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March 2008 On the orthogonal polynomials associated with a Lévy process
Josep Lluís Solé, Frederic Utzet
Ann. Probab. 36(2): 765-795 (March 2008). DOI: 10.1214/07-AOP343

Abstract

Let X={Xt, t≥0} be a càdlàg Lévy process, centered, with moments of all orders. There are two families of orthogonal polynomials associated with X. On one hand, the Kailath–Segall formula gives the relationship between the iterated integrals and the variations of order n of X, and defines a family of polynomials P1(x1), P2(x1, x2), … that are orthogonal with respect to the joint law of the variations of X. On the other hand, we can construct a sequence of orthogonal polynomials pnσ(x) with respect to the measure σ2δ0(dx)+x2ν(dx), where σ2 is the variance of the Gaussian part of X and ν its Lévy measure. These polynomials are the building blocks of a kind of chaotic representation of the square functionals of the Lévy process proved by Nualart and Schoutens. The main objective of this work is to study the probabilistic properties and the relationship of the two families of polynomials. In particular, the Lévy processes such that the associated polynomials Pn(x1, …, xn) depend on a fixed number of variables are characterized. Also, we give a sequence of Lévy processes that converge in the Skorohod topology to X, such that all variations and iterated integrals of the sequence converge to the variations and iterated integrals of X.

Citation

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Josep Lluís Solé. Frederic Utzet. "On the orthogonal polynomials associated with a Lévy process." Ann. Probab. 36 (2) 765 - 795, March 2008. https://doi.org/10.1214/07-AOP343

Information

Published: March 2008
First available in Project Euclid: 29 February 2008

zbMATH: 1149.60028
MathSciNet: MR2393997
Digital Object Identifier: 10.1214/07-AOP343

Subjects:
Primary: 60G51
Secondary: 42C05

Rights: Copyright © 2008 Institute of Mathematical Statistics

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Vol.36 • No. 2 • March 2008
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