Let $(W_t)_{t\in[0,T]}$ be a Brownian bridge such that $W_0=a$ and $W_T=b$, the probability that $\forall t\in[0,T],W_t\geqslant x$ given the parameter $x\leqslant\min(a,b)$ is well known : $$ \mathbb{P}(\forall t\in[0,T],W_t\geqslant x)=1-e^{\frac{2(x-a)(b-x)}{T}} $$ (see https://mathoverflow.ne...