Optimal stopping time problem for random walks with polynomial reward functions
Authors:
Yu. S. Mishura and V. V. Tomashyk
Translated by:
O. I. Klesov
Journal:
Theor. Probability and Math. Statist. 86 (2013), 155-167
MSC (2010):
Primary 60G40, 60G50
DOI:
https://doi.org/10.1090/S0094-9000-2013-00895-3
Published electronically:
August 20, 2013
MathSciNet review:
2986456
Full-text PDF Free Access
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Additional Information
Abstract: The optimal stopping time problem for random walks with a drift to the left and with a polynomial reward function is studied by using the Appel polynomials. An explicit form of optimal stopping times is obtained.
References
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- Evgenii B. Dynkin and Aleksandr A. Yushkevich, Markov processes: Theorems and problems, Plenum Press, New York, 1969. Translated from the Russian by James S. Wood. MR 0242252
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- Wim Schoutens, Stochastic processes and orthogonal polynomials, Lecture Notes in Statistics, vol. 146, Springer-Verlag, New York, 2000. MR 1761401
- V. V. Tomashik and Yu. S. Mishura, Optimal stopping times for random walks for polynomial reward functions, Applied Statistics. Actuarial and Financial Mathematics (2008), no. 1–2, 101–110. (Ukrainian)
- A. A. Borovkov, Veroyatnostnye protsessy v teorii massovogo obsluzhivaniya, Izdat. “Nauka”, Moscow, 1972 (Russian). MR 0315800
- O. V. Viskov, A random walk with an upper-continuous component, and the Lagrange inversion formula, Teor. Veroyatnost. i Primenen. 45 (2000), no. 1, 166–175 (Russian, with Russian summary); English transl., Theory Probab. Appl. 45 (2001), no. 1, 164–172. MR 1810980, DOI https://doi.org/10.1137/S0040585X97978105
- Wolfgang Stadje, An iterative approximation procedure for the distribution of the maximum of a random walk, Statist. Probab. Lett. 50 (2000), no. 4, 375–381. MR 1802232, DOI https://doi.org/10.1016/S0167-7152%2800%2900124-3
- Lajos Takács, Combinatorial methods in the theory of stochastic processes, John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0217858
- D. Korshunov, On distribution tail of the maximum of a random walk, Stochastic Process. Appl. 72 (1997), no. 1, 97–103. MR 1483613, DOI https://doi.org/10.1016/S0304-4149%2897%2900060-4
- A. A. Borovkov, On subexponential distributions and the asymptotics of the distribution of the maximum of sequential sums, Sibirsk. Mat. Zh. 43 (2002), no. 6, 1235–1264 (Russian, with Russian summary); English transl., Siberian Math. J. 43 (2002), no. 6, 995–1022. MR 1946226, DOI https://doi.org/10.1023/A%3A1021109132124
- S. Zakhari and S. G. Foss, On the exact asymptotics of the maximum of a random walk with increments in a class of light-tailed distributions, Sibirsk. Mat. Zh. 47 (2006), no. 6, 1265–1274 (Russian, with Russian summary); English transl., Siberian Math. J. 47 (2006), no. 6, 1034–1041. MR 2302843, DOI https://doi.org/10.1007/s11202-006-0112-8
- D. A. Korshunov, The critical case of the Cramér-Lundberg theorem on the asymptotics of the distribution of the maximum of a random walk with negative drift, Sibirsk. Mat. Zh. 46 (2005), no. 6, 1335–1340 (Russian, with Russian summary); English transl., Siberian Math. J. 46 (2005), no. 6, 1077–1081. MR 2195033, DOI https://doi.org/10.1007/s11202-005-0102-2
References
- Ĭ. Ī. Gīhman and A. V. Skorohod, Controlled stochastic processes, Springer-Verlag, New York, 1979. Translated from the Russian by Samuel Kotz. MR 544839 (80h:60081)
- Evgenii B. Dynkin and Aleksandr A. Yushkevich, Markov processes: Theorems and problems, Translated from the Russian by James S. Wood, Plenum Press, New York, 1969. MR 0242252 (39 \#3585a)
- D. A. Darling, T. Liggett, and H. M. Taylor, Optimal stopping for partial sums, Ann. Math. Statist. 43 (1972), 1363–1368. MR 0312564 (47 \#1121)
- A. A. Novikov and A. N. Shiryaev, On an effective case of the solution of the optimal stopping problem for random walks, Teor. Veroyatn. Primen. 49 (2004), no. 2, 373–382 (Russian, with Russian summary); English transl., Theory Probab. Appl. 49 (2005), no. 2, 344–354. MR 2144307 (2005m:60084), DOI https://doi.org/10.1137/S0040585X97981093
- Wim Schoutens, Stochastic processes and orthogonal polynomials, Lecture Notes in Statistics, vol. 146, Springer-Verlag, New York, 2000. MR 1761401 (2001f:60095)
1co done
- V. V. Tomashik and Yu. S. Mishura, Optimal stopping times for random walks for polynomial reward functions, Applied Statistics. Actuarial and Financial Mathematics (2008), no. 1–2, 101–110. (Ukrainian)
- A. A. Borovkov, Veroyatnostnye protsessy v teorii massovogo obsluzhivaniya, Izdat. “Nauka”, Moscow, 1972 (Russian). MR 0315800 (47 \#4349)
- O. V. Viskov, A random walk with an upper-continuous component, and the Lagrange inversion formula, Teor. Veroyatnost. i Primenen. 45 (2000), no. 1, 166–175 (Russian, with Russian summary); English transl., Theory Probab. Appl. 45 (2001), no. 1, 164–172. MR 1810980 (2001j:60089), DOI https://doi.org/10.1137/S0040585X97978105
- Wolfgang Stadje, An iterative approximation procedure for the distribution of the maximum of a random walk, Statist. Probab. Lett. 50 (2000), no. 4, 375–381. MR 1802232 (2001m:60105), DOI https://doi.org/10.1016/S0167-7152%2800%2900124-3
- Lajos Takács, Combinatorial methods in the theory of stochastic processes, John Wiley & Sons Inc., New York, 1967. MR 0217858 (36 \#947)
- D. Korshunov, On distribution tail of the maximum of a random walk, Stochastic Process. Appl. 72 (1997), no. 1, 97–103. MR 1483613 (98i:60027), DOI https://doi.org/10.1016/S0304-4149%2897%2900060-4
- A. A. Borovkov, On subexponential distributions and the asymptotics of the distribution of the maximum of sequential sums, Sibirsk. Mat. Zh. 43 (2002), no. 6, 1235–1264 (Russian, with Russian summary); English transl., Siberian Math. J. 43 (2002), no. 6, 995–1022. MR 1946226 (2004a:62031), DOI https://doi.org/10.1023/A%3A1021109132124
- S. Zakhari and S. G. Foss, On the exact asymptotics of the maximum of a random walk with increments in a class of light-tailed distributions, Sibirsk. Mat. Zh. 47 (2006), no. 6, 1265–1274 (Russian, with Russian summary); English transl., Siberian Math. J. 47 (2006), no. 6, 1034–1041. MR 2302843 (2008i:60040), DOI https://doi.org/10.1007/s11202-006-0112-8
- D. A. Korshunov, The critical case of the Cramér-Lundberg theorem on the asymptotics of the distribution of the maximum of a random walk with negative drift, Sibirsk. Mat. Zh. 46 (2005), no. 6, 1335–1340 (Russian, with Russian summary); English transl., Siberian Math. J. 46 (2005), no. 6, 1077–1081. MR 2195033 (2006j:60047), DOI https://doi.org/10.1007/s11202-005-0102-2
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Additional Information
Yu. S. Mishura
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4E, Kiev 03127, Ukraine
Email:
myus@univ.kiev.ua
V. V. Tomashyk
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4E, Kiev 03127, Ukraine
Email:
vladdislav@gmail.com
Keywords:
Appel polynomials,
random walks,
reward functions,
stopping times
Received by editor(s):
May 11, 2011
Published electronically:
August 20, 2013
Article copyright:
© Copyright 2013
American Mathematical Society
