A fully nonlinear Feynman-Kac formula with derivatives of arbitrary orders

TL;DR

This paper introduces a novel Monte Carlo algorithm that extends the Feynman-Kac formula to solve fully nonlinear parabolic PDEs with derivatives of any order, accommodating complex nonlinearities.

Contribution

It develops a new algorithm using random trees to handle nonlinearities and derivatives of arbitrary order, surpassing existing methods' capabilities.

Findings

Applicable to complex nonlinear PDEs with high-order derivatives

Provides a Monte Carlo implementation for practical use

Handles non-polynomial nonlinearities beyond standard methods

Abstract

We present an algorithm for the numerical solution of nonlinear parabolic partial differential equations. This algorithm extends the classical Feynman-Kac formula to fully nonlinear partial differential equations, by using random trees that carry information on nonlinearities on their branches. It applies to functional, non-polynomial nonlinearities that are not treated by standard branching arguments, and deals with derivative terms of arbitrary orders. A Monte Carlo numerical implementation is provided.