Semilinear Kolmogorov equations on the space of continuous functions via BSDEs

TL;DR

This paper studies semilinear Kolmogorov equations on infinite-dimensional spaces of continuous functions, using BSDEs to establish existence and regularity of solutions, with applications to control problems.

Contribution

It introduces a novel approach to analyze infinite-dimensional PDEs via BSDEs, addressing regularity issues without smoothing properties.

Findings

Established existence of solutions via BSDE representation

Developed regularity theory for solutions in infinite dimensions

Applied results to Hamilton-Jacobi-Bellman equations in control

Abstract

We deal with a class of semilinear parabolic PDEs on the space of continuous functions that arise, for example, as Kolmogorov equations associated to the infinite-dimensional lifting of path-dependent SDEs. We investigate existence of smooth solutions through their representation via forward-backward stochastic systems, for which we provide the necessary regularity theory. Because of the lack of smoothing properties of the parabolic operators at hand, solutions in general will only share the same regularity as the coefficients of the equation. To conclude we exhibit an application to Hamilton-Jacobi-Bellman equations associated to suitable optimal control problems.