Multivariate control variate technique

Question

In the control variate technique, one tries to improve convergence of the expected value of a random variable $X$, estimated from simulating a range of $n$ Monte Carlo simulations. This is done by creating a transformation with a correlated r.v. such that the expected value stays the same, but the variance of $E[X]$ is decreased for a given number of simulations. The transformation is defined as: $Z=X+\alpha (Y-E[Y])$ where $\alpha$ is optimized to minimize the variance of the combination. A closed-form solution exists for the optimal alpha, i.e. $\alpha^*=\frac{Cov(X,Y)}{ Var(Y)}$. My question is, can we write out such technique when we have multiple variables at our disposal, i.e. does there exist a multivariate extension of such technique? Many thanks!

Answer 1

Use Linear Least Squares (Regression) to determine the optimal coefficients for the control variates, as described in section 8.9 of https://artowen.su.domains/mc/Ch-var-basic.pdf, among other places.

When there is only one control variate, the Linear Least Squares approach reduces to the closed form formula you showed.

It is also possible, though less commonly used, for control variates to appear (be used) nonlinearly. In that case, Nonlinear Least Squares can be used to determine the optimal parameters.