One modification of the martingale transform and its applications to paraproducts and stochastic integrals

TL;DR

This paper introduces a modified martingale transform applicable to two martingales with different filtrations, establishing new $ ext{L}^p$ estimates and applying them to paraproducts and stochastic integrals, expanding the scope of stochastic analysis techniques.

Contribution

It proposes a novel variant of the martingale transform that works with different filtrations and derives new inequalities, with applications to paraproducts and stochastic integrals.

Findings

The modified martingale transform satisfies $ ext{L}^p$ estimates despite classical limitations.

Applications to general-dilation twisted paraproducts demonstrate the transform's utility.

Construction of stochastic integrals for non-semimartingale processes shows broader applicability.

Abstract

In this paper we introduce a variant of Burkholder's martingale transform associated with two martingales with respect to different filtrations. Even though the classical martingale techniques cannot be applied, we show that the discussed transformation still satisfies some expected $\mathrm{L}^p$ estimates. Then we apply the obtained inequalities to general-dilation twisted paraproducts, particular instances of which have already appeared in the literature. As another application we construct stochastic integrals $\int_{0}^{t}H_s d(X_s Y_s)$ associated with certain continuous-time martingales $(X_t)_{t\geq 0}$ and $(Y_t)_{t\geq 0}$. The process $(X_t Y_t)_{t\geq 0}$ is shown to be a "good integrator", although it is not necessarily a semimartingale, or even adapted to any convenient filtration.