In an article i recently read (The American Put Option and Its Critical Stock Price by David S. Bunch and Herb Johnson link) the authors presented this formula as something very general and as common knowledge
$$P = \mathop {\max }\limits_{{S_c}} \int\limits_0^T {{e^{ - rt}}(X - S_c)} fdt,\quad (S > {S_C})$$
where $P, r, T, X,$ and $Sc$ are the American put price, risk-free rate, time to maturity, exercise price, and critical stock price, respectively.Let S be the current stock price (at time $ t= 0).$ $f$, is the first-passage probability,
However i cant recall that i have seen this formula AND $f$ in the same formula, what am I missing? Where did this formula come from?