A quickest detection problem with an observation cost

TL;DR

This paper studies an optimal detection strategy for a stochastic process with observation costs, establishing a two-threshold policy based on the posterior probability of change, and compares strong and weak solution formulations.

Contribution

It introduces a novel formulation of the quickest detection problem with observation costs and explicitly characterizes the optimal two-threshold policy using the weak solution approach.

Findings

Optimal strategy is a two-threshold policy based on posterior probability.

Value functions are identical in strong and weak formulations.

Explicit thresholds are derived from model parameters.

Abstract

In the classical quickest detection problem, one must detect as quickly as possible when a Brownian motion without drift "changes" into a Brownian motion with positive drift. The change occurs at an unknown "disorder" time with exponential distribution. There is a penalty for declaring too early that the change has occurred, and a cost for late detection proportional to the time between occurrence of the change and the time when the change is declared. Here, we consider the case where there is also a cost for observing the process. This stochastic control problem can be formulated using either the notion of strong solution or of weak solution of the s.d.e. that defines the observation process. We show that the value function is the same in both cases, even though no optimal strategy exists in the strong formulation. We determine the optimal strategy in the weak formulation and show, using a form of the "principle of smooth fit" and under natural hypotheses on the parameters of the problem, that the optimal strategy takes the form of a two-threshold policy: observe only when the posterior probability that the change has already occurred, given the observations, is larger than a threshold $A\geq0$, and declare that the disorder time has occurred when this posterior probability exceeds a threshold $B\geq A$. The constants $A$ and $B$ are determined explicitly from the parameters of the problem.