On Markov processes with polynomials conditional moments

TL;DR

This paper investigates a special class of Markov processes characterized by polynomial conditional moments, providing their properties, examples, and conditions for orthogonal polynomial martingales, harnesses, and quadratic harnesses.

Contribution

It characterizes Markov processes with polynomial conditional moments, explores their properties, and identifies conditions for orthogonal polynomial martingales and harnesses.

Findings

Processes with independent increments are included.

Conditions for orthogonal polynomial martingales are established.

Examples and open questions are provided.

Abstract

We study properties of a subclass of Markov processes that have all moments that are continuous functions of the time parameter and more importantly are characterized by the property that say their $n-$th conditional moment given the past is also a polynomial of degree not exceeding $n.$ Of course all processes with independent increments with all moments belong to this class. We give characterization of them within the studied class. We indicate other examples of such process. Besides we indicate families of polynomials that have the property of constituting martingales. We also study conditions under which processes from the analysed class have orthogonal polynomial martingales and further are harnesses or quadratic harnesses. We provide examples illustrating developed theory and also provide some interesting open questions. To make paper interesting for a wider range of readers we provide short introduction formulated in the language of measures on the plane.