# Variational estimates for martingale paraproducts

## TL;DR

This paper extends bilinear variational estimates to general martingales, generalizing previous results and connecting to martingale rough path inequalities, thus broadening the applicability of these estimates in stochastic analysis.

## Contribution

It proves that bilinear variational estimates hold for general martingales, extending prior work and linking to martingale rough path inequalities.

## Key findings

- Bilinear variational estimates are valid for general martingales.
- The results generalize previous estimates to a broader class of martingales.
- Connections are established with martingale rough path inequalities.

## Abstract

We show that bilinear variational estimates of Do, Muscalu, and Thiele (arXiv:1009.5187) remain valid for a pair of general martingales with respect to the same filtration. Our result can also be viewed as an off-diagonal generalization of the Burkholder--Davis--Gundy inequality for martingale rough paths by Chevyrev and Friz (arXiv:1704.08053).