Multilevel estimation of expected exit times and other functionals of stopped diffusions

TL;DR

This paper introduces a multilevel Monte Carlo method for efficiently estimating mean exit times and related functionals of multi-dimensional Brownian diffusions, with proven complexity bounds.

Contribution

It develops a novel multilevel Monte Carlo approach for high-dimensional diffusion functionals, improving computational efficiency for PDE-related stochastic estimates.

Findings

Complexity of $O( ext{epsilon}^{-2} | ext{log epsilon}|^3)$ for root-mean-square error

Effective estimation of high-dimensional PDE solutions via stochastic methods

Theoretical analysis confirming the method's efficiency

Abstract

This paper proposes and analyses a new multilevel Monte Carlo method for the estimation of mean exit times for multi-dimensional Brownian diffusions, and associated functionals which correspond to solutions to high-dimensional parabolic PDEs through the Feynman-Kac formula. In particular, it is proved that the complexity to achieve an $\varepsilon$ root-mean-square error is $O(\varepsilon^{-2}\, |\!\log \varepsilon|^3)$.