Smoothness of martingale observables and generalized Feynman-Kac formulas

TL;DR

This paper proves the smoothness of martingale observables for certain Itô processes and derives a generalized Feynman-Kac formula, enabling smooth solutions to PDEs with degenerate diffusions and boundary stopping.

Contribution

It introduces a generalized Feynman-Kac formula under the H"ormander criterion, extending applicability to degenerate diffusions and boundary conditions.

Findings

Martingale observables are smooth under H"ormander criterion.

Generalized Feynman-Kac formula provides smooth PDE solutions with degenerate diffusions.

Application to Schramm-Loewner evolutions via Girsanov transform martingales.

Abstract

We prove that, under the H\"ormander criterion on an It\^{o} process, all its martingale observables are smooth. As a consequence, we also obtain a generalized Feynman-Kac formula providing smooth solutions to certain PDE boundary-value problems, while allowing for degenerate diffusions as well as boundary stopping (under very mild boundary regularity assumptions). We also highlight an application to a question posed on Schramm-Loewner evolutions, by making certain Girsanov transform martingales accessible via It\^{o} calculus.