Umbral Calculus, Martingales, and Associated Polynomials

Abstract

In this paper we treat with a relationship between normal martingales [Emery, M. On the Azéma Martingales; Sem. Proba. XXIII, Lec. Notes in Maths 1372; Springer, 1989; 66–87] and some special polynomial families. This problem was studied by Privault et al. [Privault, N.; Sole, J.L.; Vives, J. Chaotic Kabanov Formula for the Azéma Martingales; The Bernoulli Society for Mathematical Statistics and Probability, 2000; Vol. 6, No. 4], they showed under some assumptions which we give later, that Wiener and compensated Poisson processes are the only normal martingales which has an associated family of polynomials according to the Definition 3 in Privault et al. [Privault, N.; Sole, J.L.; Vives, J. Chaotic Kabanov Formula for the Azéma Martingales; The Bernoulli Society for Mathematical Statistics and Probability, 2000; Vol. 6, No. 4]. We study this problem without the authors assumptions in considering all normal martingales, we show more precisely that Wiener process is characterized by Wick polynomials and both Wiener and compensated Poisson processes are characterized by cross polynomials (said also of convolution type). We use here the Umbral Calculus theory [Roman, S.M. The theory of umbral calculus I. J. Math. Anal. Appl. 1982, 87, 58–115] as a new and a fast tool to solve a martingale problem.