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

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.