An Exposition of Stochastic Integrals and Their Application to Linearization Coefficients

Abstract

Stochastic integration is introduced as a tool to address the problem of finding linearization coefficients. Stochastic, off-diagonal integration against a random spectral measure is defined and its properties discussed, followed by a proof that two formulations of Ito's Lemma are equivalent. Diagonals in R<\bold>n<\super> are defined, and their relationship to partitions of {1, ..., n} is discussed. The intuitive notion of a stochastic integral along a diagonal is formalized and calculated. The relationship between partitions and diagonals is then exploited to apply Moebius inversion to stochastic integrals over different diagonals. Diagonals along which stochastic integrals may be nonzero with positive probability are shown to correspond uniquely to diagrams. This correspondence is used to prove the Diagram Formula. Ito's Lemma and the Diagram Formula are then combined to calculate the linearization coefficients for Hermite Polynomials. Finally, future work is suggested that may allow other families of linearization coefficients to be calculated.