We consider the problem of optimally stopping a Brownian bridge with an
uncertain pinning time so as to maximise the value of the process upon
stopping. Adopting a Bayesian approach, we consider a general prior
distribution of the pinning time and allow the stopper to update their belief
about this time through sequential observations of the process. Structural
properties of the optimal stopping region are shown to be qualitatively
different under different priors, however we are able to provide a sufficient
condition for a one-sided stopping region. Certain gamma and beta distributed
priors are shown to satisfy this condition and these cases are subsequently
considered in detail. In the gamma case we reveal the remarkable fact that the
optimal stopping problem becomes time homogeneous and is completely solvable in
closed form. In the beta case we find that the optimal stopping boundary takes
on a square-root form, similar to the classical solution with a known pinning
time. We also consider a two-point prior distribution in which a richer
structure emerges (with multiple optimal stopping boundaries). Furthermore,
when one of the values of the two-point prior is set to infinity (such that the
process may never pin) we observe that the optimal stopping problem is also
solvable in closed form.