L\'evy processes, martingales, reversed martingales and orthogonal polynomials

Abstract

We study class of L\'{e}vy processes having distributions being indentifiable by moments. We define system of polynomial martingales \newline $\left\{ M_{n}(X_{t},t),\mathcal{F}_{\leq t}\right\} _{n\geq 1},$ where $% \mathcal{F}_{\leq t}$ is a suitable filtration defined below. We present several properties of these martingales. Among others we show that $% M_{1}(X_{t},t)/t$ 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 $n\geq 3$ $M_{n}(X_{t},t)$ is a reversed martingale (or orthogonal polynomial) only when the L\'{e}vy 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 $M_{i},$ $i\allowbreak =\allowbreak 1,\ldots ,n$ to be reversed martingales. We analyze case $% n\allowbreak =\allowbreak 2$ in detail listing all possible cases.