The Annals of Applied Probability

On approximative solutions of multistopping problems

Andreas Faller and Ludger Rüschendorf

Full-text: Open access

Abstract

In this paper, we consider multistopping problems for finite discrete time sequences X1,  …,  Xn. m-stops are allowed and the aim is to maximize the expected value of the best of these m stops. The random variables are neither assumed to be independent not to be identically distributed. The basic assumption is convergence of a related imbedded point process to a continuous time Poisson process in the plane, which serves as a limiting model for the stopping problem. The optimal m-stopping curves for this limiting model are determined by differential equations of first order. A general approximation result is established which ensures convergence of the finite discrete time m-stopping problem to that in the limit model. This allows the construction of approximative solutions of the discrete time m-stopping problem. In detail, the case of i.i.d. sequences with discount and observation costs is discussed and explicit results are obtained.

Article information

Source
Ann. Appl. Probab., Volume 21, Number 5 (2011), 1965-1993.

Dates
First available in Project Euclid: 25 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1319576614

Digital Object Identifier
doi:10.1214/10-AAP747

Mathematical Reviews number (MathSciNet)
MR2884056

Zentralblatt MATH identifier
1251.60038

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

Keywords
Optimal multiple stopping best choice problem extreme values Poisson process

Citation

Faller, Andreas; Rüschendorf, Ludger. On approximative solutions of multistopping problems. Ann. Appl. Probab. 21 (2011), no. 5, 1965--1993. doi:10.1214/10-AAP747. https://projecteuclid.org/euclid.aoap/1319576614


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References

  • Assaf, D., Goldstein, L. and Samuel-Cahn, E. (2004). Two-choice optimal stopping. Adv. in Appl. Probab. 36 1116–1147.
  • Assaf, D., Goldstein, L. and Samuel-Cahn, E. (2006). Maximizing expected value with two stage stopping rules. In Random Walk, Sequential Analysis and Related Topics 3–27. World Sci. Publ., Hackensack, NJ.
  • Baryshnikov, Y. M. and Gnedin, A. V. (2000). Sequential selection of an increasing sequence from a multidimensional random sample. Ann. Appl. Probab. 10 258–267.
  • Bruss, F. T. (2010). On a class of optimal stopping problems with mixed constraints. Discrete Math. Theor. Comput. Sci. 12 363–380.
  • Bruss, F. T. and Delbaen, F. (2001). Optimal rules for the sequential selection of monotone subsequences of maximum expected length. Stochastic Process. Appl. 96 313–342.
  • Bruss, F. T. and Ferguson, T. S. (1997). Multiple buying or selling with vector offers. J. Appl. Probab. 34 959–973.
  • Bruss, F. T. and Rogers, L. C. G. (1991). Embedding optimal selection problems in a Poisson process. Stochastic Process. Appl. 38 267–278.
  • Chow, Y. S., Robbins, H. and Siegmund, D. (1971). Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, Boston, MA.
  • de Haan, L. and Verkade, E. (1987). On extreme-value theory in the presence of a trend. J. Appl. Probab. 24 62–76.
  • Faller, A. (2009). Approximative Lösungen von Mehrfachstoppproblemen. Dissertation, Univ. Freiburg.
  • Faller, A. and Rüschendorf, L. (2009). On approximative solutions of optimal stopping problems. Preprint, Univ. Freiburg.
  • Ferguson, T. S. (2007). Optimal stopping and applications. Electronic texts on homepage. Available at http://www.math.ucla.edu/~tom/Stopping/Contents.html.
  • Gnedin, A. V. (1996). On the full information best-choice problem. J. Appl. Probab. 33 678–687.
  • Gnedin, A. V. and Sakaguchi, M. (1992). On a best choice problem related to the Poisson process. In Strategies for Sequential Search and Selection in Real Time (Amherst, MA, 1990). Contemp. Math. 125 59–64. Amer. Math. Soc., Providence, RI.
  • Goldstein, L. and Samuel-Cahn, E. (2006). Optimal two-choice stopping on an exponential sequence. Sequential Anal. 25 351–363.
  • Haggstrom, G. W. (1967). Optimal sequential procedures when more than one stop is required. Ann. Math. Statist. 38 1618–1626.
  • Karlin, S. (1962). Stochastic models and optimal policy for selling an asset. In Studies in Applied Probability and Management Science 148–158. Stanford Univ. Press, Stanford, CA.
  • Kennedy, D. P. and Kertz, R. P. (1990). Limit theorems for threshold-stopped random variables with applications to optimal stopping. Adv. in Appl. Probab. 22 396–411.
  • Kennedy, D. P. and Kertz, R. P. (1991). The asymptotic behavior of the reward sequence in the optimal stopping of i.i.d. random variables. Ann. Probab. 19 329–341.
  • Kühne, R. and Rüschendorf, L. (2000a). Approximation of optimal stopping problems. Stochastic Process. Appl. 90 301–325.
  • Kühne, R. and Rüschendorf, L. (2000b). Optimal stopping with discount and observation costs. J. Appl. Probab. 37 64–72.
  • Kühne, R. and Rüschendorf, L. (2002). On optimal two-stopping problems. In Limit Theorems in Probability and Statistics, Vol. II (Balatonlelle, 1999) 261–271. János Bolyai Math. Soc., Budapest.
  • Kühne, R. and Rüschendorf, L. (2004). Approximate optimal stopping of dependent sequences. Theory Probab. Appl. 48 465–480.
  • Moser, L. (1956). On a problem of Cayley. Scripta Mathematica 22 289–292.
  • Nikolaev, M. L. (1999). On optimal multiple stopping of Markov sequences. Theory Probab. Appl. 43 298–306.
  • Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York.
  • Saario, V. and Sakaguchi, M. (1992). Multistop best choice games related to the Poisson process. Math. Japon. 37 41–51.
  • Sakaguchi, M. (1976). Optimal stopping problems for randomly arriving offers. Math. Japon. 21 201–217.
  • Sakaguchi, M. and Saario, V. (1995). A class of best-choice problems with full information. Math. Japon. 41 389–398.
  • Siegmund, D. O. (1967). Some problems in the theory of optimal stopping rules. Ann. Math. Statist. 38 1627–1640.