Predictable projections of conformal stochastic integrals: an application to Hermite series and to Widder's representation

TL;DR

This paper explores predictable projections of conformal Brownian motion integrals, linking Hermite polynomials and analytic functions, and applies these findings to Widder's representation for Brownian martingales.

Contribution

It extends the connection between conformal Brownian motion powers and Hermite polynomials, and applies this to characterize moments in Widder's representation.

Findings

Established predictable projections for conformal stochastic integrals.

Connected Hermite series with analytic functions in $L^p$ spaces.

Characterized moments of Widder's measure for Brownian martingales.

Abstract

In this article, we study predictable projections of stochastic integrals with respect to the conformal Brownian motion, extending the connection between powers of the conformal Brownian motion and the corresponding Hermite polynomials. As a consequence of this result, we then investigate the relation between analytic functions and $L^p$-convergent series of Hermite polynomials. Finally, our results are applied to Widder's representation for a class of Brownian martingales, retrieving a characterization for the moments of Widder's measure.