Normal martingales and polynomial families

Tom 84 / 2004

H. Hammouch Annales Polonici Mathematici 84 (2004), 93-102 MSC: 33C45, 60G44, 60G48, 60H05, 60J55. DOI: 10.4064/ap84-2-1


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.


  • H. HammouchUniversité d'Evry Val d'Essonne
    Laboratoire d'Analyse et Probabilités
    BT. Maupertuis, rue du Père Jarlan
    91025 Evry Val d'Essonne, France

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