# Convergence rate estimates of Monte-Carlo first-passage time estimates

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.

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
• Two problems: 1. $n$ is not wrt the discritization but the number of samples 2. In this case the increments are independent but are not identically distributed... (even if it were discussing the discritization of iid rvs). – AIM_BLB Jan 7 at 16:06
• You wrote “$n$ sample paths.” They are not independent? How are they produced? – Yuval Peres Jan 7 at 16:40