  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.
