By using Euler Monte Carlo discretization (for a Hull-White model) we simulate $$r(t+\Delta t)=r(t)+\lambda(\theta(t)-r(t))\Delta t+\eta\sqrt{\Delta t}Z$$ with $Z\sim N(0,1)$, $\lambda$, $\eta$ constants and $\theta(t)$ a known function up to a certain $T$ in order to estimate the expectation of a function $h(r(T))$, thus $$\frac{1}{N}\sum^{N}_{i=1}h(r_{i}(T))\rightarrow\mathbb{E}[h(r(T))]$$ as the number of paths grows or $N\rightarrow\infty$. The exact solution of $\mathbb{E}[h(r(T))]$ is not known, therefore my question is: What can be stated about the accuracy of this Euler Monte Carlo discretization with respect to the number of paths $N$?

Up till now, I found that the variance of our estimator decreases with order $N^{-1}$, since $$\mathbb{Var}[\frac{1}{N}\sum^{N}_{i=1}h(r_{i}(T))]=\frac{1}{N}\mathbb{Var}[h(r(T))]$$ This implies that the variance of the function $h(r(T))$ and thus again $\mathbb{E}[h(r(T))]$ must be known, which is not. Furhermore, the central limit theorem states that the probability distribution of the error converges to a normal distribution with mean $0$ and variance $\frac{\mathbb{Var}[h(r(T))]}{N}$, which is again unknown. As far as I known, nothing except the order of convergence of the variance of the error is known. Is there anything which can be stated about the error with respect to the amount of paths used in Monte Carlo without knowing the exact solution for $\mathbb{E}[h(r(T))]$?


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