Applied Mathematics & Optimization

, Volume 79, Issue 1, pp 145–177 | Cite as

A Verification Theorem for Optimal Stopping Problems with Expectation Constraints

  • Stefan AnkirchnerEmail author
  • Maike Klein
  • Thomas Kruse


We consider the problem of optimally stopping a continuous-time process with a stopping time satisfying a given expectation cost constraint. We show, by introducing a new state variable, that one can transform the problem into an unconstrained control problem and hence obtain a dynamic programming principle. We characterize the value function in terms of the dynamic programming equation, which turns out to be an elliptic, fully non-linear partial differential equation of second order. We prove a classical verification theorem and illustrate its applicability with several examples.


Optimal stopping Expectation constraints Dynamic programming principle Verification theorem 



We are grateful to Bruno Bouchard, Goran Peskir, Mikhail Urusov, Song Yao and Mihail Zervos for helpful comments. We thank an anonymous referee for the careful reading of the manuscript and highly appreciate her/his comments which contributed to improve our paper.


  1. 1.
    Bokanowski, O., Picarelli, A., Zidani, H.: State-constrained stochastic optimal control problems via reachability approach. SIAM J. Control Optim. 54(5), 2568–2593 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bouchard, B., Elie, R., Imbert, C.: Optimal control under stochastic target constraints. SIAM J. Control Optim., 48(5):3501–3531 (2009/10)Google Scholar
  3. 3.
    Bouchard, B. Elie, R., Touzi, N.: Stochastic target problems with controlled loss. SIAM J. Control Optim., 48(5):3123–3150 (2009/10)Google Scholar
  4. 4.
    Bouchard, B., Touzi, N.: Weak dynamic programming principle for viscosity solutions. SIAM J. Control Optim. 49(3), 948–962 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Horiguchi, M.: Markov decision processes with a stopping time constraint. Math. Methods Oper. Res. 53(2), 279–295 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus. Graduate Texts in Mathematics, vol. 113, 2nd edn. Springer, New York (1991)Google Scholar
  7. 7.
    Kennedy, D.P.: On a constrained optimal stopping problem. J. Appl. Probab. 19(3), 631–641 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Miller, C.W.: Non-linear PDE approach to time-inconsistent optimal stopping. SIAM J. Control Optim. 55(1), 557–573 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Palczewski, J., Stettner, Ł.: Infinite horizon stopping problems with (nearly) total reward criteria. Stoch. Process. Appl. 124(12), 3887–3920 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Pedersen, J.L., Peskir, G.: Optimal mean-variance selling strategies. Math. Financ. Econ. 10(2), 203–220 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Peskir, G.: Optimal detection of a hidden target: the median rule. Stoch. Process. Appl. 122(5), 2249–2263 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Peskir, G., Shiryaev, A.: Optimal stopping and free-boundary problems. Springer, New York (2006)zbMATHGoogle Scholar
  13. 13.
    Soner, H.M., Touzi, N.: Stochastic target problems, dynamic programming, and viscosity solutions. SIAM J. Control Optim. 41(2), 404–424 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Institute for MathematicsUniversity of JenaJenaGermany
  2. 2.Faculty of MathematicsUniversity of Duisburg-EssenEssenGermany

Personalised recommendations