Asymptotics of the probability distributions of the first hitting times of Bessel processes

Yuji Hamana (Kumamoto University)
Hiroyuki Matsumoto (Aoyama Gakuin University)

Abstract


The asymptotic behavior of the tail probabilities for the first hitting times of the Bessel process with arbitrary index is shown without using the explicit expressions for the distribution function obtained in the authors' previous works.

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Pages: 1-5

Publication Date: January 30, 2014

DOI: 10.1214/ECP.v19-3215

References

  • Byczkowski, T.; Ryznar, M. Hitting distributions of geometric Brownian motion. Studia Math. 173 (2006), no. 1, 19--38. MR2204460
  • Getoor, R. K.; Sharpe, M. J. Excursions of Brownian motion and Bessel processes. Z. Wahrsch. Verw. Gebiete 47 (1979), no. 1, 83--106. MR0521534
  • Gradshteyn, I. S.; Ryzhik, I. M. Table of integrals, series, and products. Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. With one CD-ROM (Windows, Macintosh and UNIX). Seventh edition. Elsevier/Academic Press, Amsterdam, 2007. xlviii+1171 pp. ISBN: 978-0-12-373637-6; 0-12-373637-4 MR2360010
  • Hamana, Yuji; Matsumoto, Hiroyuki. The probability densities of the first hitting times of Bessel processes. J. Math-for-Ind. 4B (2012), 91--95. MR3072321
  • Hamana, Yuji; Matsumoto, Hiroyuki. The probability distributions of the first hitting times of Bessel processes. Trans. Amer. Math. Soc. 365 (2013), no. 10, 5237--5257. MR3074372
  • Kent, John T. Eigenvalue expansions for diffusion hitting times. Z. Wahrsch. Verw. Gebiete 52 (1980), no. 3, 309--319. MR0576891
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7 MR1725357
  • Watson, G. N. A treatise on the theory of Bessel functions. Reprint of the second (1944) edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995. viii+804 pp. ISBN: 0-521-48391-3 MR1349110
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