Doing a project I have found in some papers that the (discretised) probability of the Brownian Bridge (which has $S(n)$ as initial value and $S(n+1)$ as final value, where S follows a Geometric Brownian Motion $dS(t)=rS(t)dt+\sigma S(t)dB(t)$, with $r$ constant drift, $\sigma$ constant volatility and $B$ standard Brownian Motion) hitting the barrier $L$ inside the interval $[t(n),t(n+1)]$ is given by:
$P(n) = exp(-2\frac{(L-S(n))*(L-S(n+1))}{(\sigma^2*S(n)*(t(n+1)-t(n))})$
I wanted to ask you if you could provide me some references where to find a proof of this statement or an idea to do it.
Thanks