Optimal Stopping of a Brownian Bridge with an Uncertain Pinning Time
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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.
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