Markov processes, polynomial martingales and orthogonal polynomials

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.