= Solution
{c}
While the <up-and-out digital put> is alive, its pricing domain is $0<S<B$, $t<T$. Its <Black-Scholes equation>, terminal <financial payoff>, and absorbing <boundary condition> are
$$
\boxed{\begin{aligned}
H_t+\tfrac12\sigma^2S^2H_{SS}+rSH_S-rH&=0,&0<S<B,\ t<T,\\
H(S,T)&=\mathbf1_{\{S<K\}},&0<S<B,\\
H(B,t)&=0,&t\leq T.
\end{aligned}}
$$
For completeness, the natural limiting condition is $\lim_{S\downarrow0}H(S,t)=e^{-r(T-t)}$, with $0\leq H(S,t)\leq e^{-r(T-t)}$. A positive <geometric Brownian motion> cannot hit zero in finite time, and from a very small starting value the survival and below-strike terminal events have probability tending to one. If the barrier has already been hit, its value is zero regardless of the current stock price.
For $K>B$, the terminal payoff tends to one from inside at $S=B$, while an already hit barrier pays zero. Thus there is a terminal-boundary corner discontinuity; the terminal condition is imposed only for $S<B$, not as a requirement of continuous extension at $(B,T)$.
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