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An optimal sequential procedure for determining the drift of a Brownian motion among three values

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  • Buonaguidi, B.

Abstract

We consider a one-dimensional Brownian motion, having a random and unobservable drift which can take one of three known values. Assuming that we monitor the position of the process in real time, the problem is to determine as soon as possible and with minimal probabilities of the wrong terminal decisions, which value the drift has taken. We derive the exact solution to the problem in the Bayesian formulation, under any prior probability distribution on the three values that the drift can assume, when the cost of observation is linear. Remarkably, the optimal stopping boundaries of the present problem are non-monotone.

Suggested Citation

  • Buonaguidi, B., 2023. "An optimal sequential procedure for determining the drift of a Brownian motion among three values," Stochastic Processes and their Applications, Elsevier, vol. 159(C), pages 320-349.
    • Handle: RePEc:eee:spapps:v:159:y:2023:i:c:p:320-349
    DOI: 10.1016/j.spa.2023.02.001
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    References listed on IDEAS

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    1. Goran Peskir, 2005. "A Change-of-Variable Formula with Local Time on Curves," Journal of Theoretical Probability, Springer, vol. 18(3), pages 499-535, July.
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    3. Johnson, P. & Pedersen, J.L. & Peskir, G. & Zucca, C., 2022. "Detecting the presence of a random drift in Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 1068-1090.
    4. Goran Peskir, 2005. "On The American Option Problem," Mathematical Finance, Wiley Blackwell, vol. 15(1), pages 169-181, January.
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