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

Christoph Reisinger, Zhenru Wang

1. Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK
• Received:2016-06-20 Revised:2016-12-09 Online:2018-03-15 Published:2018-03-15

Christoph Reisinger, Zhenru Wang. ANALYSIS OF MULTI-INDEX MONTE CARLO ESTIMATORS FOR A ZAKAI SPDE[J]. Journal of Computational Mathematics, 2018, 36(2): 202-236.

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

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