# Lévy processes

May 29, 2017 — November 17, 2021

branching
Lévy processes
point processes
probability
spatial
statmech
stochastic 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

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 $$s\rightarrow t.$$

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## 3 Subordinators

See subordinators.

## 4 Spectrally negative

Lévy processes with no positive jumps are called spectrally negative and have some nice properties as regard hitting times from below .

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## 6 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 called sparse stochastic processes and I would enjoy having a minute to sit down and understand how they work.

## 7 Bridge processes

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

## 9 Gamma process

See Gamma processes.

TBD

## 11 Subordinators

See Subordinators!

see t-processes.

## 13 References

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Doney. 2007. Fluctuation Theory for Lévy Processes: Ecole d’eté de Probabilités de Saint-Flour XXXV, 2005. Lecture Notes in Mathematics 1897.
Grigelionis. 2013. Student’s t-Distribution and Related Stochastic Processes. SpringerBriefs in Statistics.
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Jacob, and Schilling. 2001. In Lévy Processes: Theory and Applications.
Jacod, and Protter. 1988. The Annals of Probability.
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Kyprianou. 2014. Fluctuations of Lévy Processes with Applications: Introductory Lectures. Universitext.
Leonenko, Meerschaert, Schilling, et al. 2014. Communications in Applied and Industrial Mathematics.
Meerschaert, and Scheffler. 2008. Stochastic Processes and Their Applications.
Rubinstein, and Kroese. 2016. Simulation and the Monte Carlo Method. Wiley series in probability and statistics.
Sato. 1999. Lévy Processes and Infinitely Divisible Distributions.
Unser, Michael A., and Tafti. 2014. An Introduction to Sparse Stochastic Processes.
Unser, M., Tafti, Amini, et al. 2014. IEEE Transactions on Information Theory.
Unser, M., Tafti, and Sun. 2014. IEEE Transactions on Information Theory.
Upadhye, and Barman. 2020. arXiv:2004.07593 [Math].
Veillette, and Taqqu. 2010a. Statistics & Probability Letters.
———. 2010b. Methodology and Computing in Applied Probability.
Wasan. 1968. Scandinavian Actuarial Journal.
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Xu, and Darve. 2019. arXiv:1812.08883 [Cs, Stat].