On the orthogonal polynomials associated with a Lévy process

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(x 1), 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 σ2δ 0(dx) + x2 v(dx), where σ2 is the variance of the Gaussian part of X and v its Lévy measure. These polynomials are the building blocks of a kind of chaotic representation of the square functional 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 Pn (x 1,....,xn) 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. © Institute ot Mathematical Statistics, 2008.