On Markov processes with polynomial conditional moments

Abstract:

We study properties of a subclass of Markov processes that have all moments being 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 and with all moments existing, belong to this class. We give a 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 analyzed class have orthogonal polynomial martingales and further are harnesses or quadratic harnesses. Consequently we generalize some of the recent results on quadratic harnesses. We provide examples illustrating developed theory and also provide some interesting open questions. To make the paper interesting for a wider range of readers, we provide a short introduction formulated in the language of measures on the plane.