Celebratio Mathematica

Y. Hu and M. Yor: “Asymptotic studies of Brownian functionals,” pp. 187–217 in Random walks (Budapest, 13–24 July 1998). Edited by P. Révész and T. Bálint. Bolyai Society Mathematical Studies 9. János Bolyai Mathematical Society (Budapest), 1999. MR 1752895 Zbl 0973.60084 incollection

E. Csáki, Z. Shi, and M. Yor: “Fractional Brownian motions as ‘higher-order’ fractional derivatives of Brownian local times,” pp. 365–387 in Limit theorems in probability and statistics: Fourth Hungarian colloquium on limit theorems in probability and statistics (Balatonlelle, Hungary, 28 June–2 July 1999), vol. 1. Edited by I. Berkes, E. Csáki, and M. Csörgő. János Bolyai Mathematical Society (Budapest), 2002. Dedicated to Pál Révész on the occasion of his 65th birthday. MR 1979974 Zbl 1030.60073 incollection

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B. Roynette, P. Vallois, and M. Yor: “Limiting laws for long Brownian bridges perturbed by their one-sided maximum, III,” Period. Math. Hung. 50 : 1–2 (August 2005), pp. 247–280. In homage to Professors E. Csáki and P. Révész. Parts I–X have very different titles. I was published in Studia Sci. Math. Hung. 46:2 (2003); II in Studia Sci. Math. Hung. 43:3 (2006); IV in Studia Sci. Math. Hung. 44:4 (2007); V in Studia Sci. Math. Hung. 45:1 (2008); VI in ESAIM Probab. Stat. 13 (2009); VII in Ann. Inst. Henri Poincaré, Probab. Stat. 45:2 (2009); VIII in J. Funct. Anal. 255:9 (2008); IX in ESAIM Probab. Stat. 14 (2010); X in Theory Stoch. Process. 14:2 (2008). MR 2162812 Zbl 1150.60308 ArXiv math/0511102 article

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@article {key2162812m,
	AUTHOR = {Roynette, Bernard and Vallois, Pierre
		 and Yor, Marc},
	TITLE = {Limiting laws for long {B}rownian bridges
		 perturbed by their one-sided maximum,
		 {III}},
	JOURNAL = {Period. Math. Hung.},
	FJOURNAL = {Periodica Mathematica Hungarica. Journal
		 of the J\'anos Bolyai Mathematical Society},
	VOLUME = {50},
	NUMBER = {1--2},
	MONTH = {August},
	YEAR = {2005},
	PAGES = {247--280},
	DOI = {10.1007/s10998-005-0015-7},
	NOTE = {In homage to Professors E. Cs\'aki and
		 P. R\'ev\'esz. Parts I--X have very
		 different titles. I was published in
		 \textit{Studia Sci. Math. Hung.} \textbf{46}:2
		 (2003); II in \textit{Studia Sci. Math.
		 Hung.} \textbf{43}:3 (2006); IV in \textit{Studia
		 Sci. Math. Hung.} \textbf{44}:4 (2007);
		 V in \textit{Studia Sci. Math. Hung.}
		 \textbf{45}:1 (2008); VI in \textit{ESAIM
		 Probab. Stat.} \textbf{13} (2009); VII
		 in \textit{Ann. Inst. Henri Poincar\'e,
		 Probab. Stat.} \textbf{45}:2 (2009);
		 VIII in \textit{J. Funct. Anal.} \textbf{255}:9
		 (2008); IX in \textit{ESAIM Probab.
		 Stat.} \textbf{14} (2010); X in \textit{Theory
		 Stoch. Process.} \textbf{14}:2 (2008).
		 ArXiv:math/0511102. MR:2162812. Zbl:1150.60308.},
	ISSN = {0031-5303},
}

P. Salminen and M. Yor: “On hitting times of affine boundaries by reflecting Brownian motion and Bessel processes,” Period. Math. Hung. 62 : 1 (March 2011), pp. 75–101. Dedicated to Endre Csáki and Pál Révész on the occasion of their 75th birthdays. MR 2772384 Zbl 1274.60251 ArXiv 1012.2038 article

Firstly, we compute the distribution function for the hitting time of a linear time-dependent boundary \( t\mapsto a + bt \), \( a\geq 0 \), \( b\in\mathbb{R} \), by a reflecting Brownian motion. The main tool hereby is Doob’s formula which gives the probability that Brownian motion started inside a wedge does not hit this wedge. Other key ingredients are the time inversion property of Brownian motion and the time reversal property of diffusion bridges. Secondly, this methodology can also be applied for the three-dimensional Bessel process. Thirdly, we consider Bessel bridges from 0 to 0 with dimension parameter \( \delta \gt 0 \) and show that the probability that such a Bessel bridge crosses an affine boundary is equal to the probability that this Bessel bridge stays below some fixed value.

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