Orthogonal Polynomials in Stochastic Integration Theory

Abstract

In this chapter we study orthogonal polynomials in the theory of stochastic integration. Some orthogonal polynomials in stochastic theory will play the role of ordinary monomials in deterministic theory. A consequence is that related polynomial transformations of stochastic processes involved will have very simple chaotic representations. In a different context, the orthogonalization of martingales, the coefficients of some other orthogonal polynomials will play an important role. We start with a reference to deterministic integration and then search for stochastic counterparts. We look at integration with respect to Brownian motion, the compensated Poisson process, and the binomial process. Next we develop a chaotic and predictable representation theory for general Lévy processes satisfying some weak condition on its Lévy measure. It is in this representation theory that we need the concept of strongly orthogonal martingales and orthogonal polynomials come into play. Examples include the Gamma process, where again the Laguerre polynomials turn up, the Pascal process, with again the Meixner polynomials, and the Meixner process, with as expected the Meixner-Pollaczek polynomials. When we look at combinations of pure-jump Lévy processes and Brownian motion, an inner product with some additional weight in zero plays a major role. We give an example with the Laguerre-type polynomials introduced by L. Littlejohn.