# Lévy processes

Stochastic processes with independent increments, jump diffusion

May 29, 2017 — November 17, 2021

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

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 \(s\rightarrow t.\)

## 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.

## 8 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).

## 9 Gamma process

See Gamma processes.

## 10 Brownian motions

TBD

## 11 Subordinators

See Subordinators!…

## 12 Student-Lévy process

see *t*-processes.

## 13 References

*Notices of the AMS*.

*Lévy Processes and Stochastic Calculus*. Cambridge Studies in Advanced Mathematics 116.

*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.

*Stochastic Simulation: Algorithms and Analysis*.

*The Annals of Probability*.

*Proceedings of the 35th Conference on Winter Simulation: Driving Innovation*. WSC ’03.

*Advances in Applied Probability*.

*Lévy Driven Volatility Models*.

*The Annals of Applied Probability*.

*The Annals of Statistics*.

*Lévy Processes*. Cambridge Tracts in Mathematics 121.

*Electronic Communications in Probability*.

*Seminar on Stochastic Processes, 1981*. Progress in Probability and Statistics.

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

*Student’s t-Distribution and Related Stochastic Processes*. SpringerBriefs in Statistics.

*Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete*.

*The Annals of Probability*.

*Lévy Processes: Theory and Applications*.

*The Annals of Probability*.

*Foundations of Modern Probability*. Probability and Its Applications.

*Fluctuations of Lévy Processes with Applications: Introductory Lectures*. Universitext.

*Communications in Applied and Industrial Mathematics*.

*Stochastic Processes and Their Applications*.

*Simulation and the Monte Carlo Method*. Wiley series in probability and statistics.

*Lévy Processes and Infinitely Divisible Distributions*.

*An Introduction to Sparse Stochastic Processes*.

*IEEE Transactions on Information Theory*.

*IEEE Transactions on Information Theory*.

*arXiv:2004.07593 [Math]*.

*Statistics & Probability Letters*.

*Methodology and Computing in Applied Probability*.

*Scandinavian Actuarial Journal*.

*arXiv:2106.00087 [Math]*.

*arXiv:1812.08883 [Cs, Stat]*.