Algebraic polynomials and moments of stochastic integrals

Introduction. Connections between special algebraic polynomials and stochastic integrals have a long history (see Wiener [1938], Ito [1951]), and received considerable attention in stochastic analysis (Ikeda and Watanabe [1989], Carlen and Kree [1991], Borodin and Salminen [2002]). Fruitful applications of special polynomials have been found in the theory of Markov processes (Kendall [1959], Karlin and Mc-Gregor [1957]), financial mathematics (Schoutens [2000]), statistics (Diaconis and Zabell [1991]). The book Schoutens [2000] contains an extensive overview of this field of stochastic analysis and its applications. In this paper, we study a different type of applications of polynomials to stochastic integration. We show that not only properties of special systems of orthogonal polynomials can be used in stochastic analysis, but in fact that elementary properties of many general classes of polynomials lead to fruitful applications in stochastics.