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Stopping problems with an unknown state

Part of: Stochastic systems and control Stochastic processes

Published online by Cambridge University Press:  09 August 2023

Erik Ekström Open the ORCID record for Erik Ekström [Opens in a new window]  and
Yuqiong Wang
Erik Ekström*
Affiliation:
Uppsala University
Yuqiong Wang*
Affiliation:
Uppsala University
*
*Postal address: Department of Mathematics, Box 256, 751 05 Uppsala, Sweden.
*Postal address: Department of Mathematics, Box 256, 751 05 Uppsala, Sweden.
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Abstract

We extend the classical setting of an optimal stopping problem under full information to include 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, standard filtering theory can be employed to embed the problem in a Markovian framework with one additional state variable representing the posterior of the unknown state. We provide a convenient formulation of this Markovian problem, based on a measure change technique that decouples the underlying process from the new state variable. Moreover, we show by means of several novel examples that this reduced formulation can be used to solve problems explicitly.

Keywords

Optimal stoppingincomplete informationrandom time horizon

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60G35: Signal detection and filtering 93E10: Estimation and detection
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 14
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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