Representation Formula for Viscosity Solutions to a class of Nonlinear Parabolic PDEs

TL;DR

This paper derives a nonlinear extension of the Feynman--Kac formula, providing a representation for viscosity solutions to certain nonlinear parabolic PDEs using backward stochastic differential equations.

Contribution

It introduces a novel representation formula for viscosity solutions of nonlinear second order parabolic PDEs via a dynamic programming principle and backward SDEs.

Findings

Provides a new representation formula for viscosity solutions

Extends the Feynman--Kac formula to nonlinear PDEs

Utilizes backward stochastic differential equations theory

Abstract

We provide a representation formula for viscosity solutions to a class of nonlinear second order parabolic PDEs given as a sup--envelope function. This is done through a dynamic programming principle derived from Denis, Hu, Peng (2010). The formula can be seen as a nonlinear extension of the Feynman--Kac formula and is based on the backward stochastic differential equations theory.