A sequential estimation problem with control and discretionary stopping

We show that “full-bang” control is optimal in a problem that combines features of (i) sequential least-squares estimation with Bayesian updating, for a random quantity observed in a bath of white noise; (ii) bounded control of the rate at which observations are received, with a superquadratic cost per unit time; and (iii) “fast” discretionary stopping. We develop also the optimal filtering and stopping rules in this context. MSC 2020 Subject Classification: primary 62L12; secondary 60G35, 62L15, 93E11.