Mild and viscosity solutions to semilinear parabolic path-dependent PDEs
Probability
2018-11-16 v3 Analysis of PDEs
Optimization and Control
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.
Keywords
Cite
@article{arxiv.1611.08318,
title = {Mild and viscosity solutions to semilinear parabolic path-dependent PDEs},
author = {Alexander Kalinin and Alexander Schied},
journal= {arXiv preprint arXiv:1611.08318},
year = {2018}
}