Monte Carlo Errors

Under independence conditions, the standard deviation $\sigma(K)$ of the results $\theta(K)$ of a statistical model depends on the number of observations $K$ as

$$\sigma(K)=\sigma(1) / \sqrt K$$ (91)

Here $\sigma(1)$ is the "initial" standard deviation and

$$\sigma(1)=\sqrt {E (\theta(1)-\Theta(1))^2}$$ (92)

where $\Theta(1)=E \theta(1)$ and $E$ denotes the expectation.

In Monte Carlo simulation $\Theta(1)$ cannot be estimated directly using the results $\theta(1)$ obtained by the 1-th observation. Many observations are needed to reach a stationary state (one observation means one call). Therefore we repeat the Monte Carlo simulation $L$ times performing $K$ observations each time. In this case the standard error can be estimated as

$$\sigma_L^2(K)=1/(L-1) \sum_{l=1}^L (\theta_l(K)-1/L\sum_{j=1}^L\theta_j(K))^2$$ (93)

Expression (2.33) is much simpler if the exact solution $\Theta=\lim_{K \rightarrow \infty} \theta(K)$ is known

$$\sigma_L^2(K)=1/(L-1) \sum_{l=1}^L (\theta_l(K)-\Theta)^2$$ (94)

Therefore first we should consider a model with exact solution while testing a Monte Carlo procedure and only then proceed to more complicated models hoping that the MC procedure will work well enough.