Generalized Feynman-Kac Formula under volatility uncertainty

TL;DR

This paper extends the Feynman-Kac formula to settings with volatility uncertainty, showing that the G-conditional expectation of discounted payoffs solves a nonlinear PDE as a viscosity solution, enabling efficient computation.

Contribution

It generalizes the Feynman-Kac formula under volatility uncertainty without requiring Lipschitz continuity, broadening its applicability.

Findings

G-conditional expectation is a viscosity solution of a nonlinear PDE.

Provides a computationally efficient method for calculating sublinear expectations.

Extends previous results by relaxing Lipschitz continuity assumptions.

Abstract

In this paper we provide a generalization of a Feynmac-Kac formula under volatility uncertainty in presence of a linear term in the PDE due to discounting. We state our result under different hypothesis with respect to the derivation given by Hu, Ji, Peng and Song (Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion, Stochastic Processes and their Application, 124 (2)), where the Lipschitz continuity of some functionals is assumed which is not necessarily satisfied in our setting. In particular, we show that the $G$-conditional expectation of a discounted payoff is a viscosity solution of a nonlinear PDE. In applications, this permits to calculate such a sublinear expectation in a computationally efficient way.