Feynman-Kac representation of fully nonlinear PDEs and applications

TL;DR

This paper reviews the extension of the Feynman-Kac formula to fully nonlinear PDEs via backward stochastic differential equations, highlighting its theoretical foundations and applications in numerical methods, finance, and stochastic control.

Contribution

It synthesizes recent developments in nonlinear Feynman-Kac formulas and discusses their implications for solving complex PDEs and related applications.

Findings

Backward stochastic differential equations provide a probabilistic representation of nonlinear PDEs.

These representations enable new numerical methods for solving complex PDEs.

Applications include stochastic control and financial modeling under uncertainty.

Abstract

The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion. It gives then a method for solving linear PDEs by Monte Carlo simulations of random processes. The extension to (fully)nonlinear PDEs led in the recent years to important developments in stochastic analysis and the emergence of the theory of backward stochastic differential equations (BSDEs), which can be viewed as nonlinear Feynman-Kac formulas. We review in this paper the main ideas and results in this area, and present implications of these probabilistic representations for the numerical resolution of nonlinear PDEs, together with some applications to stochastic control problems and model uncertainty in finance.