On It\^{o}'s formula for elliptic diffusion processes

TL;DR

This paper extends Itô's formula for elliptic diffusion processes by reformulating it using local times instead of quadratic covariation, relaxing some regularity conditions on the function involved.

Contribution

It introduces a new representation of Itô's formula for elliptic diffusions using local times, reducing regularity assumptions on the function's derivatives.

Findings

Reformulation of Itô's formula with local times

Relaxation of continuity conditions on derivatives

Application to elliptic diffusion processes

Abstract

Bardina and Jolis [Stochastic process. Appl. 69 (1997) 83--109] prove an extension of It\^{o}'s formula for $F(X_t,t)$, where $F(x,t)$ has a locally square-integrable derivative in $x$ that satisfies a mild continuity condition in $t$ and $X$ is a one-dimensional diffusion process such that the law of $X_t$ has a density satisfying certain properties. This formula was expressed using quadratic covariation. Following the ideas of Eisenbaum [Potential Anal. 13 (2000) 303--328] concerning Brownian motion, we show that one can re-express this formula using integration over space and time with respect to local times in place of quadratic covariation. We also show that when the function $F$ has a locally integrable derivative in $t$, we can avoid the mild continuity condition in $t$ for the derivative of $F$ in $x$.