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First Passage Density of Brownian Motion with Two-sided Piecewise Linear Boundaries

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Abstract

The first passage time has many applications in fields like finance, econometrics, statistics, and biology. However, explicit formulas for the first passage density have only been obtained for a few cases. This paper derives an explicit formula for the first passage density of Brownian motion with two-sided piecewise continuous boundaries which may have some points of discontinuity. Approximations are used to obtain a simplified formula for estimating the first passage density. Moreover, the results are also generalized to the case of two-sided general nonlinear boundaries. Simulations can be easily carried out with Monte Carlo method and it is demonstrated for several typical two-sided boundaries that the proposed approximation method offers a highly accurate approximation of first passage density.

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Acknowledgements

The authors thank the editor and two anonymous reviewers for their helpful comments and suggestions.

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Authors and Affiliations

  1. Center for Applied Statistics, School of Statistics, Renmin University of China, Bei**g, 100872, P. R. China

    Zhen Yu & Mao Zai Tian

  2. School of Mathematics and Data Science, Changji University, Changji, 831100, P. R. China

    Mao Zai Tian

Authors
  1. Zhen Yu
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  2. Mao Zai Tian
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Correspondence to Mao Zai Tian.

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Conflict of Interest The authors declare no conflict of interest.

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Supported by the Fundamental Research Funds for the Central Universities, the Research Funds of Renmin University of China (Grant No. 22XNL016)

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Yu, Z., Tian, M.Z. First Passage Density of Brownian Motion with Two-sided Piecewise Linear Boundaries. Acta. Math. Sin.-English Ser. 40, 1505–1520 (2024). https://doi.org/10.1007/s10114-024-1090-0

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  • DOI: https://doi.org/10.1007/s10114-024-1090-0

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