Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 356 / 1 / f / Solution

Choose $\Delta t$ with $\lambda \Delta t\ll 1$ and $s\Delta t$ small relative to $L$ and the distance to the target. For each of many independent trajectories:

1. Set $x=y$, $v=s$, and $t=0$.
2. Propose $x_{\mathrm{new}}=x+v\Delta t$.
3. If $x_{\mathrm{new}}\le 0$, record the linearly interpolated target-crossing time and stop. If $x_{\mathrm{new}}\ge L$, reflect to $x_{\mathrm{new}}=2L-x_{\mathrm{new}}$ and set $v=-|v|$.
4. Otherwise reverse $v$ with probability $1-e^{-\lambda \Delta t}$, set $x=x_{\mathrm{new}}$ and $t=t+\Delta t$, and repeat.

The sample mean of the recorded times estimates ${\tau }^{+}(y)$. Sampling a Poisson number of reversals and their ordered times inside each step removes the at-most-one-reversal approximation, but the stated Bernoulli scheme converges as $\Delta t\to 0$.