Bayesian sequential least-squares estimation for the drift of a Wiener process

TL;DR

This paper develops a Bayesian sequential estimation method for the unknown drift of a Wiener process, focusing on optimal stopping rules to balance estimation accuracy and observation costs.

Contribution

It introduces structural properties of the optimal stopping problem for drift estimation, including monotonicity of the continuation region and conditions for one-sided stopping.

Findings

Continuation region shrinks monotonically over time.

Conditions for one-sided stopping regions are established.

Theoretical results are illustrated with concrete prior distributions.

Abstract

Given a Wiener process with unknown and unobservable drift, we try 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 fixed, positive cost per unit of observation time. In a Bayesian framework, where the unobservable drift is assumed to have a known "prior" distribution, this question reduces to choosing judiciously a stopping time for an appropriate diffusion process in natural scale. We establish structural properties of the solution for the corresponding problem of optimal stopping. In particular, we show that, regardless of the prior distribution, the continuation region is monotonically shrinking in time; and provide conditions on the prior distribution guaranteeing a one-sided stopping region. Finally, we illustrate the theoretical results through a detailed study of some concrete prior distributions.