Opuscula Math. 43, no. 2 (2023), 145-172
https://doi.org/10.7494/OpMath.2023.43.2.145

 
Opuscula Mathematica

Square-root boundaries for Bessel processes and the hitting times of radial Ornstein-Uhlenbeck processes

Yuji Hamana

Abstract. This article deals with the first hitting times of a Bessel process to a square-root boundary. We obtain the explicit form of the distribution function of the hitting time by means of zeros of the confluent hypergeometric function with respect to the first parameter. In deducing the distribution function, the time that a radial Ornstein-Uhlenbeck process reaches a certain point is very useful and plays an important role. We also give its distribution function in the case that the starting point is closer to the origin than the arrival site.

Keywords: Bessel process, confluent hypergeometric function, first hitting time, radial Ornstein-Uhlenbeck process, square-root boundary.

Mathematics Subject Classification: 60J60, 33C15, 44A10.

Full text (pdf)

  1. A.N. Borodin, P. Salminen, Handbook of Brownian Motion, 2nd ed., Birkhäuser, 2002.
  2. H. Buchholz, The Confluent Hypergeometric Function, Springer, 1969.
  3. T. Byczkowski, T. Ryznar, Hitting distribution of geometric Brownian motion, Studia Math. 173 (2006), 19-38. https://doi.org/10.4064/sm173-1-2
  4. S. Chiba, Asymptotic expansions for hitting distributions of Bessel process, Master Thesis, Tohoku University (2017) [in Japanese].
  5. Z. Ciesielski, 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. https://doi.org/10.1090/S0002-9947-1962-0143257-8
  6. D.M. Delong, Crossing probabilities for a square-root boundary for Bessel process, Comm. Statist. Theory Methods 10 (1981), 2197-2213. https://doi.org/10.1080/03610928108828182
  7. D.M. Delong, Erratum: "Crossing probabilities for a square-root boundary for Bessel process", Comm. Statist. Theory Methods 12 (1983), 1699.
  8. R.J. Elliot, J. van der Hoek, W.P. Malcolm, Pairs trading, Quantitative Finance 5 (2005), 271-276. https://doi.org/10.1080/14697680500149370
  9. N. Enriquez, C. Sabot, M. Yor, Renewal series and square-root boundaries for Bessel processes, Electron. Commun. Probab. 13 (2008), 649-652. https://doi.org/10.1214/ECP.v13-1436
  10. H. Geman, M. Yor, Bessel processes, Asian options, and perpetuities, Mathematical Finance 3 (1993), 349-375. https://doi.org/10.1111/j.1467-9965.1993.tb00092.x
  11. R.K. Getoor, M.J. Sharpe, Excursions of Brownian motion and Bessel processes, Z. Wahrsch. Verw. Gebiete 47 (1979), 83-106. https://doi.org/10.1007/BF00533253
  12. A. Göing-Jaeschke, M. Yor, A survey and some generalizations of Bessel processes, Bernoulli 9 (2003), 313-349. https://doi.org/10.3150/bj/1068128980
  13. Y. Hamana, The probability distribution of the first hitting times of radial Ornstein-Uhlenbeck processes, Studia Math. 251 (2020), 65-88. https://doi.org/10.4064/sm180410-27-12
  14. Y. Hamana, R. Kaikura, K. Shinozaki, Asymptotic expansions for the first hitting times of Bessel processes, Opuscula Math. 41 (2021), 509-537. https://doi.org/10.7494/OpMath.2021.41.4.509
  15. Y. Hamana, H. Matsumoto, The probability densities of the first hitting times of Bessel processes, J. Math-for-Industry 48 (2012), 91-95.
  16. Y. Hamana, H. Matsumoto, The probability distributions of the first hitting times of Bessel processes, Trans. Amer. Math. Soc. 365 (2013), 5237-5257. https://doi.org/10.1090/S0002-9947-2013-05799-6
  17. Y. Hamana, H. Matsumoto, Asymptotics of the probability distributions of the first hitting times of Bessel processes, Electron. Commun. Probab. 19 (2014), 1-5. https://doi.org/10.1214/ECP.v19-3215
  18. Y. Hamana, H. Matsumoto, Hitting times to spheres of Brownian motions with and without drifts, Proc. Amer. Math. Soc. 144 (2016), 5385-5396. https://doi.org/10.1090/proc/13136
  19. Y. Hamana, H. Matsumoto, Precise asymptotic formulae for the first hitting times of Bessel processes, Tokyo J. Math. 41 (2018), 603-615.
  20. Y. Hariya, Some asymptotic formulae for Bessel process, Markov Process. Related Fields 21 (2015), 293-316.
  21. P. Henrici, Applied and Computational Analysis, vol. 2, Wiley, 1991.
  22. K. Itô, H.P. McKean, Diffusion Processes and Their Sample Paths, Springer, 1974.
  23. J.T. Kent, Eigenvalue expansion for diffusion hitting times, Z. Wahrsch. Verw. Gebiete 52 (1980), 309-319. https://doi.org/10.1007/BF00538895
  24. J. Lamperti, Semi-stable Markov processes I, Z. Wahrsch. Verw. Gebiete 22 (1972), 205-225. https://doi.org/10.1007/BF00536091
  25. W. Magnus, F. Oberhettinger, R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed., Springer, 1966.
  26. D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd ed., Springer, 1999.
  27. P. Salminen, On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary, Adv. Appl. Probab. 20 (1988), 411-426. https://doi.org/10.2307/1427397
  28. T. Takemura, M. Tomisaki, Lévy measure density corresponding to inverse local t , Publ. RIMS Kyoto Univ. 49 (2013), 563-599. https://doi.org/10.4171/PRIMS/113
  29. M. Yor, On square-root boundaries for Bessel processes, and pole-seeking Brownian motion, [in:] Stochastic Analysis and Applications (Swansea, 1983), Lecture Notes in Math. 1095, Springer, 1984, pp. 100-107.
  30. M. Yor, Exponential Functionals of Brownian Motion and Related Processes, Springer, 2001.
  • Communicated by Palle E.T. Jorgensen.
  • Received: 2022-10-07.
  • Accepted: 2022-12-21.
  • Published online: 2023-03-27.
Opuscula Mathematica - cover

Cite this article as:
Yuji Hamana, Square-root boundaries for Bessel processes and the hitting times of radial Ornstein-Uhlenbeck processes, Opuscula Math. 43, no. 2 (2023), 145-172, https://doi.org/10.7494/OpMath.2023.43.2.145

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

We advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.