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Detecting the presence of a random drift in Brownian motion

Author

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  • Johnson, P.
  • Pedersen, J.L.
  • Peskir, G.
  • Zucca, C.

Abstract

Consider a standard Brownian motion in one dimension, having either a zero drift, or a non-zero drift that is randomly distributed according to a known probability law. Following the motion in real time, the problem is to detect as soon as possible and with minimal probabilities of the wrong terminal decisions, whether a non-zero drift is present in the observed motion. We solve this problem for a class of admissible laws in the Bayesian formulation, under any prior probability of the non-zero drift being present in the motion, when the passage of time is penalised linearly.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:150:y:2022:i:c:p:1068-1090
    DOI: 10.1016/j.spa.2021.05.006
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    References listed on IDEAS

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    1. Jacques du Toit & Goran Peskir, 2009. "Selling a stock at the ultimate maximum," Papers 0908.1014, arXiv.org.
    2. 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.
    3. Goran Peskir, 2005. "On The American Option Problem," Mathematical Finance, Wiley Blackwell, vol. 15(1), pages 169-181, January.
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    Cited by:

    1. 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.

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