Stochastic integration with respect to local time of the Brownian sheet and regularising properties of Brownian sheet paths

TL;DR

This paper extends stochastic local time integration to the Brownian sheet, establishing a two-parameter Itô formula and regularity bounds for operators along Brownian sheet paths, advancing stochastic calculus in multiple parameters.

Contribution

It introduces a generalized stochastic local time-space calculus for the Brownian sheet, including a two-parameter Itô formula and inequalities for regularity analysis.

Findings

Established a stochastic local time-space calculus for the Brownian sheet.

Proved a generalized two-parameter Itô formula.

Derived Davie type inequalities for the Brownian sheet.

Abstract

In this work, we generalise the stochastic local time space integration introduced in \cite{Ei00} to the case of Brownian sheet. %We develop a stochastic local time-space calculus with respect to the Brownian sheet. This allows us to prove a generalised two-parameter It\^o formula and derive Davie type inequalities for the Brownian sheet. Such estimates are useful to obtain regularity bounds for some averaging type operators along Brownian sheet curves.