A Maximal Inequality of the 2D Young Integral based on Bivariations

TL;DR

This paper establishes a new maximal inequality for the 2D Young integral based on bivariation norms, extending classical results and enabling strong approximations related to Brownian local time.

Contribution

It introduces a novel maximal inequality for 2D Young integrals using bivariation norms, complementing existing joint variation inequalities.

Findings

Derived a maximal inequality in terms of bivariation norms

Extended Young's integral theory to two-parameter settings

Applied results to approximate 2D Young integrals with Brownian local time

Abstract

In this note, we establish a novel maximal inequality of the 2D Young integral $\int_a^b\int_c^d FdG$ in terms of the $(p,q)$-bivariation norms of the section functions $x\mapsto F(x,y)$ and $y\mapsto F(x,y)$ where $G:[a,b]\times [c,d]\rightarrow \mathbb{R}$ is a controlled path satisfying finite $(p,q)$-variation conditions. The proof is reminiscent from the Young's original ideas \cite{young1} in defining two-parameter integrals in terms of $(p,q)$-finite bivariations. Our result complements the standard maximal inequality established by Towghi \cite{towghi1} in terms of joint variations. We apply the maximal inequality to get novel strong approximations for 2D Young integrals w.r.t the Brownian local time in terms of number of upcrossings of a given approximating random walk.