Variance reduction for discretised diffusions via regression

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

Truncated control variates for weak approximation schemes

“…Using Theorem 2.4, we obtain the following result (see Proposition 3.6 in [1]), which provides a bound for the discretisation error and a perfect control variate for the discretised quantity. Proposition 2.5.…”

“…The proposition below summarises important representations for the weak Euler scheme, which were derived in [1].…”

“…, J, we have for this "simplified weak Euler scheme" Discussion. In order to use the control variate M (1) ∆,T in practice, we need to estimate the unknown coefficients a j,r,s . Thus, practically implementable control variates M…”

“…. , m, that act 1 This phrase means that the discrete-time processM = (M l ) l=0,...,J , whereM0 = 0 andM l is defined like the righthand side of (2.6) but with J j=1 being replaced by l j=1 and aj,r,s byãj,r,s is a martingale, which is a straightforward calculation.…”

“…The use of control variates for computing expectations of functionals of diffusion processes via Monte Carlo was initiated by Newton [11] and further developed in Milstein and Tretyakov [8]. In Belomestny et al [1] a novel regression-based approach for the construction of control variates, which reduces the variance of the approximated functional f (X T ) was proposed. As shown in [1], the "Monte Carlo approach with the Regression-based Control Variate" (abbreviated below as "RCV approach") is able to achieve a higher order convergence of the resulting variance to zero, which in turn leads to a significant complexity reduction as compared to the SMC The work of Denis Belomestny is supported by the Russian Science Foundation project 14-50-00150.…”