On an Effective Solution of the Optimal Stopping Problem for Random Walks

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 polynomia...

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Bibliographic Details
Published in:Theory of probability and its applications 2005, Vol.49 (2), p.344-354
Main Authors: Novikov, A. A., Shiryaev, A. N.
Format: Article
Language:English
Subjects:
Applied mathematics
Polynomials
Random variables
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Summary: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.
ISSN:0040-585X
1095-7219
DOI:10.1137/S0040585X97981093