A control variate method driven by diffusion approximation
We present a control variate estimator for a quantity of interest that can be expressed as the expectation of a function of a random process, that is itself the solution of a differential equation (or a variational inequality) driven by fast mean-reverting ergodic random forces. The control variate is built with the same function and with the limit diffusion process that approximates the original random process when the mean reversion time of the driving forces goes to 0. We propose a coupling of the original process and the limit diffusion process that gives a control variate estimator with small variance. We show that the correlation between the two processes indeed goes to 1 when the mean reversion time goes to 0 and we quantify the convergence rate, which allows us to characterize the variance reduction of the proposed control variate estimator. The efficiency of the method is illustrated on a few examples.
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