Lévy measure

In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a with independent, stationary increments: it represents the motion of a point whose successive displacements are random and independent, and statistically identical over different time intervals of the same length. A Lévy process may thus be viewed as the continuous-time analog of a random walk.

The most well known examples of Lévy processes are the Wiener process, often called the Brownian motion process, and the Poisson process. Aside from Brownian motion with drift, all other proper Lévy processes have discontinuous paths.

A ${\displaystyle X=\{X_{t}:t\geq 0\}}$ is said to be a Lévy process if it satisfies the following properties:

If ${\displaystyle X}$ is a Lévy process then one may construct a version of ${\displaystyle X}$ such that ${\displaystyle t\mapsto X_{t}}$ is almost surely right continuous with left limits.

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