Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T04:28:01.415Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
*Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© Applied Probability Trust 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agranov, M. and Ortoleva, P. (2017). Stochastic choice and preferences for randomization. J. Political Econom. 125 (1), 40–68.10.1086/689774CrossRefGoogle Scholar
Cerreia-Vioglio, S., Dillenberger, D. and Ortoleva, P. (2015). Cautious expected utility and the certainty effect. Econometrica 83, 693–728.CrossRefGoogle Scholar
Cerreia-Vioglio, S., Dillenberger, D., Ortoleva, P. and Riella, G. (2019). Deliberately stochastic. American Econom. Rev. 89 (7), 2425–2445.CrossRefGoogle Scholar
Dillenberger, D. (2010). Preference of one-shot resolution of uncertainty and Allais-type behavior. Econometrica 78 (6), 1973–2004.Google Scholar
Dupuis, P. and Wang, H. (2002). Optimal stopping with random intervention times. Adv. Appl. Prob. 34 (1), 141–157.CrossRefGoogle Scholar
Fudenberg, D., Iijima, R. and Strzalecki, T. (2015). Stochastic choice and revealed perturbed utility. Econometrica 83, 2371–2409.CrossRefGoogle Scholar
Gul, F. and Pesendorfer, W. (2006). Random expected utility. Econometrica 74, 121–146.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd edn. Springer.Google Scholar
Lange, R.-J., Ralph, D. and Støre, K. (2020). Real-option valuation in multiple dimensions using Poisson optional stopping times. J. Financial Quant. Anal. 55 (2), 653–677.CrossRefGoogle Scholar
Lempa, J. (2012). Optimal stopping with information constraint. Appl. Math. Optimization 66 (2), 147–173.CrossRefGoogle Scholar
Liang, G. and Wei, W. (2016). Optimal switching at Poisson random intervention times. Discrete Contin. Dyn. Syst. Ser. B 21 (5), 1483–1505.CrossRefGoogle Scholar
Menaldi, J. L. and Robin, M. (2016). On some optimal stopping problems with constraint. SIAM J. Control Optimization 54 (5), 2650–2671.CrossRefGoogle Scholar
Petrovski, I. G. (1966). Ordinary Differential Equations, revised English edition. Prentice Hall, New Jersey.Google Scholar
Pham, H. (2009). Continuous-Time Stochastic Control and Optimization with Financial Applications. Springer.CrossRefGoogle Scholar
Rogers, L. C. G. and Zane, O. (1998). A simple model of liquidity effects. Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann, eds K. Sandmann and P Schönbucher. Springer, Berlin.Google Scholar
Scheinkman, J. A. and Xiong, W. (2003). Overconfidence and speculative bubbles. J. Political Econom. 111 (6), 1183–1218.CrossRefGoogle Scholar
Strack, P. and Viefers, P. (2019). Too proud to stop: regret in dynamic decisions. J. European Econom. Assoc. doi:10.1093/jeea/jvz073.CrossRefGoogle Scholar