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 \({\mathbb R}^d\) such that

  1. independent increments: \(X_t-X_s\) is independent of \(\{X_u\colon u\le s\}\) for any \(s<t.\)

  2. stationary increments: \(X_{s+t}-X_s\) has the same distribution as \(X_t-X_0\) for any \(s,t>0.\)

  3. continuity in probability: \(X_s\rightarrow X_t\) in probability as \(s\rightarrow t.\)

General form


Intensity measure



See change of time