Stopping problems with an unknown state

TL;DR

This paper extends classical optimal stopping problems to include unknown states affecting the process, payoff, and horizon, and provides a Markovian framework for explicit solutions using filtering and measure change techniques.

Contribution

It introduces a novel approach to solve stopping problems with unknown states by embedding them into a Markovian framework via filtering and measure change.

Findings

Reduced problem formulation simplifies solutions

Explicit solutions demonstrated through new examples

Framework handles unknown states influencing multiple aspects

Abstract

We extend the classical setting of an optimal stopping problem under full information to include for problems with an unknown state. The framework allows the unknown state to influence (i) the drift of the underlying process, (ii) the payoff functions, and (iii) the distribution of the time horizon. Since the stopper is assumed to observe the underlying process and the random horizon, this is a two-source learning problem. Assigning a prior distribution for the unknown state, filtering theory can be used to embed the problem in a Markovian framework, and we thus reduce the problem with incomplete information to a problem with complete information but with one more state-variable. We provide a convenient formulation of the reduced problem, based on a measure change technique that decouples the underlying process from the state variable representing the posterior of the unknown state. Moreover, we show by means of several new examples that this reduced formulation can be used to solve problems explicitly.