Early exercise premium for a fixed barrier ($E_B-P=\mathbb{E}[e^{-r({\tau }_B-t)}(K-B-P(B,{\tau }_B;K))1_{\{{\tau }_B\le T\}}]$)

For an alive barrier-triggered put, the difference from the corresponding European put option is the displayed discounted conditional expectation, evaluated under the risk-neutral measure. To prove it, condition the European payoff on the first hitting time or expiry and use the Strong Markov property. At a hit, continuation of the European claim is worth $P(B,{\tau }_B;K)$, whereas the fixed exercise rule pays $K-B$ immediately. On survival to expiry the two payoffs coincide. The relevant stopped quantities are bounded, so the conditional-expectation argument is justified.