Mild and viscosity solutions to semilinear parabolic path-dependent PDEs

TL;DR

This paper demonstrates that mild solutions to semilinear parabolic path-dependent PDEs are also viscosity solutions, providing a link between two weak solution concepts and establishing existence results under weak conditions.

Contribution

It proves that mild solutions are viscosity solutions for semilinear parabolic PPDEs, enabling broader existence results and applications in stochastic control.

Findings

Mild solutions are also viscosity solutions.

Existence of viscosity solutions under weak conditions.

Application to stochastic optimal control problems.

Abstract

We study and compare two concepts for weak solutions to semilinear parabolic path-dependent partial differential equations (PPDEs). The first is that of mild solutions as it appears, e.g., in the log-Laplace functionals of historical superprocesses. The aim of this paper is to show that mild solutions are also solutions in a viscosity sense. This result is motivated by the fact that mild solutions can provide value functions and optimal strategies for problems of stochastic optimal control. Since unique mild solutions exist under weak conditions, we obtain as a corollary a general existence result for viscosity solutions to semiilinear parabolic PPDEs.