Periodica Mathematica Hungarica

, Volume 62, Issue 1, pp 75–101 | Cite as

On hitting times of affine boundaries by reflecting Brownian motion and Bessel processes

  • Paavo Salminen
  • Marc Yor


Firstly, we compute the distribution function for the hitting time of a linear time-dependent boundary ta + bt, a ≥ 0, b ∈ ℝ, by a reflecting Brownian motion. The main tool hereby is Doob’s formula which gives the probability that Brownian motion started inside a wedge does not hit this wedge. Other key ingredients are the time inversion property of Brownian motion and the time reversal property of diffusion bridges. Secondly, this methodology can also be applied for the three-dimensional Bessel process. Thirdly, we consider Bessel bridges from 0 to 0 with dimension parameter δ > 0 and show that the probability that such a Bessel bridge crosses an affine boundary is equal to the probability that this Bessel bridge stays below some fixed value.

Key words and phrases

reflecting Brownian motion Bessel process hitting time linear boundary time reversal time inversion Brownian bridge Bessel bridge 

Mathematics subject classification numbers

60J65 60J60 


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  1. [1]
    M. Abramowitz and I. Stegun, Mathematical functions, Dover publications, Inc., New York, 1970.Google Scholar
  2. [2]
    M. Abundo, Some conditional crossing results of Brownian motion over a piecewise-linear boundary, Statist. Probab. Lett., 58 (2002), 131–145.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    L. Alili and P. Patie, On the first crossing times of a Brownian motion and a family of continuous curves, C. R. Math. Acad. Sci. Paris, 340 (2005), 225–228.MATHMathSciNetGoogle Scholar
  4. [4]
    L. Alili and P. Patie, Boundary crossing identities for diffusions having the time inversion property, J. Theoret. Probab., 23 (2010), 65–84.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Ph. Biane, J. Pitman and M. Yor, Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions, Bull. Amer. Math. Soc. (N.S.), 38 (2001), 435–465.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Ph. Biane and M. Yor, Valeurs principales associées aux temps locaux browniens, Bull. Sci. Math. (2), 111 (1987), 23–101.MATHMathSciNetGoogle Scholar
  7. [7]
    A. N. Borodin and P. Salminen, Handbook of Brownian motion — facts and formulae, 2nd edition, Birkhäuser, Basel, 2002.MATHGoogle Scholar
  8. [8]
    L. Breiman, First exit times from a square root boundary, Proc. 5th Berkeley Symp. Math. Stat. Probab. Vol. 2, University of California, Berkeley, 1967, 9–16.Google Scholar
  9. [9]
    K. L. Chung, Excursions in Brownian motion, Ark. Mat., 14 (1976), 155–177.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    D. M. Cifarelli and E. Regazzini, Sugli stimatori non distorti a varianza minima di una classe di funzioni di probabilità troncate, Giorn. Econom. Ann. Econom. (N.S.), 32 (1973), 492–501.MathSciNetGoogle Scholar
  11. [11]
    D. M. Delong, Crossing probabilities for a square root boundary by a Bessel process, Comm. Statist. A—Theory Methods, 10 (1981), 2197–2213.CrossRefMathSciNetGoogle Scholar
  12. [12]
    D. M. Delong, Erratum: “Crossing probabilities for a square root boundary by a Bessel process”. [Comm. Statist. A—Theory Methods, 10(1981), 2197–2213; MR 82i:62119], Comm. Statist. A—Theory Methods, 12 (1983), 1699.MathSciNetGoogle Scholar
  13. [13]
    J. L. Doob, Heuristic approach to the Kolmogorov-Smirnov theorems, Ann. Math. Statistics, 20 (1949), 393–403.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    J. Durbin, Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test, J. Appl. Probability, 8 (1971), 431–453.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    J. Durbin, The first passage density of a Brownian motion process to a curved boundary, J. Appl. Probab., 29 (1992), 291–304.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    R. Durrett and D.I. Iglehart, Functionals of Brownian meander and Brownian excursion, Ann. Probability, 5 (1977), 130–135.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    W. Feller, The asymptotic distribution of the range of sums of independent random variables, Ann. Math. Statistics, 22 (1951), 427–432.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    W. Feller, An introduction to probability theory and its applications, Vol. II., Second edition, John Wiley & Sons Inc., New York, 1971.Google Scholar
  19. [19]
    P. Groeneboom, Brownian motion with a parabolic drift and Airy functions, Probab. Theory Related Fields, 81 (1989), 79–109.CrossRefMathSciNetGoogle Scholar
  20. [20]
    D. Kennedy, The distribution of the maximum Brownian excursion, J. Appl. Probability, 13 (1976), 371–376.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    H. R. Lerche, Boundary crossing of Brownian motion, Lecture Notes in Statistics 40, Springer-Verlag, Berlin, 1986.MATHGoogle Scholar
  22. [22]
    A. Martin-Löf, The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier, J. Appl. Probab., 35 (1998), 671–682.MATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    H. Mckean, Excursions of a non-singular diffusion, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 1 (1963), 230–239.MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    G. Peskir and A.N. Shiryaev, Optimal stopping and free-boundary problems, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2006.MATHGoogle Scholar
  25. [25]
    J. Pitman and M. Yor, Bessel processes and infinitely divisible laws, Stochastic Integrals (ed.: D. Williams), Lecture Notes in Mathematics 851, Springer Verlag, Heidelberg — Berlin, 1981, 285–370.CrossRefGoogle Scholar
  26. [26]
    J. Pitman and M. Yor, The law of the maximum of a Bessel bridge, Electron. J. Probab., 4 (1999), 35 pp. (electronic).Google Scholar
  27. [27]
    J. Pitman and M. Yor, Infinitely divisible laws associated with hyperbolic functions, Canad. J. Math., 55 (2003), 292–330.MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    D. Revuz and M. Yor, Continuous martingales and Brownian motion, 3rd edition, Springer Verlag, Berlin, 1999.MATHGoogle Scholar
  29. [29]
    P. Salminen, On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary, Adv. in Appl. Probab., 20 (1988), 411–426.MATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    P. Salminen, On last exit decomposition of linear diffusions, Studia Sci. Math. Hungar., 33 (1997), 251–262.MATHMathSciNetGoogle Scholar
  31. [31]
    P. Salminen and M. Yor, On some probabilistic occurrences of the Poisson summation formula, in preparation.Google Scholar
  32. [32]
    T. H. Scheike, A boundary-crossing result for Brownian motion, J. Appl. Probab., 29 (1992), 448–453.MATHCrossRefMathSciNetGoogle Scholar
  33. [33]
    L. A. Shepp, On the integral of the absolute value of the pinned Wiener process, Ann. Probab., 10 (1982), 234–239.MATHCrossRefMathSciNetGoogle Scholar
  34. [34]
    T. Shiga and S. Watanabe, Bessel diffusions as a one-parameter family of diffusion processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 27 (1973), 37–46.MATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    W. Vervaat, A relation between Brownian bridge and Brownian excursion, Ann. Probab., 7 (1979), 143–149.MATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    D. Williams, Path decompositions and continuity of local time for one-dimensional diffusions, Proc. London Math. Soc., 28 (1974), 738–768.MATHCrossRefMathSciNetGoogle Scholar
  37. [37]
    M. Yor, On square-root boundaries for Bessel processes, and pole-seeking Brownian motion, Stochastic analysis and applications (Swansea, 1983), Lecture Notes in Mathematics 1095, Springer Verlag, Berlin, 1984, 100–107.CrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2011

Authors and Affiliations

  1. 1.Mathematical DepartmentÅbo Akademi UniversityÅboFinland
  2. 2.Laboratoire de Probabilités et Modèles aléatoiresUniversité Pierre et Marie CurieParis Cedex 05France

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