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

TBC

Intensity measure

TBC

Subordinator

See change of time