A sequential estimation problem with control and discretionary stopping

TL;DR

This paper investigates an optimal control and stopping problem involving Bayesian sequential estimation in noisy environments, deriving optimal filtering, control, and stopping strategies with applications in decision-making under uncertainty.

Contribution

It introduces a novel framework combining Bayesian estimation, controlled observation rates, and discretionary stopping, with explicit solutions for optimal policies.

Findings

Full-bang control is proven optimal under the model.

Derived explicit optimal filtering and stopping rules.

The framework addresses complex decision-making in noisy observation settings.

Abstract

We show that "full-bang" control is optimal in a problem that combines features of (i) sequential least-squares {\it estimation} with Bayesian updating, for a random quantity observed in a bath of white noise; (ii) bounded {\it control} of the rate at which observations are received, with a superquadratic cost per unit time; and (iii) "fast" discretionary {\it stopping}. We develop also the optimal filtering and stopping rules in this context.