Analysis of Multi-Index Monte Carlo Estimators for a Zakai SPDE

TL;DR

This paper introduces a space-time Multi-Index Monte Carlo estimator for Zakai SPDEs, compares its complexity with MLMC, and demonstrates its advantages through Fourier analysis and numerical tests.

Contribution

The paper develops a novel MIMC estimator for Zakai SPDEs and analyzes its complexity, showing improvements over existing MLMC methods with tailored discretizations.

Findings

MIMC has suboptimal complexity $O( ext{error}^{-2}| ext{log error}|^3)$ with standard discretization.

Optimized discretization improves MIMC complexity to $O( ext{error}^{-2}| ext{log error}|)$.

Numerical tests confirm theoretical complexity results and effectiveness of the MIMC estimator.

Abstract

In this article, we propose a space-time Multi-Index Monte Carlo (MIMC) estimator for a one-dimensional parabolic stochastic partial differential equation (SPDE) of Zakai type. We compare the complexity with the Multilevel Monte Carlo (MLMC) method of Giles and Reisinger (2012), and find, by means of Fourier analysis, that the MIMC method: (i) has suboptimal complexity of $O(\varepsilon^{-2}|\log\varepsilon|^3)$ for a root mean square error (RMSE) $\varepsilon$ if the same spatial discretisation as in the MLMC method is used; (ii) has a better complexity of $O(\varepsilon^{-2}|\log\varepsilon|)$ if a carefully adapted discretisation is used; (iii) has to be adapted for non-smooth functionals. Numerical tests confirm these findings empirically.