# Article

 Title: Stability estimating in optimal stopping problem  (English) Author: Zaitseva, Elena Language: English Journal: Kybernetika ISSN: 0023-5954 Volume: 44 Issue: 3 Year: 2008 Pages: 400-415 Summary lang: English . Category: math . 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 $\displaystyle \sup \nolimits _{x\in X}\Vert p(\cdot |x)-\widetilde{p}(\cdot |x)\Vert$, where $\Vert \cdot \Vert$ is the total variation norm. Keyword: discrete-time Markov process Keyword: optimal stopping rule Keyword: stability index Keyword: total variation metric Keyword: contractive operator Keyword: optimal asset selling MSC: 60G40 MSC: 60J10 idZBL: Zbl 1154.60326 idMR: MR2436040 . Date available: 2009-09-24T20:35:40Z Last updated: 2012-06-06 Stable URL: http://hdl.handle.net/10338.dmlcz/135859 . Reference: [1] Allart P.: Optimal stopping rules for correlated random walks with a discount.J. Appl. Prob. 41 (2004), 483–496 MR 2052586 Reference: [2] Bertsekas D. P.: Dynamic Programming: Deterministic and Stochastic Models.Prentice Hall, Englewood Cliffs, N. J. 1987 Zbl 0649.93001, MR 0896902 Reference: [3] Bertsekas D. P., Shreve S. E.: Stochastic Optimal Control: The Discrete Time Case.Academic Press, New York 1979 Zbl 0633.93001, MR 0511544 Reference: [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 MR 0932536 Reference: [5] Dijk N. M. Van, Sladký K.: Error bounds for nonnegative dynamic models.J. Optim. Theory Appl. 101 (1999), 449–474 MR 1684679 Reference: [6] Dynkin E. B., Yushkevich A. A.: Controlled Markov Process.Springer-Verlag, New York 1979 MR 0554083 Reference: [7] Favero G., Runggaldier W. J.: A robustness results for stochastic control.Systems Control Lett. 46 (2002), 91–97 MR 2010062 Reference: [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. (Translated from the Russian publication of 1989) MR 1163393 Reference: [9] Gordienko E. I., Salem F. S.: Robustness inequality for Markov control process with unbounded costs.Systems Control Lett. 33 (1998), 125–130 MR 1607814 Reference: [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, MR 1990916 Reference: [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, MR 2429561 Reference: [12] Hernández-Lerma O., Lassere J. B.: Discrete-Time Markov Control Processes: Basic Optimality Criteria.Springer-Verlag, N.Y. 1996 Reference: [13] Jensen U.: An optimal stopping problem in risk theory.Scand. Actuarial J.2 (1997), 149–159 Zbl 0888.62104, MR 1492423 Reference: [14] Meyn S. P., Tweedie R. L.: Markov Chains and Stochastic Stability.Springer-Verlag, London 1993 Zbl 1165.60001, MR 1287609 Reference: [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 MR 2193864 Reference: [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 MR 2029538 Reference: [17] Muciek B. K.: Optimal stopping of a risk process: model with interest rates.J. Appl. Prob. 39 (2002), 261–270 Zbl 1011.62111, MR 1908943 Reference: [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 MR 1484687 Reference: [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, MR 0378841 Reference: [20] Shiryaev A. N.: Optimal Stopping Rules.Springer-Verlag, New York 1978 Zbl 1138.60008, MR 2374974 Reference: [21] Shiryaev A. N.: Essential of Stochastic Finance.Facts, Models, Theory. World Scientific Publishing Co., Inc., River Edge, N.J. 1999 MR 1695318 .

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