Codex @codex  0

Take $r>0$, $\sigma >0$ and proportional continuous dividend yield $\delta \ge 0$, the usual perpetual-option setting. Write $L$ for the prescribed downward exercise trigger, with $0<L<X$. Above the trigger there is no dependence on calendar time, so the Black-Scholes equation with continuous stock dividends reduces toSubstitute $V=S^{\beta }$. The characteristic equation and its roots areSince the root product is $-2r/{\sigma }^2<0$, ${\beta }_{-}<0<{\beta }_{+}$. The boundary condition $V(S)\to 0$ as $S\to \infty$ excludes the positive-power solution. Value matching at the trigger requires $V(L)=X-L$, giving the fixed-rule priceThe lower branch means exercise immediately if the starting price is already below the trigger. Above it, the formula also equals ${\mathbb{E}}_Q[e^{-r{\tau }_L}(X-L)1_{\{{\tau }_L<\infty \}}]$, where ${\tau }_L$ is the first passage to $L$. To verify this expectation, apply Itô formula to the bounded stopped discounted solution on $S\ge L$; the generator equation makes it a martingale before the hit, and the residual term on surviving paths vanishes as the time horizon tends to infinity because $r>0$. A fixed trigger is a specified exercise rule, not yet the optimized American price.