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Constrained optimal stopping, liquidity and effort

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  • Hobson, David
  • Zeng, Matthew

Abstract

In a classical optimal stopping problem in continuous time the agent can choose any stopping time without constraint. Dupuis and Wang (2002) introduced a constraint on the class of admissible stopping times which was that they had to take values in the set of event times of an exogenous, time-homogeneous Poisson process. This can be thought of as a model of finite liquidity. In this article we extend the analysis of Dupuis and Wang to allow the agent to choose the rate of the Poisson process. Choosing a higher rate leads to a higher cost. Even for a simple model for the stopped process and a simple call-style payoff, the problem leads to a rich range of optimal behaviours which depend on the form of the cost function. Often the agent accepts the first offer — if they are not going to accept an offer then there is no point in putting in effort to generate offers, and thus there may be no offers to accept or decline — but for some set-ups this is not the case.

Suggested Citation

  • Hobson, David & Zeng, Matthew, 2022. "Constrained optimal stopping, liquidity and effort," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 819-843.
    • Handle: RePEc:eee:spapps:v:150:y:2022:i:c:p:819-843
    DOI: 10.1016/j.spa.2019.10.010
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    References listed on IDEAS

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    1. Andrew Ang & Dimitris Papanikolaou & Mark M. Westerfield, 2014. "Portfolio Choice with Illiquid Assets," Management Science, INFORMS, vol. 60(11), pages 2737-2761, November.
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    3. Yuliy Sannikov, 2008. "A Continuous-Time Version of the Principal-Agent Problem," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 75(3), pages 957-984.
    4. Avinash K. Dixit & Robert S. Pindyck, 1994. "Investment under Uncertainty," Economics Books, Princeton University Press, edition 1, number 5474.
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    1. Alvarez E., Luis H.R. & Lempa, Jukka & Saarinen, Harto & Sillanpää, Wiljami, 2024. "Solutions for Poissonian stopping problems of linear diffusions via extremal processes," Stochastic Processes and their Applications, Elsevier, vol. 172(C).

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