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The probability distributions of the first hitting times of Bessel processes


Authors: Yuji Hamana and Hiroyuki Matsumoto
Journal: Trans. Amer. Math. Soc. 365 (2013), 5237-5257
MSC (2010): Primary 60J60; Secondary 33C10, 44A10
DOI: https://doi.org/10.1090/S0002-9947-2013-05799-6
Published electronically: March 5, 2013
MathSciNet review: 3074372
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the first hitting times of the Bessel processes. We give explicit expressions for the distribution functions by means of the zeros of the Bessel functions. The resulting formula is simpler and easier to treat than the corresponding results which have already been obtained.


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  • [1] A. N. Borodin and P. Salminen, Handbook of Brownian Motion, 2nd ed., Birkhäuser, Basel, 2002. MR 1912205 (2003g:60001)
  • [2] T. Byczkowski, P. Grazyk and A. Stós, Poisson kernels of half-spaces in real hyperbolic space, Rev. Mat. Iberoam. 23 (2007), 85-126. MR 2351127 (2008h:60315)
  • [3] T. Byczkowski and M. Ryznar, Hitting distribution of geometric Brownian motion, Studia Math. 173 (2006), 19-38. MR 2204460 (2007e:60082)
  • [4] Z. Ciesielski and S. J. Taylor, First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path, Trans. Amer. Math. Soc. 103 (1962), 434-450. MR 0143257 (26:816)
  • [5] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions. Vol. 2, McGraw-Hill, New York, 1953.
  • [6] W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, 2nd ed., Wiley, New York, 1971.
  • [7] R. K. Getoor and M. J. Sharpe, Excursions of Brownian motion and Bessel processes, Z. Wahr. Ver. Gebiete 47 (1979), 83-106. MR 521534 (80b:60104)
  • [8] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed., Academic Press, Amsterdam, 2007. MR 2360010 (2008g:00005)
  • [9] Y. Hamana, On the expected volume of the Wiener sausage, J. Math. Soc. Japan 62 (2010), 1113-1136. MR 2761916 (2012b:60270)
  • [10] -, The expected volume and surface area of the Wiener sausage in odd dimensions, Osaka J. Math. 49 (2012), 853-868.
  • [11] M. G. H. Ismail, Integral representations and complete monotonicity of various quotients of Bessel functions, Canad. J. Math. 29 (1977), 1198-1207. MR 0463527 (57:3474)
  • [12] K. Itô and H. P. McKean, Diffusion Processes and Their Sample Paths, Springer-Verlag, Berlin-New York, 1974. MR 0345224 (49:9963)
  • [13] J. T. Kent, Some probabilistic properties of Bessel functions, Ann. Probab. 6 (1978), 760-770. MR 0501378 (58:18750)
  • [14] -, Eigenvalue expansion for diffusion hitting times, Z. Wahr. Ver. Gebiete 52 (1980), 309-319. MR 576891 (81i:60072)
  • [15] N. N. Lebedev, Special Functions and Their Applications, Dover, New York, 1972. MR 0350075 (50:2568)
  • [16] F. Oberhettinger and L. Badii, Tables of Laplace Transforms, Springer-Verlag, New York-Heidelberg, 1973. MR 0352889 (50:5375)
  • [17] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd ed., Springer-Verlag, Berlin, 1999. MR 1725357 (2000h:60050)
  • [18] G. N. Watson, A Treatise on the Theory of Bessel Functions, Reprinted of 2nd ed., Cambridge University Press, Cambridge, 1995. MR 1349110 (96i:33010)
  • [19] M. Yamazato, Hitting time distributions of single points for $ 1$-dimensional generalized diffusion processes, Nagoya Math. J. 119 (1990), 143-172. MR 1071905 (93c:60119)

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Additional Information

Yuji Hamana
Affiliation: Department of Mathematics, Kumamoto University, Kurokami 2-39-1, Kumamoto, Japan 860-8555

Hiroyuki Matsumoto
Affiliation: Department of Physics and Mathematics, Aoyama Gakuin University, Fuchinobe 5-10-1, Sagamihara, Japan 252-5258

DOI: https://doi.org/10.1090/S0002-9947-2013-05799-6
Keywords: Bessel process, first hitting time, Bessel functions
Received by editor(s): June 30, 2011
Received by editor(s) in revised form: July 26, 2011, and January 24, 2012
Published electronically: March 5, 2013
Additional Notes: The authors were partially supported by the Grant-in-Aid for Scientific Research (C) No. 20540121 and 23540183, Japan Society for the Promotion of Science.
Article copyright: © Copyright 2013 American Mathematical Society

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