Regression-based variance reduction approach for strong approximation schemes

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 (${\epsilon }^{-3}$) can be reduced down to ${\epsilon }^{-2}\sqrt{\mid \log (\epsilon )\mid }$ in case of the Euler scheme with $\epsilon$ being the precision to be achieved. These theoretical results are illustrated by several numerical examples.