 Solution

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 areFor completeness, the natural limiting condition is ${\lim }_{S\downarrow 0}H(S,t)=e^{-r(T-t)}$, with $0\le H(S,t)\le e^{-r(T-t)}$. 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 $(B,T)$.