 Retirement threshold with multiplicative capture risk

If a successful step in action $i$ has expected reward $a_i\ge 0$ and survival probability $q_i=1-p_i$, a finite-horizon retirement problem with zero wealth after capture has threshold ${\max }_i{a}_i{q}_i/p_i$. Above it, induction makes retirement optimal; below it, a single further step already beats retirement. The threshold does not depend on the remaining positive horizon.