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 \(s\rightarrow t.\)

General form

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Intensity measure

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Subordinators

See subordinators.

Spectrally negative

Lévy processes with no positive jumps are called spectrally negative and have some nice properties as regard hitting times from below. (Ronald A. Doney and Picard 2007)

Martingales

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Sparsity properties

In the context of stochastic differential equations, Lévy processes give a model of a driving noise that is sparse compared to the usual Wiener process driving noise model. The subcategory of such Lévy SDEs that are additionally linear are are called sparse stochastic processes (M. A. Unser and Tafti 2014; M. Unser et al. 2014; M. Unser, Tafti, and Sun 2014) and I would enjoy having a minute to sit down and understand how they work.

Bridge processes

There are various interesting uses for Lévy bridges. 🏗 For now, see bridge processes.