Lévy processes, martingales, reversed martingales and orthogonal polynomials

Abstract

We study class of Lévy processes having distributions being indentifiable by moments. We define system of polynomial martingales \newline ${\left\{M_n(X_t,t),{\mathcal{F}}_{\le t}\right\}}_{n\ge 1},$ where is a suitable filtration defined below. We present several properties of these martingales. Among others we show that is a reversed martingale as well as a harness. Main results of the paper concern the question if martingale say $M_i$ multiplied by suitable determinstic function ${\mu }_i(t)$ is a reversed martingale. We show that for $n\ge 3$ $M_n(X_t,t)$ is a reversed martingale (or orthogonal polynomial) only when the Lévy process in question is Gaussian (i.e. is a Wiener process). We study also a more general question if there are chances for a linear combination (with coefficients depending on $t)$ of martingales $M_i,$ $i=1,\dots ,n$ to be reversed martingales. We analyze case in detail listing all possible cases.

17 pages

References in corpus (3)

• On a class of free Levy laws related to a regression problem
• $q-$Wiener ($\alpha ,q)-$ Ornstein-Uhlenbeck processes. A generalization of known processes
• On the orthogonal polynomials associated with a Lévy process