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First Hitting Time Distributions for Brownian Motion and Regions with Piecewise Linear Boundaries

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  • Qinglai Dong

    (Beijing Institute of Technology
    Yanan University)

  • Lirong Cui

    (Beijing Institute of Technology)

Abstract

Explicit formulas for the first hitting time distributions for a standard Brownian motion and different regions including rectangular, triangle, quadrilateral and a region with piecewise linear boundaries are derived. Moreover, approximations to the first hitting time distribution of a standard Brownian motion with respect to regions with general nonlinear continuous boundaries are also obtained. The rules for assessing the accuracies of the approximations are given. We generalize the results of one-sided boundaries which include the general nonlinear continuous boundaries, piecewise linear boundaries, linear boundaries and constant boundaries to a region and give the relationships among the first hitting time distributions. Some numerical examples are presented to illustrate the results obtained in the paper. These formulas can be further extended to compute the first hitting time distributions of a Brownian motion with linear drift with respect to regions.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:21:y:2019:i:1:d:10.1007_s11009-018-9638-z
    DOI: 10.1007/s11009-018-9638-z
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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. Abundo, Mario, 2002. "Some conditional crossing results of Brownian motion over a piecewise-linear boundary," Statistics & Probability Letters, Elsevier, vol. 58(2), pages 131-145, June.
    3. Maria Teresa Giraudo & Laura Sacerdote & Cristina Zucca, 2001. "A Monte Carlo Method for the Simulation of First Passage Times of Diffusion Processes," Methodology and Computing in Applied Probability, Springer, vol. 3(2), pages 215-231, June.
    4. 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.
    5. Zhao, Xian & Guo, Xiaoxin & Wang, Xiaoyue, 2018. "Reliability and maintenance policies for a two-stage shock model with self-healing mechanism," Reliability Engineering and System Safety, Elsevier, vol. 172(C), pages 185-194.
    6. Doncho S. Donchev, 2010. "Brownian Motion Hitting Probabilities for General Two-Sided Square-Root Boundaries," Methodology and Computing in Applied Probability, Springer, vol. 12(2), pages 237-245, June.
    7. 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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