= Solution
Take $r>0$, $\sigma>0$ and proportional <continuous dividend yield> $\delta\ge0$, 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 to
$$
\frac12\sigma^2S^2V''+(r-\delta)SV'-rV=0.
$$
Substitute $V=S^\beta$. The characteristic equation and its roots are
$$
F(\beta)=\frac12\sigma^2\beta(\beta-1)+(r-\delta)\beta-r=0,
\qquad \beta_\pm=\frac{-(r-\delta-\sigma^2/2)\pm\Delta}{\sigma^2},
\quad \Delta=\sqrt{(r-\delta-\sigma^2/2)^2+2r\sigma^2}.
$$
Since the root product is $-2r/\sigma^2<0$, $\beta_-<0<\beta_+$. The <boundary condition> $V(S)\to0$ as $S\to\infty$ excludes the positive-power solution. Value matching at the trigger requires $V(L)=X-L$, giving the fixed-rule price
$$
\boxed{V_L(S)=\begin{cases}
X-S,&0<S\le L,\\
(X-L)(S/L)^{\beta_-},&S>L.
\end{cases}}
$$
The lower branch means exercise immediately if the starting price is already below the trigger. Above it, the formula also equals $\mathbb E_Q[e^{-r\tau_L}(X-L)\mathbf1_{\{\tau_L<\infty\}}]$, 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.
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