A Bayesian sequential soft classification problem for a Brownian motion's drift

TL;DR

This paper introduces and solves a Bayesian sequential soft classification problem for a Brownian motion's drift, characterizing the optimal stopping boundaries and their dependence on the signal-to-noise ratio.

Contribution

It formulates a novel soft classification variant of the Bayesian sequential testing problem and provides a detailed analysis of the value function and stopping boundaries.

Findings

The value function solves a free boundary problem uniquely.

Stopping boundaries depend on the signal-to-noise ratio.

Boundaries may coincide if the signal is weak.

Abstract

In this note we introduce and solve a soft classification version of the famous Bayesian sequential testing problem for a Brownian motion's drift. We establish that the value function is the unique non-trivial solution to a free boundary problem, and that the continuation region is characterized by two boundaries which may coincide if the observed signal is not strong enough. By exploiting the solution structure we are able to characterize the functional dependence of the stopping boundaries on the signal-to-noise ratio. We illustrate this relationship and compare our stopping boundaries to those derived in the classical setting.