Mathematica Eterna
Open Access

Abstract

LÈvy processes, martingales, reversed martingales and orthogonal polynomials

Pawe˜J. Szab˜owski

We study class of LÈvy processes having distributions being indentiÖable by moments. We deÖne system of polynomial martingales fMn(Xt ; t); Ftgn1 ; where Ft is a suitable Öltration deÖned below. We present several properties of these martingales. Among others we show that M1(Xt ; t)=t is a reversed martingale as well as a harness. Main results of the paper concern the question if martingale say Mi multiplied by suitable determinstic function i (t) is a reversed martingale. We show that for n  3 Mn(Xt ; t) is a reversed martingale (or orthogonal polynomial) only when the LÈvy 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 coe¢ cients depending on t) of martingales Mi ; i = 1; : : : ; n to be reversed martingales. We analyze case n = 2 in detail listing all possible cases.