Differential subordination under change of law

TL;DR

This paper establishes optimal L^2 bounds for Hilbert space valued martingales under a change of law, extending previous discrete and scalar results to the continuous-time setting with explicit Bellman functions.

Contribution

It proves the first optimal bounds for continuous-time martingales under change of law, including explicit Bellman functions for weighted estimates with jumps.

Findings

Proved optimal L^2 bounds for Hilbert space martingales under change of law.

Extended discrete scalar results to continuous-in-time martingales.

Derived explicit Bellman functions for weighted subordinate martingale estimates.

Abstract

We prove optimal ${L}^2$ bounds for a pair of Hilbert space valued differentially subordinate martingales under a change of law. The change of law is given by a process called a weight and sharpness in this context refers to the optimal growth with respect to the characteristic of the weight. The pair of martingales are adapted, uniformly integrable, and cadlag. Differential subordination is in the sense of Burkholder, defined through the use of the square bracket. In the scalar dyadic setting with underlying Lebesgue measure, this was proved by Wittwer, where homogeneity was heavily used. Recent progress by Thiele-Treil-Volberg and Lacey, independently, resloved the so-called non-homogenous case of discrete in time filtrations with two completely different proofs. The general case for continuous-in-time filtrations remained open and is adressed here. As a by-product, we give the needed explicit expression of a Bellman function of four variables for the weighted estimate of subordinate martingales with jumps.