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First Passage Time for Brownian Motion and Piecewise Linear Boundaries

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  • Zhiyong Jin

    (University of Manitoba)

  • Liqun Wang

    (University of Manitoba
    Beijing Jiaotong University)

Abstract

We propose a new approach to calculating the first passage time densities for Brownian motion crossing piecewise linear boundaries which can be discontinuous. Using this approach we obtain explicit formulas for the first passage densities and show that they are continuously differentiable except at the break points of the boundaries. Furthermore, these formulas can be used to approximate the first passage time distributions for general nonlinear boundaries. The numerical computation can be easily done by using the Monte Carlo integration, which is straightforward to implement. Some numerical examples are presented for illustration. This approach can be further extended to compute two-sided boundary crossing distributions.

Suggested Citation

  • Zhiyong Jin & Liqun Wang, 2017. "First Passage Time for Brownian Motion and Piecewise Linear Boundaries," Methodology and Computing in Applied Probability, Springer, vol. 19(1), pages 237-253, March.
    • Handle: RePEc:spr:metcap:v:19:y:2017:i:1:d:10.1007_s11009-015-9475-2
    DOI: 10.1007/s11009-015-9475-2
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    References listed on IDEAS

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    1. Molini, A. & Talkner, P. & Katul, G.G. & Porporato, A., 2011. "First passage time statistics of Brownian motion with purely time dependent drift and diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(11), pages 1841-1852.
    2. Liqun Wang & Klaus Pötzelberger, 2007. "Crossing Probabilities for Diffusion Processes with Piecewise Continuous Boundaries," Methodology and Computing in Applied Probability, Springer, vol. 9(1), pages 21-40, March.
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    Cited by:

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    2. Zhang, Ao & Wang, Zhihua & Bao, Rui & Liu, Chengrui & Wu, Qiong & Cao, Shihao, 2023. "A novel failure time estimation method for degradation analysis based on general nonlinear Wiener processes," Reliability Engineering and System Safety, Elsevier, vol. 230(C).
    3. Lee, Taeho, 2023. "Exact simulation for the first hitting time of Brownian motion and Brownian bridge," Statistics & Probability Letters, Elsevier, vol. 193(C).
    4. Qinglai Dong & Lirong Cui, 2019. "First Hitting Time Distributions for Brownian Motion and Regions with Piecewise Linear Boundaries," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 1-23, March.

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