Lévy-Sheffer and IID-Sheffer polynomials with applications to stochastic integrals

In [11] an unusual connection between orthogonal polynomials and martingales has been studied. There, all orthogonal Sheffer polynomials, were linked to a unique Lévy process, i.e., a continuous time stochastic process with stationary and independent increments. The connection between the polynomials and the Lévy process is expressed by a martingale relation. application of these martingales we show that the Charlier polynomials are the counterparts for Itôʹs integral with respect to a variant of the Poisson process of the customary powers. ler approach is possible when trying to obtain discrete time martingales from a Sheffer set. We illustrate this by for example relating Krawtchouk polynomials to partial sums of Bernoulli IID variables.