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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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    $\begingroup$ There are results concerning the convergence of the Euler scheme, specifically the scheme is $0.5$-strongly convergent and $1.0$-weakly convergent, see for example Mikosch's Elementary Stochastic Calculus, Section 3.4.1 (tip: try Google). $\endgroup$ – Daneel Olivaw May 13 at 18:04
  • $\begingroup$ To determine the order of the convergence the explicit solution of $h(r(T))$ is needed, but in my case there is no explicit fomrula for $r(T)$, therefore done by Euler discretization. The book by Mikoch's cannot help, since it states that you need the explicit expression to determine the order of convergence. $\endgroup$ – rs4rs35 May 13 at 21:16
  • $\begingroup$ My previous comment was wrong I have noticed, however the book of Mikoch only states the weak convergence for Euler discretization for a sufficient smooth function. The criteria for this sufficient smoothness is rather hard to find on the internet. $\endgroup$ – rs4rs35 May 14 at 7:50

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