Finite Horizon Decision Timing with Partially Observable Poisson Processes

TL;DR

This paper addresses optimal decision timing in systems with Poisson-based information arrivals, unobservable Markovian environments, and finite horizons, providing a solution via an optimal stopping framework for posterior likelihood processes.

Contribution

It introduces a novel approach to finite horizon decision timing problems with Poisson observations, utilizing a piecewise-deterministic process for posterior likelihoods.

Findings

Method is numerically simple to implement.

Applicable to investment, reliability, and technology adoption models.

Provides illustrative numerical examples.

Abstract

We study decision timing problems on finite horizon with Poissonian information arrivals. In our model, a decision maker wishes to optimally time her action in order to maximize her expected reward. The reward depends on an unobservable Markovian environment, and information about the environment is collected through a (compound) Poisson observation process. Examples of such systems arise in investment timing, reliability theory, Bayesian regime detection and technology adoption models. We solve the problem by studying an optimal stopping problem for a piecewise-deterministic process which gives the posterior likelihoods of the unobservable environment. Our method lends itself to simple numerical implementation and we present several illustrative numerical examples.