Alex Karrila: Smoothness of martingale observables and generalized Feynman–Kac formulas

Abstract:

In this talk I will explicate connections between three closely related concepts:

1. parabolic linear PDEs of the form (1) \(G(x)f(x, t) + \frac{df}{dt}(x,t) +h(x,t)=0\), where \(G(x)\) is a positive semi-definite second-order operator in the spatial variables;
2. Feynman–Kac formulas, representing, in some cases, solutions to boundary-value problems for (1) as conditional expectations in terms of an Ito process \(X_t\), in the simplest case with the generator \(G(x) + \frac{d}{dt}\);
3. martingale observables, i.e., in the simplest case functions \(f(x,t)\) such that \(f(X_t,t)\) is a local martingale.

Our main result is that, assuming only the classic Hörmander criterion on the Ito process \(X\) (no ellipticity, no boundedness of the diffusion coefficients, no infinite life-time), all its martingale observables are smooth. As a consequence, we also obtain a comparatively general Feynman–Kac type formula, that provides smooth solutions to boundary-value problems for (1), while allowing for degenerate diffusions, unbounded coefficients, as well as a spatial boundary, under very mild assumptions on boundary regularity. Another application (and the speaker's original motivation) comes from Schramm–Loewner evolutions, for which the result makes a certain Girsanov transform martingale accessible via Ito calculus.

Joint work with Lauri Viitasaari (Aalto U., Finland).