Bayesian sequential least-squares estimation for the drift of a Wiener process
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Given a Wiener process with unknown and unobservable drift, we seek to estimate this drift as effectively but also as quickly as possible, in the presence of a quadratic penalty for the estimation error and of a linearly growing cost for the observation duration. In a Bayesian framework, this question reduces to choosing judiciously a stopping time for an appropriate diffusion process in natural scale; we provide structural properties of the solution for the corresponding problem of optimal stopping. In particular, regardless of the prior distribution, the continuation region is monotonically shrinking in time. Moreover, conditions on the prior distribution that guarantee a one-sided boundary are provided.
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