Monte Carlo estimation of the solution of fractional partial differential equations

TL;DR

This paper introduces Monte Carlo methods for numerically solving fractional PDEs, providing error bounds, fluctuation estimates, confidence intervals, and convergence rates, supported by numerical experiments.

Contribution

It develops a probabilistic Monte Carlo framework for fractional PDEs, including error analysis, fluctuation estimation, and convergence rate results, which are novel in this context.

Findings

Error bounds between exact and approximate solutions

Central limit theorem-based fluctuation estimates

Convergence rates with Berry-Esseen bounds

Abstract

The paper is devoted to the numerical solutions of fractional PDEs based on its probabilistic interpretation, that is, we construct approximate solutions via certain Monte Carlo simulations. The main results represent the upper bound of errors between the exact solution and the Monte Carlo approximation, the estimate of the fluctuation via the appropriate central limit theorem(CLT) and the construction of confidence intervals. Moreover, we provide rates of convergence in the CLT via Berry-Esseen type bounds. Concrete numerical computations and illustrations are included.