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


Intensity measure



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.

Gamma process

See Gamma processes.

Brownian motions



See Subordinators!

Student-Lévy process

see t-processes.


Applebaum, David. 2004. Lévy Processes — from Probability to Finance and Quantum Groups.” Notices of the AMS 51 (11): 1336–47.
———. 2009. Lévy Processes and Stochastic Calculus. 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge ; New York: Cambridge University Press.
Arras, Benjamin, and Christian Houdré. 2019. 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.
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.
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.
Barndorff-Nielsen, Ole Eiler, and Robert Stelzer. 2011. Multivariate supOU Processes.” The Annals of Applied Probability 21 (1): 140–82.
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.
Barndorff-Nielsen, Ole E, and Neil Shephard. 2012. Basics of Lévy Processes.” In Lévy Driven Volatility Models, 70.
Belomestny, Denis. 2011. Statistical Inference for Time-Changed Lévy Processes via Composite Characteristic Function Estimation.” The Annals of Statistics 39 (4): 2205–42.
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 ….
Borovkov, Konstantin, and Zaeem Burq. 2001. Kendall’s Identity for the First Crossing Time Revisited.” Electronic Communications in Probability 6: 91–94.
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.
Doney, Ronald A. 2007. 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.
Grigelionis, Bronius. 2013. Student’s t-Distribution and Related Stochastic Processes. SpringerBriefs in Statistics. Berlin, Heidelberg: Springer Berlin Heidelberg.
Grosswald, E. 1976. The Student t-Distribution of Any Degree of Freedom Is Infinitely Divisible.” Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete 36 (2): 103–9.
Ismail, Mourad E. H. 1977. Bessel Functions and the Infinite Divisibility of the Student \(t\)- Distribution.” The Annals of Probability 5 (4): 582–85.
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.
Jacod, Jean, and Philip Protter. 1988. Time Reversal on Levy Processes.” The Annals of Probability 16 (2): 620–41.
Kallenberg, Olav. 2002. Foundations of Modern Probability. 2nd ed. Probability and Its Applications. New York: Springer-Verlag.
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).
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.
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. 1999. Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
Unser, Michael A., and Pouya Tafti. 2014. An Introduction to Sparse Stochastic Processes. New York: Cambridge University Press.
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.
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.
Upadhye, Neelesh S., and Kalyan Barman. 2020. A Unified Approach to Stein’s Method for Stable Distributions.” arXiv:2004.07593 [Math], July.
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.
———. 2010b. Numerical Computation of First-Passage Times of Increasing Lévy Processes.” Methodology and Computing in Applied Probability 12 (4): 695–729.
Wasan, M. T. 1968. On an Inverse Gaussian Process.” Scandinavian Actuarial Journal 1968 (1-2): 69–96.
Wolpert, Robert L. 2021. Lecture Notes on Stationary Gamma Processes.” arXiv:2106.00087 [Math], May.
Xu, Kailai, and Eric Darve. 2019. Calibrating Multivariate Lévy Processes with Neural Networks.” arXiv:1812.08883 [Cs, Stat], September.

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