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 (ε−3 in the case of a first order scheme and ε−2.5 in the case of a second order scheme) can be reduced down to ε−2+δ for any δ∈[0,0.25) with ε being the precision to be achieved. These theoretical results are illustrated by several numerical examples.