New isometry of Krall-Laguerre orthogonal polynomials in martingale spaces

TL;DR

This paper introduces a new isometry linking orthogonal polynomials derived from Uvarov transformations of Laguerre weights to Teugels martingales, expanding the class of orthogonal martingales for Lévy processes, especially the Gamma process.

Contribution

It presents a novel isometry between polynomial spaces and martingale spaces using Uvarov-transformed Laguerre weights, extending orthogonal martingale constructions.

Findings

Infinite sets of strongly orthogonal martingales for c in (-∞,0)

New isometry between polynomial and martingale spaces

Application to Gamma process

Abstract

Sets of orthogonal martingales are importants because they can be used as stochastic integrators in a kind of chaotic representation property, see [20]. In this paper, we revisited the problem studied by W. Schoutens in [21], investigating how an inner product derived from an Uvarov transformation of the Laguerre weight function is used in the orthogonalization procedure of a sequence of martingales related to a certain L\'evy process, called Teugels Martingales. Since the Uvarov transformation depends by a c<0, we are able to provide infinite sets of strongly orthogonal martingales, each one for every c in (-infty,0). In a similar fashion of [21], we introduce a suitable isometry between the space of polynomials and the space of linear combinations of Teugels martingales as well as the general orthogonalization procedure. Finally, the new construction is applied to the Gamma process.