Adaptive Multilevel Monte Carlo for Probabilities

TL;DR

This paper introduces an adaptive multilevel Monte Carlo method that efficiently estimates probabilities involving discontinuous functionals of hierarchical approximations, improving computational complexity over standard approaches.

Contribution

The paper develops a general adaptive framework that achieves optimal MLMC complexities for discontinuous functionals, broadening applicability beyond smooth cases.

Findings

Numerical experiments demonstrate improved efficiency in risk estimation and option pricing.

The method achieves near-optimal computational complexity for probability estimation with discontinuous functionals.

The framework is versatile and applicable to a wide range of problems involving hierarchical approximations.

Abstract

We consider the numerical approximation of $\mathbb{P}[G\in \Omega]$ where the $d$-dimensional random variable $G$ cannot be sampled directly, but there is a hierarchy of increasingly accurate approximations $\{G_\ell\}_{\ell\in\mathbb{N}}$ which can be sampled. The cost of standard Monte Carlo estimation scales poorly with accuracy in this setup since it compounds the approximation and sampling cost. A direct application of Multilevel Monte Carlo improves this cost scaling slightly, but returns sub-optimal computational complexities since estimation of the probability involves a discontinuous functional of $G_\ell$. We propose a general adaptive framework which is able to return the MLMC complexities seen for smooth or Lipschitz functionals of $G_\ell$. Our assumptions and numerical analysis are kept general allowing the methods to be used for a wide class of problems. We present numerical experiments on nested simulation for risk estimation, where $G = \mathbb{E}[X|Y]$ is approximated by an inner Monte Carlo estimate. Further experiments are given for digital option pricing, involving an approximation of a $d$-dimensional SDE.