= Solution
Choose $\Delta t$ with $\lambda\Delta t\ll1$ 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 $x_{\rm new}=x+v\Delta t$.
3. If $x_{\rm new}\leq0$, record the linearly interpolated target-crossing time and stop. If $x_{\rm new}\geq L$, reflect to $x_{\rm new}=2L-x_{\rm new}$ and set $v=-|v|$.
4. Otherwise reverse $v$ with probability $1-e^{-\lambda\Delta t}$, set $x=x_{\rm new}$ 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 $\Delta t\to0$.
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