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The control variates method, used as a variance reduction technique for Monte Carlo simulations, takes one new variable $t$, correlated to the random variable $m$ to estimate (using the same notations as Wikipedia), and $t$ should be chosen as an a variable with the same mean than $m$.

Why cannot we use a Cholesky decomposition to generate $t$, as an artificially correlated variable to $m$, ie: $$t = \rho \times m + \sqrt{1-\rho^2}\times z$$ with $z$ another independent Gaussian random variable and $\rho$ the correlation coefficient.

I couldn't find any reference to this anywhere, while this seems to be the easiest way to get the new correlated variable.

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    $\begingroup$ I am unable to see how this $t$ would qualify as a control statistic. According to your link, "Suppose we calculate another statistic $t$ such that $ {\mathbb {E}}\left[t\right]=\tau$ is a known value." The expectation of your $t$ is $\rho\mu+\sqrt{1-\rho^2}E[z].$ If you know that, you could solve for $\mu$ and therefore have no need of simulation at all! $\endgroup$ – whuber May 7 at 13:34

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