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January  2017, 13(1): 399-411. doi: 10.3934/jimo.2016023

Optimal stopping problems with restricted stopping times

 1 Department of Statistics and Actuarial Science, East China Normal University, Shanghai, China 2 Department of Mathematics, Zhejiang Normal University, China 3 Department of Applied Finance and Actuarial Studies, Macquarie University, Sydney, Australia

* Corresponding author: Prof. Xianyi Wu, E-mail: xywu@stat.ecnu.edu.cn

Received  May 2015 Published  March 2016

Fund Project: This research was partially supported by the Natural Science Foundation of China under Grant No. 71371074, 111 Project Grant No. B14019 and the Australian Research Council Discovery Project Grant No. DP1094153.

This paper provides a general ground for the problems of optimal stopping times over the families of partially available (or restricted) stopping times. It subsumes the classical framework in continuous-time, discrete-time, as well as semi-Markov settings as special cases. We model the problem by a restricted pool of stopping times meeting certain natural conditions and present its solution by means of Snell's envelope technique that extends the classical results. We further extend this type of problems to the stochastic processes indexed by partially ordered set.

Citation: Wenqing Bao, Xianyi Wu, Xian Zhou. Optimal stopping problems with restricted stopping times. Journal of Industrial & Management Optimization, 2017, 13 (1) : 399-411. doi: 10.3934/jimo.2016023
References:
 [1] J. A. Bather and H. Chernoff, Sequential decisions in the control of a spaceship, Proceeding of Fifth Berkeley Symposium on Mathematical Statistics and Probability, 3 (1966), 181-207.   Google Scholar [2] A. Bensoussan, On the theory of option pricing, Acta Applicandae Mathematicae, 2 (1984), 139-158.   Google Scholar [3] J.-M. Bismut and B. Skalli, Temps d'arr$\hat{e}$t th$\hat{e}$orie g$\hat{e}$n$\hat{e}$rale de processus et processus de Markov, Z. Wasrscheinlichkeitstheorie Verw. Gebiete, 39 (1977), 301-313.  doi: 10.1007/BF01877497.  Google Scholar [4] M. J. Brennan and E. S. Schwartz, Evaluating natural rescource investments, Journal of Business, 58 (1985), 135-157.   Google Scholar [5] M. Broadie and P. Glasserman, Pricing American-style securities using simulation, Journal of Economic Dynamics and Control, 21 (1997), 1323-1352.  doi: 10.1016/S0165-1889(97)00029-8.  Google Scholar [6] X. Cai, X. Wu and X. Zhou, Dynamically optimal policies for stochastic scheduling subject 84 to breakdown-repeat breakdowns, IEEE Transactions on Automation Science and Engineering, 85 (2005), 158-172.   Google Scholar [7] X. Cai, X. Wu and X. Zhou, Stochastic scheduling subject to preemptive-repeat breakdowns with incomplete information, Operations Research, 57 (2009), 1236-1249.  doi: 10.1287/opre.1080.0660.  Google Scholar [8] Y. S. Chow, H. Robbins and D. Siegmund, Great Expectations, The Theory of Optimal Stopping Houghton Mifflin, Boston, 1971.  Google Scholar [9] P. Dupuis and H. Wang, Optimal stopping with random intervention times, Advances in Applied Probability, 34 (2002), 141-157.  doi: 10.1239/aap/1019160954.  Google Scholar [10] E. B. Dynkin, Optimal choice of a stopping time for a Markov process, Dokl. Akad. Nauk USSR, 150 (1963), 238-240.   Google Scholar [11] A. G. Fakeev, Optimal stopping of a Markov process, Theory of Probability and Its Applications, 15 (1970), 324-331.  doi: 10.1137/1116076.  Google Scholar [12] A. G. Fakeev, Optimal stopping rules for processes with continuous parameter, Theory of Probability and Its Applications, 16 (1971), 694-696.  doi: 10.1137/1115039.  Google Scholar [13] A. G. Fakeev, On optimal stopping rules for stochastic processes with continuous parameter, Theory of Probability and Its Applications, 18 (1973), 304-311.  doi: 10.1137/1115039.  Google Scholar [14] J. Gittins, K. Glazebrook and R. Weber, Multi-Armed Bandit Allocation Indices, 2nd edition, John Wiley & Sons, Ltd., 2011. Google Scholar [15] H. Kaspi and A. Mandelbaum, Multi-armed bandits in discrete and continuous time, Annals of Applied Probability, 8 (1998), 1270-1290.  doi: 10.1214/aoap/1028903380.  Google Scholar [16] I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer-Verlag, New York, 1998. doi: 10.1007/b98840.  Google Scholar [17] N. E. Karoui, Les aspects probabilities du contrôle stochastique, Lecture Notes in Mathematics, 876, Springer-Verlag, Berlin, 1981, 73-238. Google Scholar [18] N. E. Karoui and I. Karatzas, Dynamic allocation problems in continuous time, Annals of Applied Probability, 4 (1994), 255-286.  doi: 10.1214/aoap/1177005062.  Google Scholar [19] U. Krengel and L. Sucheston, Stopping rules and tactics for processes indexed by a directed set, Journal of Multivariate Analysis, 11 (1981), 199-229.  doi: 10.1016/0047-259X(81)90109-3.  Google Scholar [20] G. F. Lawler and R. J. Vanderbei, Markov strategies for optimal control problems indexed by a partially ordered set, The Annals of Probability, 11 (1983), 642-647.  doi: 10.1214/aop/1176993508.  Google Scholar [21] A. Mandelbaum and R. J. Vanderbei, Optimal stopping and supermartingales over partially ordered sets, Probability Theory and Related Fields, 57 (1981), 253-264.  doi: 10.1007/BF00535493.  Google Scholar [22] M. P. McKean, A free boundary problem for the heat equation arising from a problem in Mathematical Economics, Industrial Management Review, 6 (1965), 32-39.   Google Scholar [23] J. F. Mertens, Processus stochastiques g$\acute{e}$n$\acute{e}$raux et surmartingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 22 (1972), 45-48.  doi: 10.1007/BF00538905.  Google Scholar [24] J. Neveu, Discrete-Parameter Martingales, English translation, North-Holland, Amsterdam and American Elsevier, New York, 1975.  Google Scholar [25] D. Nualart, Randomized stopping points and optimal stopping on the plane, The Annals of Probability, 20 (1992), 883-900.  doi: 10.1214/aop/1176989810.  Google Scholar [26] B. Oksendal and A. Sulem, Optimal consumption and portfolio with both fixed and proportional transaction costs, SIAM Journal on Control and Optimization, 40 (2002), 1765-1790.  doi: 10.1137/S0363012900376013.  Google Scholar [27] G. Peskir and A. N. Shiryaev, Optimal Stopping and Free-Boundary Problems, Lectures in Mathematics, ETH Zürich, Birkhäuser, 2006.  Google Scholar [28] H. Pham and P. Tankov, A coupled system of integrodifferential equations arising in liquidity risk model, Applied Mathematics and Optimization, 59 (2009), 147-173.  doi: 10.1007/s00245-008-9046-9.  Google Scholar [29] C. Rogers and O. Zane, A simple model of liquidity effects, in Advances in Finance and Stochastics, Essays in Honour of Dieter Sondermann (eds. K. Sandmann and P. Schoenbucher), Springer, Berlin, 2002, 161-176.  Google Scholar [30] A. N. Shiryaev, Statistical Sequential Analysis, (in Russian) Nauka, Moscow, 1976.  Google Scholar [31] A. N. Shiryaev, Optimal Stopping Rules, Springer-Verlag, New York, 1978.  Google Scholar [32] I. L. Snell, Applications of martingale system theory, Transactions of the American Mathematical Society, 73 (1953), 293-312.   Google Scholar [33] M. E. Thompson, Continuous parameter optimal stopping problems, Z. Wahrsheinlichkeitstheorie und Verw. Gebiete, 19 (1971), 302-318.  doi: 10.1007/BF00535835.  Google Scholar [34] A. Wald, Sequential Analysis, Wiley, New York, 1947.  Google Scholar [35] J. B. Walsh, Optional increasing paths, in Processus Aléatoire a Deux Indices, Lecture Notes in Mathematics, 863, Springer, Berlin, 1981, 172-201.  Google Scholar

show all references

References:
 [1] J. A. Bather and H. Chernoff, Sequential decisions in the control of a spaceship, Proceeding of Fifth Berkeley Symposium on Mathematical Statistics and Probability, 3 (1966), 181-207.   Google Scholar [2] A. Bensoussan, On the theory of option pricing, Acta Applicandae Mathematicae, 2 (1984), 139-158.   Google Scholar [3] J.-M. Bismut and B. Skalli, Temps d'arr$\hat{e}$t th$\hat{e}$orie g$\hat{e}$n$\hat{e}$rale de processus et processus de Markov, Z. Wasrscheinlichkeitstheorie Verw. Gebiete, 39 (1977), 301-313.  doi: 10.1007/BF01877497.  Google Scholar [4] M. J. Brennan and E. S. Schwartz, Evaluating natural rescource investments, Journal of Business, 58 (1985), 135-157.   Google Scholar [5] M. Broadie and P. Glasserman, Pricing American-style securities using simulation, Journal of Economic Dynamics and Control, 21 (1997), 1323-1352.  doi: 10.1016/S0165-1889(97)00029-8.  Google Scholar [6] X. Cai, X. Wu and X. Zhou, Dynamically optimal policies for stochastic scheduling subject 84 to breakdown-repeat breakdowns, IEEE Transactions on Automation Science and Engineering, 85 (2005), 158-172.   Google Scholar [7] X. Cai, X. Wu and X. Zhou, Stochastic scheduling subject to preemptive-repeat breakdowns with incomplete information, Operations Research, 57 (2009), 1236-1249.  doi: 10.1287/opre.1080.0660.  Google Scholar [8] Y. S. Chow, H. Robbins and D. Siegmund, Great Expectations, The Theory of Optimal Stopping Houghton Mifflin, Boston, 1971.  Google Scholar [9] P. Dupuis and H. Wang, Optimal stopping with random intervention times, Advances in Applied Probability, 34 (2002), 141-157.  doi: 10.1239/aap/1019160954.  Google Scholar [10] E. B. Dynkin, Optimal choice of a stopping time for a Markov process, Dokl. Akad. Nauk USSR, 150 (1963), 238-240.   Google Scholar [11] A. G. Fakeev, Optimal stopping of a Markov process, Theory of Probability and Its Applications, 15 (1970), 324-331.  doi: 10.1137/1116076.  Google Scholar [12] A. G. Fakeev, Optimal stopping rules for processes with continuous parameter, Theory of Probability and Its Applications, 16 (1971), 694-696.  doi: 10.1137/1115039.  Google Scholar [13] A. G. Fakeev, On optimal stopping rules for stochastic processes with continuous parameter, Theory of Probability and Its Applications, 18 (1973), 304-311.  doi: 10.1137/1115039.  Google Scholar [14] J. Gittins, K. Glazebrook and R. Weber, Multi-Armed Bandit Allocation Indices, 2nd edition, John Wiley & Sons, Ltd., 2011. Google Scholar [15] H. Kaspi and A. Mandelbaum, Multi-armed bandits in discrete and continuous time, Annals of Applied Probability, 8 (1998), 1270-1290.  doi: 10.1214/aoap/1028903380.  Google Scholar [16] I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer-Verlag, New York, 1998. doi: 10.1007/b98840.  Google Scholar [17] N. E. Karoui, Les aspects probabilities du contrôle stochastique, Lecture Notes in Mathematics, 876, Springer-Verlag, Berlin, 1981, 73-238. Google Scholar [18] N. E. Karoui and I. Karatzas, Dynamic allocation problems in continuous time, Annals of Applied Probability, 4 (1994), 255-286.  doi: 10.1214/aoap/1177005062.  Google Scholar [19] U. Krengel and L. Sucheston, Stopping rules and tactics for processes indexed by a directed set, Journal of Multivariate Analysis, 11 (1981), 199-229.  doi: 10.1016/0047-259X(81)90109-3.  Google Scholar [20] G. F. Lawler and R. J. Vanderbei, Markov strategies for optimal control problems indexed by a partially ordered set, The Annals of Probability, 11 (1983), 642-647.  doi: 10.1214/aop/1176993508.  Google Scholar [21] A. Mandelbaum and R. J. Vanderbei, Optimal stopping and supermartingales over partially ordered sets, Probability Theory and Related Fields, 57 (1981), 253-264.  doi: 10.1007/BF00535493.  Google Scholar [22] M. P. McKean, A free boundary problem for the heat equation arising from a problem in Mathematical Economics, Industrial Management Review, 6 (1965), 32-39.   Google Scholar [23] J. F. Mertens, Processus stochastiques g$\acute{e}$n$\acute{e}$raux et surmartingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 22 (1972), 45-48.  doi: 10.1007/BF00538905.  Google Scholar [24] J. Neveu, Discrete-Parameter Martingales, English translation, North-Holland, Amsterdam and American Elsevier, New York, 1975.  Google Scholar [25] D. Nualart, Randomized stopping points and optimal stopping on the plane, The Annals of Probability, 20 (1992), 883-900.  doi: 10.1214/aop/1176989810.  Google Scholar [26] B. Oksendal and A. Sulem, Optimal consumption and portfolio with both fixed and proportional transaction costs, SIAM Journal on Control and Optimization, 40 (2002), 1765-1790.  doi: 10.1137/S0363012900376013.  Google Scholar [27] G. Peskir and A. N. Shiryaev, Optimal Stopping and Free-Boundary Problems, Lectures in Mathematics, ETH Zürich, Birkhäuser, 2006.  Google Scholar [28] H. Pham and P. Tankov, A coupled system of integrodifferential equations arising in liquidity risk model, Applied Mathematics and Optimization, 59 (2009), 147-173.  doi: 10.1007/s00245-008-9046-9.  Google Scholar [29] C. Rogers and O. Zane, A simple model of liquidity effects, in Advances in Finance and Stochastics, Essays in Honour of Dieter Sondermann (eds. K. Sandmann and P. Schoenbucher), Springer, Berlin, 2002, 161-176.  Google Scholar [30] A. N. Shiryaev, Statistical Sequential Analysis, (in Russian) Nauka, Moscow, 1976.  Google Scholar [31] A. N. Shiryaev, Optimal Stopping Rules, Springer-Verlag, New York, 1978.  Google Scholar [32] I. L. Snell, Applications of martingale system theory, Transactions of the American Mathematical Society, 73 (1953), 293-312.   Google Scholar [33] M. E. Thompson, Continuous parameter optimal stopping problems, Z. Wahrsheinlichkeitstheorie und Verw. Gebiete, 19 (1971), 302-318.  doi: 10.1007/BF00535835.  Google Scholar [34] A. Wald, Sequential Analysis, Wiley, New York, 1947.  Google Scholar [35] J. B. Walsh, Optional increasing paths, in Processus Aléatoire a Deux Indices, Lecture Notes in Mathematics, 863, Springer, Berlin, 1981, 172-201.  Google Scholar
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