking, a Lévy process is a Brownian motion which may jump, the times, length and direction of the jumps being random variables. A prototypical example would be $X(t)=B(t)+N(t)$ where $B(t)$ is the standard Brownian motion and $N(t)$ is a Compound Poisson process, the trajectory described by typical sample path of this process would look like the union of several disconnected Brownian motion paths.

Definition

A stochastic process $X(t)$ with values in $\mathbb{R}^d$ is said to be a Lévy process if

1.For any sequence $0 \leq t_1 < t_2 <...<t_n$ the random variables $X(t_0),X(t_1)-X(t_0),...,X(t_n)-X(t_{n-1})$ are independent.

2.For any positive times $s\leq t$ the random variables $X(t-s)$ and $X(t)-X(s)$ have the same probability law.


Lévy-Khintchine Formula

Semigroup and infinitesimal generator

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