Multi-level Picard approximations of high-dimensional semilinear parabolic differential equations with gradient-dependent nonlinearities

TL;DR

This paper introduces a multi-level Picard approximation method for high-dimensional semilinear parabolic PDEs with gradient-dependent nonlinearities, achieving polynomial complexity in dimension and accuracy.

Contribution

It provides a novel approximation scheme with proven polynomial computational complexity for complex high-dimensional PDEs with gradient dependencies.

Findings

Polynomial complexity in dimension and accuracy for the approximation

Applicable to high-dimensional PDEs in financial modeling

Theoretical proof of convergence and efficiency

Abstract

Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) have a wide range of applications. In particular, high-dimensional PDEs with gradient-dependent nonlinearities appear often in the state-of-the-art pricing and hedging of financial derivatives. In this article we prove that semilinear heat equations with gradient-dependent nonlinearities can be approximated under suitable assumptions with computational complexity that grows polynomially both in the dimension and the reciprocal of the accuracy.