On the orthogonal polynomials associated with a Lévy process

Abstract

Let $X=\{X_t, t\ge0\}$ 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 $P_1(x_1), P_2(x_1,x_2),...$ that are orthogonal with respect to the joint law of the variations of $X$. On the other hand, we can construct a sequence of orthogonal polynomials $p^σ_n(x)$ with respect to the measure $σ^2δ_0(dx)+x^2 ν(dx)$, where $σ^2$ 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 $P_n(x_1,...,x_n)$ 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 $X$.