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Suppose that $B_t$ is a Brownian bridge from $a$ to $b$ on an interval $[0,T]$ (i.e., a Brownian motion $W_t$ such that $W_0=a$ conditioned to satisfy $W_T=b$).

Let $T_c=\inf\{t\leq T:B_t=c\}\land T$ be the hitting time of some point $c\in\mathbb R$ for the bridge in question. Does there exist exact formulas for the cdf/density of $T_c$, similar to the distribution of hitting times for a standard Brownian motion?

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