Integration with respect to local time and Ito's formula for smooth nondegenerate martingales

TL;DR

This paper extends Itô's formula to nondegenerate Brownian martingales with functions having locally integrable derivatives, expressing the correction term via local time integrals.

Contribution

It introduces a new version of Itô's formula for nondegenerate martingales involving local time, broadening the class of functions and processes it applies to.

Findings

Itô's formula is valid for nondegenerate Brownian martingales with certain functions.

The correction term in Itô's formula can be represented as an integral over local time.

The approach generalizes classical Itô calculus to more complex stochastic processes.

Abstract

We show an It\^ o's formula for nondegenerate Brownian martingales $X_t=\int_0^t u_s dW_s$ and functions $F(x,t)$ with locally integrable derivatives in $t$ and $x$. We prove that one can express the additional term in It\^o's s formula as an integral over space and time with respect to local time.