Hitting times to spheres of Brownian motions with and without drifts
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- by Yuji Hamana and Hiroyuki Matsumoto PDF
- Proc. Amer. Math. Soc. 144 (2016), 5385-5396 Request permission
Abstract:
Explicit formulae for the densities of the first hitting times to the sphere of Brownian motions with drifts and the asymptotic behavior of the tail probabilities are shown. For this purpose we present an explicit formula for the Laplace transform of the joint distribution of the first hitting time to a sphere and the hitting position, which is different from the known formulae in the literature and is of independent interest.References
- M. Aizenman and B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (1982), no. 2, 209–273. MR 644024, DOI 10.1002/cpa.3160350206
- 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 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
- Gerald B. Folland, Introduction to partial differential equations, 2nd ed., Princeton University Press, Princeton, NJ, 1995. MR 1357411
- 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
- Yuji Hamana and Hiroyuki Matsumoto, Asymptotics of the probability distributions of the first hitting times of Bessel processes, Electron. Commun. Probab. 19 (2014), no. 5, 5. MR 3164752, DOI 10.1214/ECP.v19-3215
- Yuji Hamana and Hiroyuki Matsumoto, A formula for the expected volume of the Wiener sausage with constant drift, Forum Math. (to appear), available at arXiv:1512.03514[Math.PR].
- 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 T. Kent, Eigenvalue expansions for diffusion hitting times, Z. Wahrsch. Verw. Gebiete 52 (1980), no. 3, 309–319. MR 576891, DOI 10.1007/BF00538895
- Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966. MR 0232968
- Jim Pitman and Marc Yor, Bessel processes and infinitely divisible laws, Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980) Lecture Notes in Math., vol. 851, Springer, Berlin, 1981, pp. 285–370. MR 620995
- Daniel W. Stroock, On the growth of stochastic integrals, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 18 (1971), 340–344. MR 287622, DOI 10.1007/BF00535035
- 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, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110
- J. G. Wendel, Hitting spheres with Brownian motion, Ann. Probab. 8 (1980), no. 1, 164–169. MR 556423
- Chuancun Yin, The joint distribution of the hitting time and place to a sphere or spherical shell for Brownian motion with drift, Statist. Probab. Lett. 42 (1999), no. 4, 367–373. MR 1707182, DOI 10.1016/S0167-7152(98)00231-4
- Chuancun Yin and Chunwei Wang, Hitting time and place of Brownian motion with drift, Open Stat. Prob. J. 1 (2009), 38–42. MR 2520345, DOI 10.2174/1876527000901010038
Additional Information
- Yuji Hamana
- Affiliation: Department of Mathematics, Kumamoto University, Kurokami 2-39-1, Kumamoto 860-8555, Japan
- MR Author ID: 326929
- Email: hamana@kumamoto-u.ac.jp
- Hiroyuki Matsumoto
- Affiliation: Department of Physics and Mathematics, Aoyama Gakuin University, Fuchinobe 5-10-1, Sagamihara 252-5258, Japan
- MR Author ID: 220140
- Email: matsu@gem.aoyama.ac.jp
- Received by editor(s): April 28, 2015
- Received by editor(s) in revised form: October 20, 2015, December 13, 2015, and February 4, 2016
- Published electronically: June 3, 2016
- Additional Notes: This work was partially supported by the Grant-in-Aid for Scientific Research (C) No.24540181 and No.26400144 of the Japan Society for the Promotion of Science (JSPS)
- Communicated by: Mark M. Meerschaert
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 5385-5396
- MSC (2010): Primary 60J65; Secondary 60J60, 60G40
- DOI: https://doi.org/10.1090/proc/13136
- MathSciNet review: 3556280