Askey--Wilson polynomials, quadratic harnesses and martingales

TL;DR

This paper constructs Markov processes using Askey--Wilson polynomials, revealing their orthogonality measures and demonstrating their role as martingale polynomials with specific regression and variance properties.

Contribution

It introduces a novel connection between Askey--Wilson polynomials and Markov processes with quadratic harness properties, expanding their applications.

Findings

Markov processes with linear regressions and quadratic conditional variances constructed

Askey--Wilson polynomials serve as orthogonal martingale polynomials for these processes

New insights into the structure of quadratic harnesses and their polynomial martingales

Abstract

We use orthogonality measures of Askey--Wilson polynomials to construct Markov processes with linear regressions and quadratic conditional variances. Askey--Wilson polynomials are orthogonal martingale polynomials for these processes.