The Living Thing / Notebooks :

Lévy processes

Processes with independent, well-behaved increments.

Specific examples of interest include Gamma processes, Brownian Motions, continuous time branching processes etc.

Let's start with George Lowther:

Continuous-time stochastic processes with stationary independent increments are known as Lévy processes. […]it was seen that processes with independent increments are described by three terms — the covariance structure of the Brownian motion component, a drift term, and a measure describing the rate at which jumps occur. Being a special case of independent increments processes, the situation with Lévy processes is similar.[…]

A d-dimensional Lévy process X is a stochastic process taking values in such that

  1. independent increments: is independent of for any

  2. stationary increments: has the same distribution as for any

  3. continuity in probability: in probability as

General form

TBC

Intensity measure

TBC

Subordinator

See change of time