Bellman function method for general operators on martingales: arbitrary regular filtrations

TL;DR

This paper extends the Bellman function method to general operators on martingales, demonstrating its applicability to arbitrary regular filtrations beyond the previously assumed special filtrations.

Contribution

It generalizes the Bellman function approach for vector-valued martingales to any regular filtration, removing previous restrictions.

Findings

Bellman function method applies to all regular filtrations

Extension of Gundy's extrapolation theorem to general filtrations

Broader applicability of martingale inequalities

Abstract

It has been recently shown that the Bellman function method can be applied in the general context of Gundy's extrapolation theorem for vector-valued martingales. But the additional assumption has been made that martingales are adapted to a certain special filtration. Here it is shown that those results can be extended to any regular filtration.