Control variates for variance-reduced ratio of means estimators

The control variates method is a classical variance reduction technique for Monte Carlo estimators that exploits correlated auxiliary variables without introducing bias. In many applications, the quantity of interest can be expressed as a ratio of expectations. We propose a variance-reduced estimator for such ratios, which applies control variates to both the numerator and the denominator. The control variate coefficients are optimized jointly to minimize the variance of the resulting estimator. This approach theoretically guarantees variance reduction and naturally extends to approximate control variates. Simulation studies show significant variance reduction, particularly when correlations between variables and control variates are strong. The practical value of the method is illustrated with a multi-fidelity aircraft design use case.