Electronic Communications in Probability

Optimal stopping and the sufficiency of randomized threshold strategies

Vicky Henderson, David Hobson, and Matthew Zeng

Full-text: Open access

Abstract

In a classical optimal stopping problem the aim is to maximize the expected value of a functional of a diffusion evaluated at a stopping time. This note considers optimal stopping problems beyond this paradigm. We study problems in which the value associated to a stopping rule depends on the law of the stopped process. If this value is quasi-convex on the space of attainable laws then it is well known that it is sufficient to restrict attention to the class of threshold strategies. However, if the objective function is not quasi-convex, this may not be the case. We show that, nonetheless, it is sufficient to restrict attention to mixtures of threshold strategies.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 30, 11 pp.

Dates
Received: 3 August 2017
Accepted: 12 March 2018
First available in Project Euclid: 3 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1525312854

Digital Object Identifier
doi:10.1214/18-ECP125

Mathematical Reviews number (MathSciNet)
MR3798241

Zentralblatt MATH identifier
1390.60155

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Keywords
optimal stopping threshold strategies randomised strategies

Rights
Creative Commons Attribution 4.0 International License.

Citation

Henderson, Vicky; Hobson, David; Zeng, Matthew. Optimal stopping and the sufficiency of randomized threshold strategies. Electron. Commun. Probab. 23 (2018), paper no. 30, 11 pp. doi:10.1214/18-ECP125. https://projecteuclid.org/euclid.ecp/1525312854


Export citation

References

  • [1] Azéma J. and M. Yor, 1979, Une solution simple au problème de Skorokhod.Sem. de Prob. XIII, 90-115.
  • [2] Barberis N., 2012, A Model of Casino Gambling, Management Science, 58, 35-51.
  • [3] Camerer, C. F., and T. Ho, 1994, Violations of the betweenness axiom and nonlinearity in probabilities, Journal of Risk and Uncertainty, 8, 167-196.
  • [4] Cerreia-Vioglio, S., D.Dillenberger, P. Ortoleva, and G. Riella, 2017, Deliberately Stochastic, Working paper, Columbia University. SSRN Working paper, id2977320.
  • [5] Dayanik S. and I. Karatzas, 2003, On the optimal stopping problem for one-dimensional diffusions. Stoc. Proc. & Appl, 107, 2, 173-212.
  • [6] Durrett R., 1991, Probability: Theory and Examples. Wadsworth, Pacific Grove, California.
  • [7] Ebert S. and P. Strack, 2015, Until the Bitter End: On Prospect Theory in a Dynamic Context. American Economic Review, 105(4), 1618-1633.
  • [8] Hall W.J., 1968, On the Skorokhod embedding theorem. Technical Report 33 Stanford University, Department of Statistics.
  • [9] Henderson V., D. Hobson and M. Zeng, 2017, Cautious Stochastic Choice, Optimal Stopping and Deliberate Randomization, SSRN Working paper, id3118906.
  • [10] Henderson V., Hobson D. and A.S.L.Tse, 2017, Randomized Strategies and Prospect Theory in a Dynamic Context, Journal of Economic Theory, 168, 287-300.
  • [11] He X., S. Hu, J. Obloj and X.Y. Zhou, 2017, Path dependent and randomized strategies in Barberis’ Casino Gambling model, Operations Research, 65, 1, 97-103.
  • [12] Hirsch F., C. Profetta, B. Roynette and M. Yor, 2011, Constructing self-similar martingales via two Skorokhod embeddings. Sem. de Prob. XLIII 451-503 LNM 2006, Springer-Verlag, Berlin.
  • [13] Loomes G. and R. Sugden, 1982, Regret theory: An alternative theory of rational choice under uncertainty, Economic Journal, 92, 805-824.
  • [14] Machina M., 1985, Stochastic Choice Functions Generated from Deterministic Preferences over Lotteries, Economic Journal, 95, 379, 575-594.
  • [15] Quiggin J., 1982, A Theory of Anticipated Utility, Journal of Economic Behaviour and Organisation, 3, 323-343.
  • [16] Rogers L.C.G. and D. Williams, 2000, Diffusions, Markov Processes and Martingales: Itô Calculus Wiley, Chichester.
  • [17] Rogozin B.A., 1996, On the distribution of functionals related to boundary problems for processes with independent increments. Th. Prob. Appl., 11, 580-591.
  • [18] Skorokhod A.V., 1965, Studies in the theory of random processes, Addison-Wesley, Reading, Mass..
  • [19] Strack P. and P. Viefers, 2017, Too Proud to Stop: Regret in Dynamic Decisions, SSRN Working paper, id2465840.
  • [20] Tversky, A. and D. Kahneman, 1992, Advances in Prospect Theory: Cumulative Representation of Uncertainty, Journal of Risk and Uncertainty, 5, 297-323.
  • [21] Wakker P., 2010, Prospect Theory for Risk and Ambiguity, Cambridge University Press.
  • [22] Xu Z.Q. and X.Y. Zhou, 2013, Optimal stopping under probability distortion. Ann. Appl. Prob., 23, 1, 251-282.