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Teor. Veroyatnost. i Primenen., 2004, Volume 49, Issue 2, Pages 373–382 (Mi tvp227)  

This article is cited in 22 scientific papers (total in 22 papers)

Short Communications

On an effective solution of the optimal stopping problem for random walks

A. A. Novikova, A. N. Shiryaevb

a University of Technology, Sydney
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We find a solution of the optimal stopping problem for the case when a reward function is an integer power function of a random walk on an infinite time interval. It is shown that an optimal stopping time is a first crossing time through a level defined as the largest root of Appell's polynomial associated with the maximum of the random walk. It is also shown that a value function of the optimal stopping problem on the finite interval $\{0,1\ldots T\}$ converges with an exponential rate as $T\to\infty$ to the limit under the assumption that jumps of the random walk are exponentially bounded.

Keywords: optimal stopping, random walk, rate of convergence, Appell polynomials.

DOI: https://doi.org/10.4213/tvp227

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English version:
Theory of Probability and its Applications, 2005, 49:2, 344–354

Bibliographic databases:

Received: 01.07.2002

Citation: A. A. Novikov, A. N. Shiryaev, “On an effective solution of the optimal stopping problem for random walks”, Teor. Veroyatnost. i Primenen., 49:2 (2004), 373–382; Theory Probab. Appl., 49:2 (2005), 344–354

Citation in format AMSBIB
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    2. V. I. Arkin, A. D. Slastnikov, “A Variational Approach to Optimal Stopping Problems for Diffusion Processes”, Theory Probab. Appl., 53:3 (2009), 467–480  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. Deligiannidis G., Le H., Utev S., “Optimal stopping for processes with independent increments, and applications”, J. Appl. Probab., 46:4 (2009), 1130–1145  crossref  mathscinet  zmath  isi  elib  scopus
    4. Klesov A., “Rate of convergence for certain optimal stopping problems”, Publ Math Debrecen, 76:3–4 (2010), 317–328  mathscinet  zmath  isi
    5. Abbring J.H., “Identification of Dynamic Discrete Choice Models”, Annual Review of Economics, 2, 2010, 367–394  crossref  isi  scopus
    6. Salminen P., “Optimal stopping, Appell polynomials, and Wiener-Hopf factorization”, Stochastics-An International Journal of Probability and Stochastic Processes, 83:4–6 (2011), 611–622  crossref  mathscinet  zmath  isi  scopus
    7. Boyarchenko S., Levendorskii S., “Optimal Stopping in Levy Models for Nonmonotone Discontinuous Payoffs”, SIAM J Control Optim, 49:5 (2011), 2062–2082  crossref  mathscinet  zmath  isi  elib  scopus
    8. Abbring J.H., “Mixed Hitting-Time Models”, Econometrica, 80:2 (2012), 783–819  crossref  mathscinet  zmath  isi  elib  scopus
    9. Christensen S., “Phase-Type Distributions and Optimal Stopping for Autoregressive Processes”, J. Appl. Probab., 49:1 (2012), 22–39  crossref  mathscinet  zmath  isi  elib  scopus
    10. Christensen S., Salminen P., Bao Quoc Ta, “Optimal Stopping of Strong Markov Processes”, Stoch. Process. Their Appl., 123:3 (2013), 1138–1159  crossref  mathscinet  zmath  isi  scopus
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    12. Ta B.Q., “Probabilistic Approach To Appell Polynomials”, Expo. Math., 33:3 (2015), 269–294  crossref  mathscinet  zmath  isi  scopus
    13. Mordecki E. Mishura Yu., “Optimal Stopping for Lévy Processes with One-Sided Solutions”, SIAM J. Control Optim., 54:5 (2016), 2553–2567  crossref  mathscinet  zmath  isi  elib  scopus
    14. Christensen S., Salminen P., “Impulse control and expected suprema”, Adv. Appl. Probab., 49:1 (2017), 238–257  crossref  mathscinet  isi  scopus
    15. Christensen S., “An effective method for the explicit solution of sequential problems on the real line”, Seq. Anal., 36:1 (2017), 2–18  crossref  mathscinet  zmath  isi  scopus
    16. Tartakovsky A.G., “Discussion on ?An effective method for the explicit solution of sequential problems on the real line? by S?ren Christensen”, Seq. Anal., 36:1 (2017), 19–23  crossref  mathscinet  zmath  isi  scopus
    17. Wijegunawardana P., Ojha V., Gera R., Soundarajan S., “Sampling Dark Networks to Locate People of Interest”, Soc. Netw. Anal. Min., 8:1 (2018), 15  crossref  isi  scopus
    18. Christensen S. Irle A., “A General Method For Finding the Optimal Threshold in Discrete Time”, Stochastics, 91:5 (2019), 728–753  crossref  isi
    19. Lin Y.-Sh., Yao Y.-Ch., “One-Sided Solutions For Optimal Stopping Problems With Logconcave Reward Functions”, Adv. Appl. Probab., 51:1 (2019), 87–115  crossref  mathscinet  isi
    20. V. I. Arkin, “Optimality of threshold stopping times for diffusion processes”, Theory Probab. Appl., 65:3 (2020), 341–358  mathnet  crossref  crossref  isi  elib
    21. O. V. Zverev, V. M. Khametov, E. A. Shelemekh, “Optimal stopping time for geometric random walks with power payoff function”, Autom. Remote Control, 81:7 (2020), 1192–1210  mathnet  crossref  crossref  isi  elib
    22. Bao Quoc Ta, “Appell Polynomilas Associated With Levy Processes and Applications”, Proc. Rom. Acad. Ser. A-Math. Phys., 21:3 (2020), 221–230  mathscinet  isi
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