Smoothness of martingale observables and generalized Feynman-Kac formulas

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.