Mild and viscosity solutions to semilinear parabolic path-dependent PDEs
File(s)
Working paper
Author(s)
Kalinin, Alexander
Schied, Alexander
Type
Working Paper
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.
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.
Date Issued
2018-11-14
Citation
2018
Identifier
http://arxiv.org/abs/1611.08318v3
Subjects
math.PR
math.PR
math.AP
math.OC
60H10, 60H30, 60J68, 35D40, 35D30, 35K10, 93E20