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

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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.

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References

  • Abundo M (2002) Some conditional crossing results of Brownian motion over a piecewise-linear boundary. Stat Prob Lett 58(2):131–145

    Article  MathSciNet  MATH  Google Scholar 

  • Che X, Dassios A (2013) Stochastic boundary crossing probabilities for the Brownian motion. J Appl Probab 50(2):419–429

    Article  MathSciNet  MATH  Google Scholar 

  • Cox DR, Miller HD (1977) The theory of stochastic processes. CRC Press, Boca Raton

    MATH  Google Scholar 

  • Cui LR, Huang JB, Li Y (2016) Degradation models with wiener diffusion processes under calibrations. IEEE Trans Reliab 65(2):613–623

    Article  Google Scholar 

  • Daniels HE (1969) The minimum of a stationary Markov process superimposed on a U-Shaped trend. J Appl Probab 6(2):399–408

    Article  MathSciNet  MATH  Google Scholar 

  • Daniels HE (1996) Approximating the first crossing-time density for a curved boundary. Bernoulli 2(2):133–143

    Article  MathSciNet  MATH  Google Scholar 

  • Donchev DS (2010) Brownian motion hitting probabilities for general two-sided square-root boundaries. Methodol Comput Appl Probab 12(2):237–245

    Article  MathSciNet  MATH  Google Scholar 

  • Dong QL, Cui LR (2017) A study on stochastic degradation process models under different types of failure thresholds. (Submitted)

  • Doob JL (1949) Heuristic approach to the Kolmogorov-Smirnov theorems. Ann Math Stat 20(3):393–403

    Article  MathSciNet  MATH  Google Scholar 

  • Durbin J (1971) Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test. J Appl Probab 8(3):431–453

    Article  MathSciNet  MATH  Google Scholar 

  • Durbin J, Williams D (1992) The first-passage density of the Brownian motion process to a curved boundary. J Appl Probab 29(2):291–304

    Article  MathSciNet  MATH  Google Scholar 

  • Ferebee B (1983) An asymptotic expansion for one-sided Brownian exit densities. Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete 63(1):1–15

    Article  MathSciNet  MATH  Google Scholar 

  • Fu JC, Wu T (2010) Linear and nonlinear boundary crossing probabilities for Brownian motion and related processes. J Appl Probab 47(4):1058–1071

    Article  MathSciNet  MATH  Google Scholar 

  • Gao HD, Cui LR, Kong DJ (2017) Reliability analysis for a Wiener degradation process model under changing failure thresholds. Submitted

  • Giraudo MT, Sacerdote L (1999) An improved technique for the simulation of first passage times for diffusion processes. Commun Stat Simul Comput 28(4):1135–1163

    Article  MathSciNet  MATH  Google Scholar 

  • Giraudo MT, Sacerdote L, Zucca C (2001) A Monte Carlo method for the simulation of first passage times of diffusion processes. Methodol Comput Appl Probab 3(2):215–231

    Article  MathSciNet  MATH  Google Scholar 

  • Herrmann S, Tanré E (2016) The first-passage time of the Brownian motion to a curved boundary: an algorithmic approach. SIAM J Sci Comput 38(1):A196–A215

    Article  MathSciNet  MATH  Google Scholar 

  • Jin Z, Wang L (2017) First passage time for Brownian motion and piecewise linear boundaries. Methodol Comput Appl Probab 19:237–253

    Article  MathSciNet  MATH  Google Scholar 

  • Karatzas I, Shreve SE (1991) Brownian motion and stochastic calculus. Springer, New York

    MATH  Google Scholar 

  • Kong DJ, Balakrishnan N, Cui LR (2017) Two-phase degradation process model with abrupt jump at change point governed by wiener process. IEEE Trans Reliab 66(4):1345–1360

    Article  Google Scholar 

  • Molini A, Talkner P, Katul GG et al (2011) First passage time statistics of Brownian motion with purely time dependent drift and diffusion. Physica A: Stat Mech Appl 390:1841–1852

    Article  MathSciNet  MATH  Google Scholar 

  • Niederreiter H (1992) Random number generation and quasi-Monte Carlo methods. Society for Industrial and Applied Mathematics, Philadelphia

    Book  MATH  Google Scholar 

  • Novikov A, Frishling V, Kordzakhia N (1999) Approximations of boundary crossing probabilities for a Brownian motion. J Appl Probab 36(4):1019–1030

    Article  MathSciNet  MATH  Google Scholar 

  • Peškir G, Shiryaev AN (1998) On the Brownian first-passage time over a one-sided stochastic boundary. Theory Prob Appl 42(3):444–453

    Article  MathSciNet  Google Scholar 

  • Pötzelberger K, Wang L (2001) Boundary crossing probability for Brownian motion. J Appl Probab 38(01):152–164

    Article  MathSciNet  MATH  Google Scholar 

  • Ripley BD (1987) Stochastic simulation. Wiley, New York

    Book  MATH  Google Scholar 

  • Siegmund D (1986) Boundary crossing probabilities and statistical applications. Ann Stat 14(2):361–404

    Article  MathSciNet  MATH  Google Scholar 

  • Strassen V (1967) Almost-sure behavior of sums of independent random variables and martingales. Berkeley symposium on mathematical statistics & probability. The Regents of the University of California

  • Vondraček Z (2000) Asymptotics of first-passage time over a one-sided stochastic boundary. J sTheor Probab 13(1):279–309

    Article  MathSciNet  MATH  Google Scholar 

  • Wang L, Pötzelberger K (1997) Boundary crossing probability for Brownian motion and general boundaries. J Appl Probab 34(1):54–65

    Article  MathSciNet  MATH  Google Scholar 

  • Wang L, Pötzelberger K (2007) Crossing probabilities for diffusion processes with piecewise continuous boundaries. Methodol Comput Appl Probab 9(1):21–40

    Article  MathSciNet  MATH  Google Scholar 

  • Wang J, Zhao X, Guo X et al (2018) Analyzing the research subjects and hot topics of power system reliability through the Web of Science from 1991 to 2015. Renew Sust Energ Rev 82:700–713

    Article  Google Scholar 

  • Zhao X, Guo X, Wang X (2018) Reliability and maintenance policies for a two-stage shock model with self-healing mechanism. Reliability Engineering & System Safety, 172

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Acknowledgments

The research was supported by the National Natural Science Foundation of China under Grant 71631001.

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Authors and Affiliations

  1. School of Management & Economics, Beijing Institute of Technology, Beijing, China

    Qinglai Dong & Lirong Cui

  2. School of Mathematics & Computer Science, Yanan University, Shaanxi, China

    Qinglai Dong

Authors
  1. Qinglai Dong
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  2. Lirong Cui
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Correspondence to Qinglai Dong.

Additional information

This work was supported by the National Natural Science Foundation of China under Grant 71631001.

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Dong, Q., Cui, L. First Hitting Time Distributions for Brownian Motion and Regions with Piecewise Linear Boundaries. Methodol Comput Appl Probab 21, 1–23 (2019). https://doi.org/10.1007/s11009-018-9638-z

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  • DOI: https://doi.org/10.1007/s11009-018-9638-z

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