Setup
Let $X_t$ be a $d$-dimensional diffusion process solving the Ito-stochastic differential equation $$ X_t = x+ \int_0^t f(X_t,u_t)dt + \int_0^t \sigma dW_t, $$ where $x \in \mathbb{R}^d$, $u_t$ is predictable, $f(\cdot,\cdot)$ is locally Lipschitz, $\sigma \sigma^T$ is positive-definite, and $W_t$ is a $d$-dimensional Browinian motion.
If $D$ is a connected open neighbourhood of $x$ with smooth boundary, and suppose that we would like to estimate the distribution of the first exit time $$ \tau \triangleq \left\{ t >0: \, X_t \not\in D \right\} = \left\{ t >0: \, X_t \in \partial D \right\} , $$ via Monte-Carlo. Let us denote $\tau^n$ the empirical distribution approximating $\tau$ from producing $n$ sample paths of $X_t$.
Question:
Are there known estimates on the convergence speed of $\tau^n$ to $\tau$ (in any reasonable sense)? For example an estimate on the first moment the form: $$ \|\tau^n - \mathbb{E}[\tau]\| \leq C r(n), $$ where $r$ is some nice lsc rate function and $C>0$ is some universal constant.
