Proceedings of the Steklov Institute of Mathematics

, Volume 287, Issue 1, pp 299–307 | Cite as

On the existence of solutions of unbounded optimal stopping problems

Article

Abstract

Known conditions of existence of solutions of optimal stopping problems for Markov processes assume that payoff functions are bounded in some sense. In this paper we prove weaker conditions which are applicable to unbounded payoff functions. The results obtained are applied to the optimal stopping problem for a Brownian motion with the payoff function G(τ,Gτ)=|Gτ|-c/(1-τ).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Breakwell and H. Chernoff, “Sequential tests for the mean of a normal distribution. II: Large t,” Ann. Math. Stat. 35, 162–173 (1964).CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    H. Chernoff, “Sequential tests for the mean of a normal distribution,” in Proc. Fourth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California Press, Berkeley, 1961), Vol. 1, pp. 79–91.Google Scholar
  3. 3.
    H. Chernoff, “Sequential tests for the mean of a normal distribution. III: Small t,” Ann. Math. Stat. 36, 28–54 (1965).CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    H. Chernoff, “Sequential tests for the mean of a normal distribution. IV: Discrete case,” Ann. Math. Stat. 36, 55–68 (1965).CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Y. S. Chow, H. Robbins, and D. Siegmund, Great Expectations: The Theory of Optimal Stopping (Houghton Mifflin, Boston, 1971).MATHGoogle Scholar
  6. 6.
    E. B. Dynkin, Foundations of the Theory of Markov Proceses (Fizmatgiz, Moscow, 1959). Engl. transl.: Theory of Markov Processes (Pergamon, Oxford, 1961).Google Scholar
  7. 7.
    N. V. Krylov, Controlled Diffusion Processes (Nauka, Moscow, 1977; Springer, New York, 1980).MATHGoogle Scholar
  8. 8.
    G. Peskir and A. Shiryaev, Optimal Stopping and Free-Boundary Problems (Birkhäuser, Basel, 2006).MATHGoogle Scholar
  9. 9.
    D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd ed. (Springer, Berlin, 2005).MATHGoogle Scholar
  10. 10.
    A. N. Shiryaev, Optimal Stopping Rules (Springer, New York, 1978).MATHGoogle Scholar
  11. 11.
    A. N. Shiryaev, Probability, 3rd ed. (MTsNMO, Moscow, 2004) [in Russian].Google Scholar
  12. 12.
    M. V. Zhitlukhin, “Sequential methods of testing statistical hypotheses and detecting changepoints,” Cand. Sci. (Phys.-Math.) Dissertation (Steklov Math. Inst., Moscow, 2013).Google Scholar
  13. 13.
    M. V. Zhitlukhin and A. A. Muravlev, “On Chernoff’s hypotheses testing problem for the drift of a Brownian motion,” Teor. Veroyatn. Primen. 57(4), 778–788 (2012) [Theory Probab. Appl. 57, 708–717 (2013)].CrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.International Laboratory of Quantitative FinanceNational Research University Higher School of EconomicsMoscowRussia
  3. 3.Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia

Personalised recommendations