$C^{ 0,1}$ -It{\^o} chain rules and generalized solutions of parabolic PDEs

TL;DR

This paper develops an Itô formula for processes with finite quadratic variation and derives a chain rule for solutions of parabolic PDEs, extending stochastic calculus tools to broader classes of functions and solutions.

Contribution

It introduces a new Itô formula for finite quadratic variation processes and a chain rule for quasi-strong solutions of parabolic PDEs, broadening stochastic calculus applications.

Findings

Established an Itô formula for processes with finite quadratic variation.

Derived an explicit chain rule for quasi-strong solutions of parabolic PDEs.

Extended stochastic calculus to functions with less regularity.

Abstract

In this paper we first establish an It\^o formula for a finite quadratic variation process $X$ expanding $f(t,X_t),$ when $f$ is of class $C^2$ in space and is absolutely continuous in time. Second, via a Fukushima-Dirichlet decomposition we obtain an explicit chain rule for $f(t,X_t)$, when $X$ is a continuous semimartingale and $f$ is a ``quasi-strong solution'' (in the sense of approximation of classical solutions) of a parabolic PDE.