Solving linear parabolic rough partial differential equations

TL;DR

This paper develops a regression Monte Carlo algorithm for solving linear rough partial differential equations driven by deterministic rough paths, with a rigorous convergence analysis and practical simulation results.

Contribution

It introduces a novel Monte Carlo method for rough PDEs driven by deterministic rough paths and provides a comprehensive convergence analysis.

Findings

The algorithm effectively approximates solutions to rough PDEs.

Convergence is proven under new bounds for solution derivatives.

Simulation results demonstrate practical applicability.

Abstract

We study linear rough partial differential equations in the setting of [Friz and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear parabolic partial differential equation driven by a deterministic rough path $\mathbf{W}$ of H\"older regularity $\alpha$ with $1/3 < \alpha \le 1/2$. Based on a stochastic representation of the solution of the rough partial differential equation, we propose a regression Monte Carlo algorithm for spatio-temporal approximation of the solution. We provide a full convergence analysis of the proposed approximation method which essentially relies on the new bounds for the higher order derivatives of the solution in space. Finally, a comprehensive simulation study showing the applicability of the proposed algorithm is presented.