**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. 



Convergence rate estimates of Monte-Carlo first-passage time estimates
Assuming $D$ is bounded, the best estimate that holds almost surely is given by the law of the iterated logarithm <https://en.m.wikipedia.org/wiki/Law_of_the_iterated_logarithm>


