Proceedings of the Japan Academy, Series A, Mathematical Sciences

Hitting times to spheres of Brownian motions with drifts starting from the origin

Yuji Hamana

Abstract

We investigate the first hitting times to spheres of Brownian motions with constant drifts. In the case when the Brownian motion starts from a point in $\mathbf{R}^{d}$ except for the origin, an explicit formula for the density function of the hitting time has been obtained. When the starting point is the origin, we represent the density function by means of the density of the hitting time of the Brownian motion without the drift.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 95, Number 4 (2019), 37-39.

Dates
First available in Project Euclid: 1 April 2019

https://projecteuclid.org/euclid.pja/1554084023

Digital Object Identifier
doi:10.3792/pjaa.95.37

Mathematical Reviews number (MathSciNet)
MR3934984

Citation

Hamana, Yuji. Hitting times to spheres of Brownian motions with drifts starting from the origin. Proc. Japan Acad. Ser. A Math. Sci. 95 (2019), no. 4, 37--39. doi:10.3792/pjaa.95.37. https://projecteuclid.org/euclid.pja/1554084023

References

• 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.
• R. K. Getoor and M. J. Sharpe, Excursions of Brownian motion and Bessel processes, Z. Wahrsch. Verw. Gebiete 47 (1979), no. 1, 83–106.
• I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Academic Press, Amsterdam, 2007.
• Y. Hamana and H. Matsumoto, The probability densities of the first hitting times of Bessel processes, J. Math-for-Ind. 4B (2012), 91–95.
• Y. Hamana and H. Matsumoto, Hitting times to spheres of Brownian motions with and without drifts, Proc. Amer. Math. Soc. 144 (2016), no. 12, 5385–5396.
• J. Kent, Some probabilistic properties of Bessel functions, Ann. Probab. 6 (1978), no. 5, 760–770.
• J. T. Kent, Eigenvalue expansions for diffusion hitting times, Z. Wahrsch. Verw. Gebiete 52 (1980), no. 3, 309–319.
• S. C. Port and C. J. Stone, Brownian motion and classical potential theory, Academic Press, New York, 1978.
• G. N. Watson, A treatise on the theory of Bessel functions, reprint of the 2nd (1944) ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995.
• M. Yamazato, Hitting time distributions of single points for 1-dimensional generalized diffusion processes, Nagoya Math. J. 119 (1990), 143–172.
• C. Yin and C. Wang, Hitting time and place of Brownian motion with drift, Open Stat. Prob. J. 1 (2009), 38–42.