= Solution
{c}
The <barrier-triggered put> exercises at the particular <stopping time> $\tau_B\wedge T$. The <American put option> optimizes over all allowed exercise times, so
$$
\boxed{E_B(S,t)\leq A(S,t;K).}
$$
It can be strictly smaller, but can never exceed the American value. Equality is possible if immediate exercise is already triggered and optimal; a fixed barrier is not generally the optimal finite-maturity exercise rule.
The comparison with the <European put option> can go in either direction. By the <early exercise premium for a fixed barrier>,
$$
E_B(S,t)-P(S,t;K)=\mathbb E_{\mathbb Q}\!\left[e^{-r(\tau_B-t)}
\{K-B-P(B,\tau_B;K)\}\mathbf1_{\{\tau_B\leq T\}}\mid S_t=S\right].
$$
Thus an early payment gains interest but sacrifices remaining put time value; the expression inside braces need not have a fixed sign for arbitrary barriers.
For a concrete general construction take current spot $S\geq K$, $r>0$, $\sigma>0$, and $t<T$. Set $\gamma=2r/\sigma^2$ and $b_\infty=K\gamma/(1+\gamma)$, the <perpetual put option> exercise boundary. For $0<B<b_\infty$, the perpetual value at $B$ equals $K-B$ and strictly dominates the finite-maturity European value. The premium formula and positive probability of hitting $B$ therefore give \b[$E_B>P$]. Strictness follows from the perpetual verification argument: before a fixed positive expiry, its discounted value loses drift $rK$ whenever the stock is below $b_\infty$, with positive expected occupation time when started at $B<b_\infty$.
Conversely, for $B<K$ sufficiently close to $K$, every possible payment of the exploding put is at most $K-B$, whether paid at touch or on surviving to expiry. Hence $0\leq E_B\leq K-B\to0$, while $P(S,t;K)>0$. This gives \b[$E_B<P$] for such barriers, and also strict inequality below the American value.
These examples establish both possibilities in general, not a claim that every fixed initial state admits both. If $S<b_\infty$ and only alive barriers $B<S$ are allowed, all those barriers are in the same perpetual stopping region and the premium comparison is nonnegative.
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