Abstract
An optimal stopping problem of a Markov process with infinite horizon is considered. For the case of discrete time and finite number m of states Sonin proposed an algorithm which allows to find the value function and the stopping set in no more than 2(m−1) steps. The algorithm is based on a modification of a Markov chain on each step, related to the elimination of the states which definitely belong to the continuation set. To solve the problem with arbitrary state space and to have possibility of a generalization to a continuous time, the procedure was modified in Presman (Stochastics 83(4–6):467–475, 2011). The modified procedure was based on a sequential modification of the payoff function for the same chain in such a way that the value function is the same for both problems and the modified payoff function is greater than the initial one on some set and is equal to it on the complement. In this paper, we give some examples showing that the procedure can be generalized to continuous time.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bronstein, A.L., Hughston, L.P., Pistorius, M.R., Zervos, M.: Discretionary stopping of one-dimensional Ito diffusions with a staircase reward function. J. Appl. Probab. 43, 984–996 (2006)
Dayanik, S., Karatzas, I.: On the optimal stopping problem for one-dimensional diffusions. Stoch. Process. Appl. 107, 173–212 (2003)
Feldman, R., Valdez-Flores, C.: Applied Probability and Stochastic Processes. PWS, Boston (1995)
Peskir, P., Shiryaev, A.N.: Optimal Stopping and Free-Boundary Problems. Birkhauser, Basel (2006)
Presman, E.L.: On Sonin’s algorithm for solution of the optimal stopping problem. In: Proceedings of the Fourth International Conference on Control Problems (January 26–30, 2009), pp. 300–309. Institute of Control Sciences (2009)
Presman, E.L.: A new approach to the solution of optimal stopping problem in a discrete time. Stochastics 83(4–6), 467–475 (2011)
Presman, E.L., Sonin, I.M.: On optimal stopping of random sequences modulated by Markov chain. Theory Probab. Appl. 54(3), 534–542 (2009)
Salminen, P.: Optimal stopping of one-dimensional diffusions. Math. Nachr. 124, 85–101 (1985)
Shiryayev, A.N.: Statistical Sequential Analysis: Optimal Stopping Rules. Nauka, Moscow (1969) (in Russian). English translation of the second edition: Shiryayev, A.N.: Optimal Stopping Rules, Springer, Berlin, 1978
Sonin, I.M.: Two simple theorems in the problems of optimal stopping. In: Proc. 8th INFORMS Applied Probability Conference, Atlanta, Georgia, p. 27 (1995)
Sonin, I.M.: The elimination algorithm for the problem of optimal stopping. Math. Methods Oper. Res. 49, 111–123 (1999)
Sonin, I.M.: The state reduction and related algorithms and their applications to the study of Markov chains, graph theory and the optimal stopping problem. Adv. Math. 145, 159–188 (1999)
Sonin, I.M.: Optimal stopping of Markov chains and recursive solution of Poisson and Bellman equations. In: Kabanov, Yu., Liptser, R., Stoyanov, J. (eds.) From Stochastic Calculus to Mathematical Finance. The Shiryaev Festschrift, pp. 609–621. Springer, Berlin (2006)
Acknowledgements
The author would like to thank V.I. Arkin, A.D. Slastnikov for useful discussions, I.M. Sonin, Yu.M. Kabanov and anonymous referees for very valuable remarks and suggestions, and one of the referees for drawing his attention to the paper [1].
This work was partly supported by RFBR grant 10-01-00767-a.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Presman, E. (2014). Solution of Optimal Stopping Problem Based on a Modification of Payoff Function. In: Kabanov, Y., Rutkowski, M., Zariphopoulou, T. (eds) Inspired by Finance. Springer, Cham. https://doi.org/10.1007/978-3-319-02069-3_23
Download citation
DOI: https://doi.org/10.1007/978-3-319-02069-3_23
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-02068-6
Online ISBN: 978-3-319-02069-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)