Lévy processes

May 29, 2017 — November 17, 2021

Lévy processes
point processes
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

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 \(s\rightarrow t.\)

1 General form


2 Intensity measure


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 (Doney 2007).

5 As martingales


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 (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.

7 Bridge processes

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

9 Gamma process

See Gamma processes.

10 Brownian motions


11 Subordinators

See Subordinators!

12 Student-Lévy process

see t-processes.

13 References

Applebaum. 2004. Lévy Processes — from Probability to Finance and Quantum Groups.” Notices of the AMS.
———. 2009. Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics 116.
Arras, and 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.
Asmussen, and Glynn. 2007. Stochastic Simulation: Algorithms and Analysis.
Aurzada, and Dereich. 2009. Small Deviations of General Lévy Processes.” The Annals of Probability.
Avramidis, L’Ecuyer, and 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. WSC ’03.
Barndorff-Nielsen, Ole E., Pedersen, and Sato. 2001. Multivariate Subordination, Self-Decomposability and Stability.” Advances in Applied Probability.
Barndorff-Nielsen, Ole E, and Shephard. 2012. Basics of Lévy Processes.” In Lévy Driven Volatility Models.
Barndorff-Nielsen, Ole Eiler, and Stelzer. 2011. Multivariate supOU Processes.” The Annals of Applied Probability.
Belomestny. 2011. Statistical Inference for Time-Changed Lévy Processes via Composite Characteristic Function Estimation.” The Annals of Statistics.
Bertoin. 1996. Lévy Processes. Cambridge Tracts in Mathematics 121.
———. 2000. Subordinators, Lévy Processes with No Negative Jumps, and Branching Processes.
Borovkov, and Burq. 2001. Kendall’s Identity for the First Crossing Time Revisited.” Electronic Communications in Probability.
Çinlar, and Jacod. 1981. Representation of Semimartingale Markov Processes in Terms of Wiener Processes and Poisson Random Measures.” In Seminar on Stochastic Processes, 1981. Progress in Probability and Statistics.
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.
Grosswald. 1976. The Student t-Distribution of Any Degree of Freedom Is Infinitely Divisible.” Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete.
Ismail. 1977. Bessel Functions and the Infinite Divisibility of the Student \(t\)- Distribution.” The Annals of Probability.
Jacob, and Schilling. 2001. Lévy-Type Processes and Pseudodifferential Operators.” In Lévy Processes: Theory and Applications.
Jacod, and Protter. 1988. Time Reversal on Levy Processes.” The Annals of Probability.
Kallenberg. 2002. Foundations of Modern Probability. Probability and Its Applications.
Kyprianou. 2014. Fluctuations of Lévy Processes with Applications: Introductory Lectures. Universitext.
Leonenko, Meerschaert, Schilling, et al. 2014. Correlation Structure of Time-Changed Lévy Processes.” Communications in Applied and Industrial Mathematics.
Meerschaert, and Scheffler. 2008. Triangular Array Limits for Continuous Time Random Walks.” 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. A Unified Formulation of Gaussian Vs Sparse Stochastic Processes - Part II: Discrete-Domain Theory.” IEEE Transactions on Information Theory.
Unser, M., Tafti, and Sun. 2014. A Unified Formulation of Gaussian Vs Sparse Stochastic Processes—Part I: Continuous-Domain Theory.” IEEE Transactions on Information Theory.
Upadhye, and Barman. 2020. A Unified Approach to Stein’s Method for Stable Distributions.” arXiv:2004.07593 [Math].
Veillette, and Taqqu. 2010a. Using Differential Equations to Obtain Joint Moments of First-Passage Times of Increasing Lévy Processes.” Statistics & Probability Letters.
———. 2010b. Numerical Computation of First-Passage Times of Increasing Lévy Processes.” Methodology and Computing in Applied Probability.
Wasan. 1968. On an Inverse Gaussian Process.” Scandinavian Actuarial Journal.
Wolpert. 2021. Lecture Notes on Stationary Gamma Processes.” arXiv:2106.00087 [Math].
Xu, and Darve. 2019. Calibrating Multivariate Lévy Processes with Neural Networks.” arXiv:1812.08883 [Cs, Stat].