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Optimally stopping the sample mean of a wiener process with an unknown drift

Author

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  • Simons, Gordon
  • Yao, Yi-Ching

Abstract

It is well known that optimally stopping the sample mean of a standard Wiener process is associated with a square root boundary. It is shown that when W(t) is replaced by X(t) = W(t) + [theta]t with [theta] normally distributed N([mu], [sigma]2) and independently of the Wiener process, the optimal stopping problem is equivalent to the time-truncated version of the original problem. It is also shown that the problem of optimally stopping (b + X(t))/(a + t), with constants a > 0 and b, is equivalent to the time-truncated version of the original problem or the one-arm bandit problem depending on whether [sigma]2 a-1. Furthermore, the optimal stopping region changes drastically as the prior parameters ([mu], [sigma]2) are slightly perturbed in a neighborhood of (, ).

Suggested Citation

  • Simons, Gordon & Yao, Yi-Ching, 1989. "Optimally stopping the sample mean of a wiener process with an unknown drift," Stochastic Processes and their Applications, Elsevier, vol. 32(2), pages 347-354, August.
    • Handle: RePEc:eee:spapps:v:32:y:1989:i:2:p:347-354
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