ANALYSIS OF MULTI-INDEX MONTE CARLO ESTIMATORS FOR A ZAKAI SPDE

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(epsilon(-2)vertical bar log epsilon vertical bar(3)) for a root mean square error (RMSE) epsilon if the same spatial discretisation as in the MLMC method is used; (ii) has a better complexity of O(epsilon(-2)vertical bar log epsilon vertical bar epsilon) if a carefully adapted discretisation is used; (iii) has to be adapted for non-smooth functionals. Numerical tests confirm these findings empirically.