Lévy processes

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Stochastic processes with i.i.d. increments over disjoint intervals of the same length, i.e. which arise from divisible distributions. Specific examples of interest include Gamma processes, Brownian motions, certain branching processes, non-negative processes…

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 \(\Lambda(\cdot)\) is a stochastic process indexed by \(\bb{R}\) taking values in \({\mathbb R}^d\) such that it possesses

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

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

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