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

TL;DR

This paper investigates polynomial martingales associated with Lévy processes, revealing that higher-order polynomial martingales are reversed martingales only for Gaussian processes, and explores conditions for linear combinations to be reversed martingales.

Contribution

It characterizes when polynomial martingales of Lévy processes are reversed martingales, showing this occurs only for Gaussian processes, and analyzes linear combinations of such martingales.

Findings

Higher-order polynomial martingales are reversed martingales only for Gaussian processes.

The process $M_1(t)/t$ is a reversed martingale and a harness.

Conditions for linear combinations of martingales to be reversed martingales are established.

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.