On the orthogonal polynomials associated with Levy process

• 成果类型: Article
• 署名作者: Sole, Josep Lluis; Utzet, Frederic
• 署名单位: Autonomous University of Barcelona
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/07-AOP343
• 发表日期: 2008
• 页码: 765-795
• 关键词: integrals

摘要

Let X = {X-t, t >= 0} be a cadlag Levy 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 P-1(x(t)), P-2(x(1), x(2)).... that are orthogonal with respect to the joint law of the variations of X. On the other hand, we can F construct a sequence of orthogonal polynomials p(n)(sigma)(x) with respect to the measure (sigma(2)delta(0)(dx) +x(2) v(dx), where sigma(2) is the variance of the Gaussian part of X and v its Levy measure. These polynomials are the building blocks of a kind of chaotic representation of the square functionals of the Levy 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 Levy processes such that the associated polynomials P-n (x(1),...,x(n)) depend on a fixed number of variables are characterized. Also, we give a sequence of Levy 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.