Pathwise It\^o Calculus for Rough Paths and Rough PDEs with Path Dependent Coefficients

TL;DR

This paper develops a pathwise Itô calculus framework for rough paths and rough PDEs with path-dependent coefficients, enabling analysis similar to classical stochastic calculus and addressing less regular coefficients.

Contribution

It introduces path derivatives for controlled paths in rough path theory, extending the calculus to non-geometric rough paths and rough PDEs with path-dependent, less regular coefficients.

Findings

Established a theory for rough integration with path derivatives.

Applied the framework to rough PDEs with path-dependent coefficients.

Provided tools for viscosity solutions of stochastic PDEs.

Abstract

This paper introduces the path derivatives, in the spirit of Dupire's functional It\^o calculus, for the controlled paths in the rough path theory with possibly non-geometric rough paths. The theory allows us to deal with rough integration and rough PDEs in the same manner as standard stochastic calculus. We next study rough PDEs with coefficients depending on the rough path itself, which corresponds to stochastic PDEs with random coefficients. Such coefficients is less regular in the time variable and is not covered in the existing literature. The results are useful for studying viscosity solutions of stochastic PDEs.