Normal martingales and polynomial families

H. Hammouch¹

¹ Université d'Evry Val d'Essonne Laboratoire d'Analyse et Probabilités BT. Maupertuis, rue du Père Jarlan 91025 Evry Val d'Essonne, France

Annales Polonici Mathematici, Tome 84 (2004) no. 2, pp. 93-102

Résumé

Wiener and compensated Poisson processes, as normal martingales, are associated to classical sequences of polynomials, namely Hermite polynomials for the first one and Charlier polynomials for the second. The problem studied in this paper is to find if there exist other normal martingales which are associated to classical sequences of polynomials. Privault, Solé and Vives [5] solved this problem via the quantum Kabanov formula under some assumptions on the normal martingales considered. We solve the problem without these assumptions and we give a complete study of this subject in Section 2. In Section 3 we introduce the notion of algebraic process and we prove that Azéma martingales are infinitely algebraic.

Détail

DOI : 10.4064/ap84-2-1

Keywords: wiener compensated poisson processes normal martingales associated classical sequences polynomials namely hermite polynomials first charlier polynomials second problem studied paper there exist other normal martingales which associated classical sequences polynomials privault sol vives solved problem via quantum kabanov formula under assumptions normal martingales considered solve problem without these assumptions complete study subject section section introduce notion algebraic process prove martingales infinitely algebraic