Markov processes, polynomial martingales and orthogonal polynomials

TL;DR

This paper explores stochastic processes with polynomial regression properties, establishing the existence of polynomial martingales that encode the process's distribution and characterizing conditions for properties like independent increments and Levy processes.

Contribution

It introduces a framework linking polynomial martingales to process properties, providing new conditions for independence, Levy processes, and harness structures.

Findings

Existence of polynomial martingales encoding process distribution

Conditions for processes to have independent increments and be Levy processes

Characterization of harness and quadratic harness properties

Abstract

We study general properties for the family of stochastic processes with polynomial regression property, that is that every conditional moment of the process is a polynomial. It turns out that then there exists a family of polynomial martingales $\left\{ M_{n}(X_{t},t)\right\}_{n\geq1}$ that contains complete information on the distribution (both marginal and transitional) of the process. We specify conditions expressed in terms of $M_{n}^{\prime}s$ under which a given process has independent increments and further is a Levy process, contains reversed martingales, is a harness or quadratic harness. We also give conditions under which some of these martingales are also reversed martingales.