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On Estimation for Brownian Motion Governed by Telegraph Process with Multiple Off States

Author

Listed:
  • V. Pozdnyakov

    (University of Connecticut)

  • L. M. Elbroch
  • C. Hu

    (University of Connecticut)

  • T. Meyer

    (University of Connecticut)

  • J. Yan

    (University of Connecticut
    University of Connecticut)

Abstract

Brownian motion whose infinitesimal variance changes according to a three-state continuous-time Markov Chain is studied. This Markov Chain can be viewed as a telegraph process with one on state and two off states. We first derive the distribution of occupation time of the on state. Then the result is used to develop a likelihood estimation procedure when the stochastic process at hand is observed at discrete, possibly irregularly spaced time points. The likelihood function is evaluated with the forward algorithm in the general framework of hidden Markov models. The analytic results are confirmed with simulation studies. The estimation procedure is applied to analyze the position data from a mountain lion.

Suggested Citation

  • V. Pozdnyakov & L. M. Elbroch & C. Hu & T. Meyer & J. Yan, 2020. "On Estimation for Brownian Motion Governed by Telegraph Process with Multiple Off States," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1275-1291, September.
    • Handle: RePEc:spr:metcap:v:22:y:2020:i:3:d:10.1007_s11009-020-09774-1
    DOI: 10.1007/s11009-020-09774-1
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    References listed on IDEAS

    as
    1. Vladimir Pozdnyakov & L. Mark Elbroch & Anthony Labarga & Thomas Meyer & Jun Yan, 2019. "Discretely Observed Brownian Motion Governed by Telegraph Process: Estimation," Methodology and Computing in Applied Probability, Springer, vol. 21(3), pages 907-920, September.
    2. Antonio Di Crescenzo & Shelemyahu Zacks, 2015. "Probability Law and Flow Function of Brownian Motion Driven by a Generalized Telegraph Process," Methodology and Computing in Applied Probability, Springer, vol. 17(3), pages 761-780, September.
    3. Marco Corazza & Florence Legros & Cira Perna & Marilena Sibillo, 2017. "Mathematical and Statistical Methods for Actuarial Sciences and Finance," Post-Print hal-01776135, HAL.
    4. Alessandro Gregorio & Stefano Iacus, 2008. "Parametric estimation for the standard and geometric telegraph process observed at discrete times," Statistical Inference for Stochastic Processes, Springer, vol. 11(3), pages 249-263, October.
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    Cited by:

    1. Surya Teja Eada & Vladimir Pozdnyakov & Jun Yan, 2025. "Discretely observed Brownian motion governed by telegraph signal process: Estimation and application to finance," Statistical Inference for Stochastic Processes, Springer, vol. 28(1), pages 1-17, April.

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