Algebraic polynomials and moments of stochastic integrals

TL;DR

This paper introduces an algebraic approach to estimate moments of stochastic integrals, leveraging polynomial root properties, and provides a new proof of a key inequality in stochastic analysis.

Contribution

It presents a novel algebraic method for moment estimates and offers a new proof of a fundamental inequality for stochastic integrals.

Findings

New algebraic method for moment estimates

Alternative proof of Burkholder-Davis-Gundy inequality

Potential for broader applications in stochastic analysis

Abstract

We propose an algebraic method for proving estimates on moments of stochastic integrals. The method uses qualitative properties of roots of algebraic polynomials from certain general classes. As an application, we give a new proof of a variation of the Burkholder-Davis-Gundy inequality for the case of stochastic integrals with respect to real locally square integrable martingales. Further possible applications and extensions of the method are outlined.