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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

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

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

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

## General form

🏗

## 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
(Doney 2007).

## As martingales

🏗

## 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
(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.

## Recommended readings

Albin’s lectures based on Sato (1999) seems pretty good, as does Sat’s bok itself. Also Bertoin (2000) is good. Kyprianou (2014) is good introductory tratement. People recommend Applebaum (2009) a lot, and it is a fine and readable book but it does not emphasise areas I personally need (SDEs, non-negative processes), so I do not actually ever use it after chapter 2 or so. If I wanted to be excessively general I would probably go for Kallenberg (2002). M. A. Unser and Tafti (2014) has a theory of linear Lévy SDEs from the perspective of signal processing whicih is an unusual and helpful angle. If you want to get straight to business, perhaps favour the minimalist introduction in the stochastic simulation textbooks (Asmussen and Glynn 2007; Rubinstein and Kroese 2016) or, my favourite, (Aurzada and Dereich 2009).

## Gamma process

See Gamma processes.

## Brownian motions

TBD

## Subordinators

See Subordinators!…

## Student-Lévy process

see *t*-processes.

## References

*Notices of the AMS*51 (11): 1336–47.

*Lévy Processes and Stochastic Calculus*. 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge ; New York: Cambridge University Press.

*On Stein’s Method for Infinitely Divisible Laws with Finite First Moment*. Edited by Benjamin Arras and Christian Houdré. SpringerBriefs in Probability and Mathematical Statistics. Cham: Springer International Publishing.

*Stochastic Simulation: Algorithms and Analysis*. 2007 edition. New York: Springer.

*The Annals of Probability*37 (5): 2066–92.

*Proceedings of the 35th Conference on Winter Simulation: Driving Innovation*, 319–26. WSC ’03. New Orleans, Louisiana: Winter Simulation Conference.

*The Annals of Applied Probability*21 (1): 140–82.

*Advances in Applied Probability*33 (1): 160–87.

*Lévy Driven Volatility Models*, 70.

*The Annals of Statistics*39 (4): 2205–42.

*Lévy Processes*. Cambridge Tracts in Mathematics 121. Cambridge ; New York: Cambridge University Press.

*Subordinators, Lévy Processes with No Negative Jumps, and Branching Processes*. University of Aarhus. Centre for Mathematical Physics and Stochastics ….

*Electronic Communications in Probability*6: 91–94.

*Seminar on Stochastic Processes, 1981*, edited by E. Çinlar, K. L. Chung, and R. K. Getoor, 159–242. Progress in Probability and Statistics. Boston, MA: Birkhäuser.

*Fluctuation Theory for Lévy Processes: Ecole d’eté de Probabilités de Saint-Flour XXXV, 2005*. Vol. 1897. Lecture Notes in Mathematics 1897. Berlin ; New York: Springer.

*Student’s t-Distribution and Related Stochastic Processes*. SpringerBriefs in Statistics. Berlin, Heidelberg: Springer Berlin Heidelberg.

*Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete*36 (2): 103–9.

*The Annals of Probability*5 (4): 582–85.

*Lévy Processes: Theory and Applications*, edited by Ole E. Barndorff-Nielsen, Sidney I. Resnick, and Thomas Mikosch, 139–68. Boston, MA: Birkhäuser.

*The Annals of Probability*16 (2): 620–41.

*Foundations of Modern Probability*. 2nd ed. Probability and Its Applications. New York: Springer-Verlag.

*Fluctuations of Lévy Processes with Applications: Introductory Lectures*. Second edition. Universitext. Heidelberg: Springer.

*Communications in Applied and Industrial Mathematics*6 (1).

*Stochastic Processes and Their Applications*118 (9): 1606–33.

*Simulation and the Monte Carlo Method*. 3 edition. Wiley series in probability and statistics. Hoboken, New Jersey: Wiley.

*Lévy Processes and Infinitely Divisible Distributions*. Cambridge University Press.

*An Introduction to Sparse Stochastic Processes*. New York: Cambridge University Press.

*IEEE Transactions on Information Theory*60 (5): 3036–51.

*IEEE Transactions on Information Theory*60 (3): 1945–62.

*arXiv:2004.07593 [Math]*, July.

*Statistics & Probability Letters*80 (7): 697–705.

*Methodology and Computing in Applied Probability*12 (4): 695–729.

*Scandinavian Actuarial Journal*1968 (1-2): 69–96.

*arXiv:2106.00087 [Math]*, May.

*arXiv:1812.08883 [Cs, Stat]*, September.

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