The $C^{0,1}$ It\^o-Ventzell formula for weak Dirichlet processes

TL;DR

This paper extends the Itô-Ventzell formula to continuous weak Dirichlet processes with $C^{0,1}$ regularity, enabling new applications in SPDEs, quadratic variation formulas, and financial mathematics.

Contribution

It introduces a generalized Itô-Ventzell formula for weak Dirichlet processes with $C^{0,1}$ regularity, broadening its applicability.

Findings

Provides a representation for solutions of time-dependent elliptic SPDEs

Derives new formulas for quadratic variations

Relaxes assumptions in financial mathematics models

Abstract

This paper proves an extension of the It\^o-Ventzell formula that applies to stochastic flows in $C^{0,1}$ for continuous weak Dirichlet processes. We apply this theorem, for example, to give a representation result for strong solutions of time-dependent elliptic SPDEs, to derive formulas for quadratic variations, and to relax assumptions in a financial mathematics context.