Analysis of various estimators for multi-dimensional Zakai equations

We first consider a one-dimensional stochastic partial differential equation (SPDE) of Zakai type describing a large credit portfolio. Specifically, we construct estimators of linear functionals of the solution from an implicit Milstein scheme on a space-time mesh. We compare the complexity of a multi-index Monte Carlo (MIMC) approach with the multilevel Monte Carlo (MLMC) method of Giles and Reisinger (2012), and find, by means of Fourier analysis, that the MIMC method has slightly improved complexity O(ε-2|log ε|) for a root mean square error (RMSE) ε if a carefully adapted discretisation is used. Then, we propose an implicit finite difference scheme for a two-dimensional parabolic SPDE of Zakai type, based on a Milstein approximation to the stochastic integral and an alternating direction implicit (ADI) discretisation of the elliptic term. We prove its mean-square stability and convergence in L2 of first order in time and second order in space, by Fourier analysis, in the presence of Dirac initial data. Next, we analyse the accuracy and computational complexity of estimators for linear functionals of the solution to the 2-d SPDE, coupled with the sparse combination technique and MLMC. We find, by detailed Fourier analysis, that for a RMSE ε, MLMC with sparse combination has the optimal complexity O(ε-2), whereas MLMC on regular grids has O(ε-2(log ε)2), standard MC with sparse combination O(ε-7/2(|log ε|)5/2), and MC on regular grids O(ε-4). We give a discussion of the higher-dimensional setting without detailed proofs, which suggests that MLMC with sparse combination always leads to the optimal complexity. Finally, we consider a particular two-dimensional SPDE with fast mean-reverting volatility on a timescale O(ε-1), and study the small ε asymptotics. We find an asymptotic expansion of the solution to the SPDE as ε → 0 and conclude from numerical experiments the convergence order 1=2 of the leading term and order 1 after inclusion of the first correction term.