A control variate method driven by diffusion approximation

TL;DR

This paper introduces a control variate estimator using diffusion approximation for stochastic processes driven by fast mean-reverting forces, improving variance reduction in expectation estimation.

Contribution

It proposes a coupling method to efficiently link the original process with its diffusion limit, quantifying correlation and convergence for variance reduction.

Findings

Correlation approaches one as mean reversion time decreases

The method achieves significant variance reduction in examples

Convergence rate of the coupling is explicitly characterized

Abstract

In this paper we examine a control variate estimator for a quantity that can be expressed as the expectation of a functional of a random process, that is itself the solution of a differential equation driven by fast mean-reverting ergodic forces. The control variate is the expectation of the same functional for the limit diffusion process that approximates the original process when the mean-reversion time goes to zero. To get an efficient control variate estimator, we propose a coupling method to build the original process and the limit diffusion process. We show that the correlation between the two processes indeed goes to one when the mean reversion time goes to zero and we quantify the convergence rate, which makes it possible to characterize the variance reduction of the proposed control variate method. The efficiency of the method is illustrated on a few examples.