On stationary Markov processes with polynomial conditional moments

Abstract

We study a class of stationary Markov processes with marginal distributions identifiable by moments such that every conditional moment of degree say $m$ is a polynomial of degree at most $m\;\text{.}\;$ We show that then under some additional, natural technical assumption there exists a family of orthogonal polynomial martingales. More precisely we show that such a family of processes is completely characterized by the sequence $\{(\alpha_{n} ,p_{n})\}_{n \geq 0}$ where $\alpha_{n}^{ \prime } s$ are some positive reals while $p_{n}^{ \prime } s$ are some monic orthogonal polynomials. Paper of Bakry\&Mazet(2003) assures that under some additional mild technical conditions each such sequence generates some stationary Markov process with polynomial regression. We single out two important subclasses of the considered class of Markov processes. The class of harnesses that we characterize completely. The second one constitutes of the processes that have independent regression property and are stationary. Processes with independent regression property so to say generalize ordinary Ornstein-Uhlenbeck (OU) processes or can also be understood as time scale transformations of L{\'e}vy processes. We list several properties of these processes. In particular we show that if these process are time scale transforms of L{\'e}vy processes then they are not stationary unless we deal with classical OU- process. Conversely, time scale transformations of stationary processes with independent regression property are not L{\'e}vy unless we deal with classical OU process.