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On the trivariate joint distribution of Brownian motion and its maximum and minimum

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  • Choi, ByoungSeon
  • Roh, JeongHo

Abstract

The trivariate joint probability density function of Brownian motion and its maximum and minimum can be expressed as an infinite series of normal probability density functions. In this letter, we show that the infinite series converges uniformly, and satisfies the Fokker–Planck equation. Also, we express it as a product form using Jacobi’s triple product identity, and present some error bounds of a finite series approximation of the infinite series.

Suggested Citation

  • Choi, ByoungSeon & Roh, JeongHo, 2013. "On the trivariate joint distribution of Brownian motion and its maximum and minimum," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1046-1053.
    • Handle: RePEc:eee:stapro:v:83:y:2013:i:4:p:1046-1053
    DOI: 10.1016/j.spl.2012.12.015
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    Cited by:

    1. Budhi Surya & Wenyuan Wang & Xianghua Zhao & Xiaowen Zhou, 2020. "Parisian excursion with capital injection for draw-down reflected Levy insurance risk process," Papers 2005.09214, arXiv.org.
    2. Liang-Ching Lin & Li-Hsien Sun, 2019. "Modeling financial interval time series," PLOS ONE, Public Library of Science, vol. 14(2), pages 1-20, February.
    3. Suk Joon Byun & Jung‐Soon Hyun & Woon Jun Sung, 2021. "Estimation of stochastic volatility and option prices," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 41(3), pages 349-360, March.
    4. Choi, ByoungSeon & Kim, Chansoo & Kang, Hyuk & Choi, M.Y., 2020. "General solutions of the heat equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 539(C).
    5. Kurt Riedel, 2021. "The value of the high, low and close in the estimation of Brownian motion," Statistical Inference for Stochastic Processes, Springer, vol. 24(1), pages 179-210, April.

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