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

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 $(X_t{)}_{t\ge 0}$ and $(Y_t{)}_{t\ge 0}$. The process $(X_t{Y}_t{)}_{t\ge 0}$ is shown to be a "good integrator", although it is not necessarily a semimartingale, or even adapted to any convenient filtration.

18 pages; v3: accepted for publication, references updated, research support acknowledged

References in corpus (2)

• Bellman function technique for multilinear estimates and an application to generalized paraproducts
• Sobolev norm estimates for a class of bilinear multipliers

Cited by in corpus (6)

• Norm-variation of ergodic averages with respect to two commuting transformations
• Trilinear smoothing inequalities and a variant of the triangular Hilbert transform
• Density theorems for anisotropic point configurations
• Convergence of ergodic-martingale paraproducts
• T(1) theorem for dyadic singular integral forms associated with hypergraphs
• Large dilates of hypercube graphs in the plane