Exit times densities of the Bessel process

Serafin, Grzegorz

doi:10.1090/proc/13419

Exit times densities of the Bessel process
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by Grzegorz Serafin

Proc. Amer. Math. Soc. 145 (2017), 3165-3178

DOI: https://doi.org/10.1090/proc/13419

Published electronically: January 6, 2017

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Abstract:

We examine the density functions of the first exit times of the Bessel process from the intervals $[0,1)$ and $(0,1)$. First, we express them by means of the transition density function of the killed process. Using that relationship we provide precise estimates and asymptotics of the exit time densities. In particular, the results hold for the first exit time of the $n$-dimensional Brownian motion from a ball.

References

Paolo Baldi, Enrico Casadio Tarabusi, Alessandro Figà-Talamanca, and Marc Yor, Non-symmetric hitting distributions on the hyperbolic half-plane and subordinated perpetuities, Rev. Mat. Iberoamericana 17 (2001), no. 3, 587–605. MR 1900896, DOI 10.4171/RMI/305
Kamil Bogus and Jacek Małecki, Sharp estimates of transition probability density for Bessel process in half-line, Potential Anal. 43 (2015), no. 1, 1–22. MR 3361787, DOI 10.1007/s11118-015-9461-x
K. Bogus and J. Małecki, Heat kernel estimates for the Bessel differential operator in half-line, Math. Nachr. (2016), DOI:10.1002/mana.201500163.
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
T. Byczkowski, J. Małecki, and M. Ryznar, Hitting times of Bessel processes, Potential Anal. 38 (2013), no. 3, 753–786. MR 3034599, DOI 10.1007/s11118-012-9296-7
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
Daniel Dufresne, The distribution of a perpetuity, with applications to risk theory and pension funding, Scand. Actuar. J. 1-2 (1990), 39–79. MR 1129194, DOI 10.1080/03461238.1990.10413872
Hélyette Geman and Marc Yor, Quelques relations entre processus de Bessel, options asiatiques et fonctions confluentes hypergéométriques, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), no. 6, 471–474 (French, with English summary). MR 1154389
Anja Göing-Jaeschke and Marc Yor, A survey and some generalizations of Bessel processes, Bernoulli 9 (2003), no. 2, 313–349. MR 1997032, DOI 10.3150/bj/1068128980
I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press, New York-London, 1965. Fourth edition prepared by Ju. V. Geronimus and M. Ju. Ceĭtlin; Translated from the Russian by Scripta Technica, Inc; Translation edited by Alan Jeffrey. MR 0197789
Yuji Hamana and Hiroyuki Matsumoto, The probability densities of the first hitting times of Bessel processes, J. Math-for-Ind. 4B (2012), 91–95. MR 3072321
Yuji Hamana and Hiroyuki Matsumoto, The probability distributions of the first hitting times of Bessel processes, Trans. Amer. Math. Soc. 365 (2013), no. 10, 5237–5257. MR 3074372, DOI 10.1090/S0002-9947-2013-05799-6
Samuel Karlin and James McGregor, Classical diffusion processes and total positivity, J. Math. Anal. Appl. 1 (1960), 163–183. MR 121844, DOI 10.1016/0022-247X(60)90020-2
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
Andrea Laforgia, Bounds for modified Bessel functions, J. Comput. Appl. Math. 34 (1991), no. 3, 263–267. MR 1102583, DOI 10.1016/0377-0427(91)90087-Z
Jacek Małecki, Grzegorz Serafin, and Tomasz Zorawik, Fourier-Bessel heat kernel estimates, J. Math. Anal. Appl. 439 (2016), no. 1, 91–102. MR 3474351, DOI 10.1016/j.jmaa.2016.02.051
Hiroyuki Matsumoto and Marc Yor, Exponential functionals of Brownian motion. I. Probability laws at fixed time, Probab. Surv. 2 (2005), 312–347. MR 2203675, DOI 10.1214/154957805100000159
S. A. Molčanov and E. Ostrovskiĭ, Symmetric stable processes as traces of degenerate diffusion processes. , Teor. Verojatnost. i Primenen. 14 (1969), 127–130 (Russian, with English summary). MR 0247668
Andrzej Pyć, Grzegorz Serafin, and Tomasz Żak, Supremum distribution of Bessel process of drifting Brownian motion, Probab. Math. Statist. 35 (2015), no. 2, 201–222. MR 3433649
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
Kôhei Uchiyama, Asymptotics of the densities of the first passage time distributions for Bessel diffusions, Trans. Amer. Math. Soc. 367 (2015), no. 4, 2719–2742. MR 3301879, DOI 10.1090/S0002-9947-2014-06155-2
G. N. Watson, Treatise on the theory of Bessel functions, Cambridge University Press, 1994.
Marc Yor, Exponential functionals of Brownian motion and related processes, Springer Finance, Springer-Verlag, Berlin, 2001. With an introductory chapter by Hélyette Geman; Chapters 1, 3, 4, 8 translated from the French by Stephen S. Wilson. MR 1854494, DOI 10.1007/978-3-642-56634-9

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Exit times densities of the Bessel process
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Abstract: