Error analysis of truncated expansion solutions to high-dimensional parabolic PDEs

TL;DR

This paper analyzes an expansion method for high-dimensional parabolic PDEs, deriving sharp error bounds and demonstrating convergence in option pricing applications.

Contribution

It provides the first rigorous error bounds for the method in the constant coefficient case and characterizes their applicability to non-smooth options.

Findings

Error bounds are sharp and theoretically justified.

Numerical results confirm convergence speed matches predictions.

Method is effective for problems with few dominant principal components.

Abstract

We study an expansion method for high-dimensional parabolic PDEs which constructs accurate approximate solutions by decomposition into solutions to lower-dimensional PDEs, and which is particularly effective if there are a low number of dominant principal components. The focus of the present article is the derivation of sharp error bounds for the constant coefficient case and a first and second order approximation. We give a precise characterisation when these bounds hold for (non-smooth) option pricing applications and provide numerical results demonstrating that the practically observed convergence speed is in agreement with the theoretical predictions.