# Cholesky decomposition in control variates method (Monte Carlo variance reduction technique)

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.

• 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! – whuber May 7 '19 at 13:34