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Stability estimating in optimal stopping problem. (English) Zbl 1154.60326
Summary: We consider the optimal stopping problem for a discrete-time Markov process on a Borel state space $$X$$. It is supposed that an unknown transition probability $$p(\cdot |x)$$, $$x\in X$$, is approximated by the transition probability $$\widetilde{p}(\cdot |x)$$, $$x\in X$$, and the stopping rule $$\widetilde{\tau}_*$$, optimal for $$\widetilde{p}$$, is applied to the process governed by $$p$$. We found an upper bound for the difference between the total expected cost, resulting when applying $$\widetilde{\tau}_*$$, and the minimal total expected cost. The bound given is a constant times $$\sup_{x\in X}\|p(\cdot |x)-\widetilde{p}(\cdot |x)\|$$, where $$\|\cdot\|$$ is the total variation norm.

##### MSC:
 60G40 Stopping times; optimal stopping problems; gambling theory
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##### References:
 [1] Allart P.: Optimal stopping rules for correlated random walks with a discount. J. Appl. Prob. 41 (2004), 483-496 · Zbl 1071.60026 · doi:10.1239/jap/1082999080 [2] Bertsekas D. P.: Dynamic Programming: Deterministic and Stochastic Models. Prentice Hall, Englewood Cliffs, N. J. 1987 · Zbl 0649.93001 [3] Bertsekas D. P., Shreve S. E.: Stochastic Optimal Control: The Discrete Time Case. Academic Press, New York 1979 · Zbl 0633.93001 [4] Dijk N. M. Van: Perturbation theory for unbounded Markov reward process with applications to queueing systems. Adv. in Appl. Probab. 20 (1988), 99-111 · Zbl 0642.60099 · doi:10.2307/1427272 [5] Dijk N. M. Van, Sladký K.: Error bounds for nonnegative dynamic models. J. Optim. Theory Appl. 101 (1999), 449-474 · Zbl 0946.90106 · doi:10.1023/A:1021749829267 [6] Dynkin E. B., Yushkevich A. A.: Controlled Markov Process. Springer-Verlag, New York 1979 [7] Favero G., Runggaldier W. J.: A robustness results for stochastic control. Systems Control Lett. 46 (2002), 91-97 · Zbl 0994.93008 · doi:10.1016/S0167-6911(02)00121-4 [8] Gordienko E. I.: An estimate of the stability of optimal control of certain stochastic and deterministic systems. J. Soviet Math. 59 (1992), 891-899. ( · Zbl 1267.49026 · doi:10.1007/BF01099115 [9] Gordienko E. I., Salem F. S.: Robustness inequality for Markov control process with unbounded costs. Systems Control Lett. 33 (1998), 125-130 · Zbl 0902.93068 · doi:10.1016/S0167-6911(97)00077-7 [10] Gordienko E. I., Yushkevich A. A.: Stability estimates in the problem of average optimal switching of a Markov chain. Math. Methods Oper. Res. 57 (2003), 345-365 · Zbl 1116.90401 · doi:10.1007/s001860200258 [11] Gordienko E. I., Lemus-Rodríguez E., Montes-de-Oca R.: Discounted cost optimality problem: stability with respect to weak metrics. In press in: Math. Methhods Oper. Res. (2008) · Zbl 1166.60041 · doi:10.1007/s00186-007-0171-z [12] Hernández-Lerma O., Lassere J. B.: Discrete-Time Markov Control Processes: Basic Optimality Criteria. Springer-Verlag, N.Y. 1996 [13] Jensen U.: An optimal stopping problem in risk theory. Scand. Actuarial J.2 (1997), 149-159 · Zbl 0888.62104 · doi:10.1080/03461238.1997.10413984 [14] Meyn S. P., Tweedie R. L.: Markov Chains and Stochastic Stability. Springer-Verlag, London 1993 · Zbl 1165.60001 · doi:10.1017/CBO9780511626630 [15] Montes-de-Oca R., Salem-Silva F.: Estimates for perturbations of an average Markov decision process with a minimal state and upper bounded by stochastically ordered Markov chains. Kybernetika 41 (2005), 757-772 · Zbl 1249.90313 · www.kybernetika.cz · eudml:33786 [16] Montes-de-Oca R., Sakhanenko, A., Salem-Silva F.: Estimate for perturbations of general discounted Markov control chains. Appl. Math. 30 (2003), 287-304 · Zbl 1055.90086 · doi:10.4064/am30-3-4 [17] Muciek B. K.: Optimal stopping of a risk process: model with interest rates. J. Appl. Prob. 39 (2002), 261-270 · Zbl 1011.62111 · doi:10.1239/jap/1025131424 [18] Müller A.: How does the value function of a Markov decision process depend on the transition probabilities? Math. Oper. Res. 22 (1997), 872-885 · Zbl 0892.90178 · doi:10.1287/moor.22.4.872 [19] Schäl M.: Conditions for optimality in dynamic programming and for the limit of $$n$$-stage optimal policies to be optimal. Z. Wahrsch. verw. Gebiete 32 (1975), 179-196 · Zbl 0316.90080 · doi:10.1007/BF00532612 [20] Shiryaev A. N.: Optimal Stopping Rules. Springer-Verlag, New York 1978 · Zbl 1138.60008 · doi:10.1007/978-3-540-74011-7 [21] Shiryaev A. N.: Essential of Stochastic Finance. Facts, Models, Theory. World Scientific Publishing Co., Inc., River Edge, N.J. 1999 · Zbl 0926.62100
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