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

  • M. V. ZhitlukhinEmail author
  • A. N. Shiryaev


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,G τ)=|G τ|-c/(1-τ).


Brownian Motion Function Versus STEKLOV Institute Lower Semicontinuous Standard Brownian Motion 
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  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).CrossRefzbMATHMathSciNetGoogle 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).CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    H. Chernoff, “Sequential tests for the mean of a normal distribution. IV: Discrete case,” Ann. Math. Stat. 36, 55–68 (1965).CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Y. S. Chow, H. Robbins, and D. Siegmund, Great Expectations: The Theory of Optimal Stopping (Houghton Mifflin, Boston, 1971).zbMATHGoogle 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).zbMATHGoogle Scholar
  8. 8.
    G. Peskir and A. Shiryaev, Optimal Stopping and Free-Boundary Problems (Birkhäuser, Basel, 2006).zbMATHGoogle Scholar
  9. 9.
    D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd ed. (Springer, Berlin, 2005).zbMATHGoogle Scholar
  10. 10.
    A. N. Shiryaev, Optimal Stopping Rules (Springer, New York, 1978).zbMATHGoogle 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

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© 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

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