\(\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}}\)

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 \(s\rightarrow t.\)

For a more thorough presentation, see e.g. (Applebaum 2009; Kyprianou 2014; Sato, Ken-Iti, and Katok 1999; Bertoin 1996, 2000). My favourite introductory treatment is Kyprianou (2014), although 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, [AurzadaSmall2009].

## General form

🏗

## Intensity measure

🏗

## 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

🏗

## 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.

Applebaum, David. 2004. “Lévy Processes—from Probability to Finance and Quantum Groups.” *Notices of the AMS* 51 (11): 12. https://core.ac.uk/download/pdf/50531.pdf.

———. 2009. *Lévy Processes and Stochastic Calculus*. 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge ; New York: Cambridge University Press.

Asmussen, Søren, and Peter W. Glynn. 2007. *Stochastic Simulation: Algorithms and Analysis*. 2007 edition. New York: Springer.

Aurzada, Frank, and Steffen Dereich. 2009. “Small Deviations of General Lévy Processes.” *The Annals of Probability* 37 (5): 2066–92. https://doi.org/10.1214/09-AOP457.

Avramidis, Athanassios N., Pierre L’Ecuyer, and Pierre-Alexandre Tremblay. 2003. “New Simulation Methodology for Finance: Efficient Simulation of Gamma and Variance-Gamma Processes.” In *Proceedings of the 35th Conference on Winter Simulation: Driving Innovation*, 319–26. WSC ’03. New Orleans, Louisiana: Winter Simulation Conference. http://www-perso.iro.umontreal.ca/~lecuyer/myftp/papers/wsc03vg.pdf.

Barndorff-Nielsen, Ole Eiler, and Robert Stelzer. 2011. “Multivariate supOU Processes.” *The Annals of Applied Probability* 21 (1): 140–82. https://doi.org/10.1214/10-AAP690.

Barndorff-Nielsen, Ole E., Jan Pedersen, and Ken-Iti Sato. 2001. “Multivariate Subordination, Self-Decomposability and Stability.” *Advances in Applied Probability* 33 (1): 160–87. https://doi.org/10.1017/S0001867800010685.

Barndorff-Nielsen, Ole E, and Neil Shephard. 2012. “Basics of Lévy Processes.” In *Lévy Driven Volatility Models*, 70. https://pdfs.semanticscholar.org/fe80/07ba98fafa23ddca98ac5d9b1cf1732f70bd.pdf.

Belomestny, Denis. 2011. “Statistical Inference for Time-Changed Lévy Processes via Composite Characteristic Function Estimation.” *The Annals of Statistics* 39 (4): 2205–42. https://doi.org/10.1214/11-AOS901.

Bertoin, Jean. 1996. *Lévy Processes*. Cambridge Tracts in Mathematics 121. Cambridge ; New York: Cambridge University Press.

———. 2000. *Subordinators, Lévy Processes with No Negative Jumps, and Branching Processes*. University of Aarhus. Centre for Mathematical Physics and Stochastics …. http://www.maphysto.dk/oldpages/events/LevyBranch2000/notes/bertoin.pdf.

Borovkov, Konstantin, and Zaeem Burq. 2001. “Kendall’s Identity for the First Crossing Time Revisited.” *Electronic Communications in Probability* 6: 91–94. https://doi.org/10.1214/ECP.v6-1038.

Çinlar, E., and J. Jacod. 1981. “Representation of Semimartingale Markov Processes in Terms of Wiener Processes and Poisson Random Measures.” In *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. https://doi.org/10.1007/978-1-4612-3938-3_8.

Doney, Ronald A., and Jean Picard, eds. 2007. “Spectrally Negative Lévy Processes.” In *Fluctuation Theory for Lévy Processes: Ecole d’Eté de Probabilités de Saint-Flour XXXV - 2005*, 95–113. Lecture Notes in Mathematics. Berlin, Heidelberg: Springer. https://doi.org/10.1007/978-3-540-48511-7_9.

Doney, Ronald A, and Jean Picard. 2007. *Fluctuation Theory for Lévy Processes: Ecole d’Eté de Probabilités de Saint-Flour XXXV - 2005*. Berlin, Heidelberg: Springer-Verlag Berlin Heidelberg. https://doi.org/10.1007/978-3-540-48511-7.

Jacob, Niels, and René L. Schilling. 2001. “Lévy-Type Processes and Pseudodifferential Operators.” In *Lévy Processes: Theory and Applications*, edited by Ole E. Barndorff-Nielsen, Sidney I. Resnick, and Thomas Mikosch, 139–68. Boston, MA: Birkhäuser. https://doi.org/10.1007/978-1-4612-0197-7_7.

Jacod, Jean, and Philip Protter. 1988. “Time Reversal on Levy Processes.” *The Annals of Probability* 16 (2): 620–41. https://doi.org/10.1214/aop/1176991776.

Kyprianou, Andreas E. 2014. *Fluctuations of Lévy Processes with Applications: Introductory Lectures*. Second edition. Universitext. Heidelberg: Springer.

Leonenko, Nikolai N, Mark M Meerschaert, René L Schilling, and Alla Sikorskii. 2014. “Correlation Structure of Time-Changed Lévy Processes.” *Communications in Applied and Industrial Mathematics* 6 (1). https://doi.org/10.1685/journal.caim.483.

Meerschaert, Mark M., and Hans-Peter Scheffler. 2008. “Triangular Array Limits for Continuous Time Random Walks.” *Stochastic Processes and Their Applications* 118 (9): 1606–33. https://doi.org/10.1016/j.spa.2007.10.005.

Rubinstein, Reuven Y., and Dirk P. Kroese. 2016. *Simulation and the Monte Carlo Method*. 3 edition. Wiley Series in Probability and Statistics. Hoboken, New Jersey: Wiley.

Sato, Ken-iti, Sato Ken-Iti, and A. Katok. 1999. *Lévy Processes and Infinitely Divisible Distributions*. Cambridge University Press.

Schilling, René L. 2016. “An Introduction to Lévy and Feller Processes. Advanced Courses in Mathematics - CRM Barcelona 2014,” March. http://arxiv.org/abs/1603.00251.

Unser, Michael A., and Pouya Tafti. 2014. *An Introduction to Sparse Stochastic Processes*. New York: Cambridge University Press. http://www.sparseprocesses.org/sparseprocesses-123456.pdf.

Unser, M., P. D. Tafti, A. Amini, and H. Kirshner. 2014. “A Unified Formulation of Gaussian Vs Sparse Stochastic Processes - Part II: Discrete-Domain Theory.” *IEEE Transactions on Information Theory* 60 (5): 3036–51. https://doi.org/10.1109/TIT.2014.2311903.

Unser, M., P. D. Tafti, and Q. Sun. 2014. “A Unified Formulation of Gaussian Vs Sparse Stochastic Processes—Part I: Continuous-Domain Theory.” *IEEE Transactions on Information Theory* 60 (3): 1945–62. https://doi.org/10.1109/TIT.2014.2298453.

Veillette, Mark, and Murad S. Taqqu. 2010a. “Using Differential Equations to Obtain Joint Moments of First-Passage Times of Increasing Lévy Processes.” *Statistics & Probability Letters* 80 (7): 697–705. https://doi.org/10.1016/j.spl.2010.01.002.

———. 2010b. “Numerical Computation of First-Passage Times of Increasing Lévy Processes.” *Methodology and Computing in Applied Probability* 12 (4): 695–729. https://doi.org/10.1007/s11009-009-9158-y.