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Discretely Observed Brownian Motion Governed by Telegraph Process: Estimation

Author

Listed:
  • Vladimir Pozdnyakov

    (University of Connecticut)

  • L. Mark Elbroch

    (Panthera)

  • Anthony Labarga

    (University of Connecticut)

  • Thomas Meyer

    (University of Connecticut
    University of Connecticut)

  • Jun Yan

    (University of Connecticut
    University of Connecticut)

Abstract

A Brownian motion whose infinitesimal variance alternates according to a telegraph process is considered. This stochastic process can be employed to model a variety of real-word situations, such as animal movement in ecology and stochastic volatility in mathematical finance. The main goal is to develop an estimation procedure for the underlying model parameters when the process is observed at discrete, possibly irregularly spaced time points. The sequence of observations is not Markov, but the sequence of the state of the telegraph process, if observed, is Markov. The observed sequence is therefore from a hidden Markov model. Likelihood inference is developed via dynamic programming, and is demonstrated to have much higher efficiency than the composite likelihood approach that was applied in an earlier work. The model is applied to model the movement of a mountain lion.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:21:y:2019:i:3:d:10.1007_s11009-017-9547-6
    DOI: 10.1007/s11009-017-9547-6
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    References listed on IDEAS

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    1. Fouque,Jean-Pierre & Papanicolaou,George & Sircar,Ronnie & Sølna,Knut, 2011. "Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives," Cambridge Books, Cambridge University Press, number 9780521843584, October.
    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. A. Crescenzo & E. Nardo & L. M. Ricciardi, 2005. "Simulation of First-Passage Times for Alternating Brownian Motions," Methodology and Computing in Applied Probability, Springer, vol. 7(2), pages 161-181, June.
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    Cited by:

    1. 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.

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