The probability distributions of the first hitting times of Bessel processes

Hamana, Yuji; Matsumoto, Hiroyuki

doi:10.1090/S0002-9947-2013-05799-6

The probability distributions of the first hitting times of Bessel processes
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by Yuji Hamana and Hiroyuki Matsumoto PDF

Trans. Amer. Math. Soc. 365 (2013), 5237-5257 Request permission

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.

References

Andrei N. Borodin and Paavo Salminen, Handbook of Brownian motion—facts and formulae, 2nd ed., Probability and its Applications, Birkhäuser Verlag, Basel, 2002. MR 1912205, DOI 10.1007/978-3-0348-8163-0
Tomasz Byczkowski, Piotr Graczyk, and Andrzej Stós, Poisson kernels of half-spaces in real hyperbolic spaces, Rev. Mat. Iberoam. 23 (2007), no. 1, 85–126. MR 2351127, DOI 10.4171/RMI/487
T. Byczkowski and M. Ryznar, Hitting distributions of geometric Brownian motion, Studia Math. 173 (2006), no. 1, 19–38. MR 2204460, DOI 10.4064/sm173-1-2
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 143257, DOI 10.1090/S0002-9947-1962-0143257-8
A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions. Vol. 2, McGraw-Hill, New York, 1953.
W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, 2nd ed., Wiley, New York, 1971.
R. K. Getoor and M. J. Sharpe, Excursions of Brownian motion and Bessel processes, Z. Wahrsch. Verw. Gebiete 47 (1979), no. 1, 83–106. MR 521534, DOI 10.1007/BF00533253
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger; With one CD-ROM (Windows, Macintosh and UNIX). MR 2360010
Yuji Hamana, On the expected volume of the Wiener sausage, J. Math. Soc. Japan 62 (2010), no. 4, 1113–1136. MR 2761916
—, The expected volume and surface area of the Wiener sausage in odd dimensions, Osaka J. Math. 49 (2012), 853–868.
Mourad E. H. Ismail, Integral representations and complete monotonicity of various quotients of Bessel functions, Canadian J. Math. 29 (1977), no. 6, 1198–1207. MR 463527, DOI 10.4153/CJM-1977-119-5
Kiyosi Itô and Henry P. McKean Jr., Diffusion processes and their sample paths, Die Grundlehren der mathematischen Wissenschaften, Band 125, Springer-Verlag, Berlin-New York, 1974. Second printing, corrected. MR 0345224
John Kent, Some probabilistic properties of Bessel functions, Ann. Probab. 6 (1978), no. 5, 760–770. MR 0501378
John T. Kent, Eigenvalue expansions for diffusion hitting times, Z. Wahrsch. Verw. Gebiete 52 (1980), no. 3, 309–319. MR 576891, DOI 10.1007/BF00538895
N. N. Lebedev, Special functions and their applications, Dover Publications, Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; Unabridged and corrected republication. MR 0350075
Fritz Oberhettinger and Larry Badii, Tables of Laplace transforms, Springer-Verlag, New York-Heidelberg, 1973. MR 0352889
Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357, DOI 10.1007/978-3-662-06400-9
G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110
Makoto Yamazato, Hitting time distributions of single points for $1$-dimensional generalized diffusion processes, Nagoya Math. J. 119 (1990), 143–172. MR 1071905, DOI 10.1017/S0027763000003172