Variance reduction for discretised diffusions via regression

In this paper we present a novel approach towards variance reduction for discretised diffusion processes. The proposed approach involves specially constructed control variates and allows for a significant reduction in the variance for the terminal functionals. In this way the complexity order of the standard Monte Carlo algorithm (${\epsilon }^{-3}$ in the case of a first order scheme and ${\epsilon }^{-2.5}$ in the case of a second order scheme) can be reduced down to ${\epsilon }^{-2+\delta }$ for any $\delta \in [0,0.25)$ with $\epsilon$ being the precision to be achieved. These theoretical results are illustrated by several numerical examples.