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Randomised rules for stopping problems

Part of: Stochastic processes Operations research and management science

Published online by Cambridge University Press:  04 September 2020

David Hobson  and
Matthew Zeng
David Hobson*
Affiliation:
University of Warwick
Matthew Zeng*
Affiliation:
University of Warwick
*
*Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email: d.hobson@warwick.ac.uk
*Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email: d.hobson@warwick.ac.uk
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Abstract

In a classical, continuous-time, optimal stopping problem, the agent chooses the best time to stop a stochastic process in order to maximise the expected discounted return. The agent can choose when to stop, and if at any moment they decide to stop, stopping occurs immediately with probability one. However, in many settings this is an idealistic oversimplification. Following Strack and Viefers we consider a modification of the problem in which stopping occurs at a rate which depends on the relative values of stopping and continuing: there are several different solutions depending on how the value of continuing is calculated. Initially we consider the case where stopping opportunities are constrained to be event times of an independent Poisson process. Motivated by the limiting case as the rate of the Poisson process increases to infinity, we also propose a continuous-time formulation of the problem where stopping can occur at any instant.

Keywords

Randomised stoppingoptimal stoppingPoisson process constrained optimal stoppingstochastic choice

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 90B50: Management decision making, including multiple objectives
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 3 , September 2020 , pp. 981 - 1004
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Copyright
© Applied Probability Trust 2020

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