Some new Bellman functions and subordination by orthogonal martingales in $L^{p}, 1<p\le 2$

TL;DR

This paper develops new Bellman functions using Monge-Ampère equations to estimate the $L^p$ norms of orthogonally subordinated martingales in a two-dimensional Brownian motion setting, for $1<p extless=2$.

Contribution

It introduces several novel Bellman functions, distinct from Burkholder's, to provide $L^p$ norm estimates for orthogonal martingales with subordination, expanding the analytical toolkit.

Findings

Constructed new Bellman functions via Monge-Ampère equations.

Derived $L^p$ norm estimates for orthogonal subordinated martingales.

Extended the theory to the case $1<p extless=2$ with explicit bounds.

Abstract

Given two martingales on the filtration generated by two dimensional Brownian motion, we want to estimate the $L^p$ norm of the subordinated one if we have some extra orthogonality property available. We construct several new Bellman functions, very different from Burkholder's function, and using them give an estimate of $L^p$ norm of a subordinated martingale, if the dominating martingale is orthogonal and $1<p\le 2$. We use Monge--Ampere equation to construct these Bellman functions.