Analysis of Multi-Index Monte Carlo Estimators for a Zakai SPDE
This work provides a theoretical and empirical analysis of MIMC for a specific SPDE, showing it can outperform MLMC under certain conditions, but the results are incremental and domain-specific.
The paper proposes a Multi-Index Monte Carlo estimator for a Zakai SPDE and analyzes its complexity, finding suboptimal performance with standard discretization but improved complexity with adapted discretization, achieving O(ε^{-2}|log ε|) for RMSE ε.
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.