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 ${\left\{M_n(X_t,t)\right\}}_{n\ge 1}$ 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.

References in corpus (3)

• On a class of free Levy laws related to a regression problem
• Conditional moments of q-Meixner processes
• A few remarks on quadratic harnesses

Cited by in corpus (3)

• On positivity of orthogonal series and its applications in probability
• On the families of polynomials forming a part of the Askey--Wilson scheme and their probabilistic applications
• Stationary, Markov, stochastic processes with polynomial conditional moments and continuous paths