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

✍ Scribed by Wim Schoutens

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 318 KB
• Volume: 99
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(98)00170-8

✦ Synopsis

In [11] an unusual connection between orthogonal polynomials and martingales has been studied. There, all orthogonal Sheffer polynomials, were linked to a unique Lfvy process, i.e., a continuous time stochastic process with stationary and independent increments. The connection between the polynomials and the Lfvy process is expressed by a martingale relation.

As an application of these martingales we show that the Charlier polynomials are the counterparts for Itf's integral with respect to a variant of the Poisson process of the customary powers.

A simpler 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 liD variables. (~) 1998 Elsevier Science B.V. All rights reserved.