Variance reduction for discretised diffusions via regression

TL;DR

This paper introduces a new variance reduction technique for discretised diffusion processes using control variates, significantly improving the efficiency of Monte Carlo simulations for estimating terminal functionals.

Contribution

The paper proposes a novel control variate method that reduces variance and computational complexity in Monte Carlo simulations of discretised diffusions.

Findings

Variance reduction achieved with control variates

Complexity order improved to .2+.25 for any  in [0,0.25)

Numerical examples confirm theoretical improvements

Abstract

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 ($\varepsilon^{-3}$ in the case of a first order scheme and $\varepsilon^{-2.5}$ in the case of a second order scheme) can be reduced down to $\varepsilon^{-2+\delta}$ for any $\delta\in [0,0.25)$ with $\varepsilon$ being the precision to be achieved. These theoretical results are illustrated by several numerical examples.