Skip to main content Accessibility help

On first exit times and their means for Brownian bridges

  • Christel Geiss (a1), Antti Luoto (a1) and Paavo Salminen (a2)


For a Brownian bridge from 0 to y, we prove that the mean of the first exit time from the interval $\left( -h,h \right),h>0$ , behaves as ${\mathrm{O}}(h^2)$ when $h \downarrow 0$ . Similar behaviour is also seen to hold for the three-dimensional Bessel bridge. For the Brownian bridge and three-dimensional Bessel bridge, this mean of the first exit time has a puzzling representation in terms of the Kolmogorov distribution. The result regarding the Brownian bridge is applied to provide a detailed proof of an estimate needed by Walsh to determine the convergence of the binomial tree scheme for European options.


Corresponding author

* Postal address: Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35 (MaD), FI-40014, Finland.
** Email address:
*** Email address:
**** Postal address: Faculty of Science and Engineering, Åbo Akademi University, Domkyrkotorget 3, 20500 Åbo, Finland. Email address:


Hide All
[1] Biane, P., Pitman, J. and Yor, M. (2001). Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions. Bull. Amer. Math. Soc. (N.S.) 38, 435465.
[2] Borodin, A. N. and Salminen, P. (2015). Handbook of Brownian Motion: Facts and Formulae, 2nd edn (Probability and Its Applications). Birkhäuser, Basel.
[3] Christensen, S. and Lindensjö, K. (2018). On time-inconsistent stopping problems and mixed strategy stopping times. Available at arXiv:1804.07018.
[4] Chung, K. L. and Walsh, J. B. (2005). Markov Processes, Brownian Motion, and Time Symmetry, 2nd edn (Grundlehren der Mathematischen Wissenschaften 249). Springer, New York.
[5] Doob, J. L. (1949). Heuristic approach to the Kolmogorov–Smirnov theorems. Ann. Math. Stat. 20, 393403.
[6] Fitzsimmons, P., Pitman, J. and Yor, M. (1993). Markovian bridges: construction, Palm interpretation, and splicing. In Seminar on Stochastic Processes (Progress in Probability 32), pp. 101134. Springer.
[7] Gradshteyn, I. S. and Ryzhik, I. M. (2014). Table of Integrals, Series, and Products, 8th edn. Elsevier/Academic Press, Amsterdam.
[8] Itô, K. and McKean, H. P. (1974). Diffusion Processes and their Sample Paths (Grundlehren der Mathematischen Wissenschaften 125). Springer, Berlin and New York.
[9] Knight, F. B. (1969). Brownian local times and taboo processes. Trans. Amer. Math. Soc. 143, 173185.
[10] Kolmogorov, A. N. (1933). Sulla determinazione empirica delle leggi di probabilità. Giorn. Ist. Ital. Attuari 4, 111.
[11] Kroese, D., Taimre, T. and Botev, Z. (2011). Handbook of Monte Carlo Methods (Wiley Series in Probability and Statistics). Wiley, New York.
[12] Luoto, A. (2017). Time-dependent weak rate of convergence for functions of generalized bounded variation. Available at arXiv:1609.05768v3.
[13] Pitman, J. and Yor, M. (1999). The law of the maximum of a Bessel bridge. Electron. J. Prob. 4, 115.
[14] Salminen, P. (1997). On last exit decomposition of linear diffusions. Studia Sci. Math. Hungar. 33, 251262.
[15] Salminen, P. and Yor, M. (2011). On hitting times of affine boundaries by reflecting Brownian motion and Bessel processes. Period. Math. Hungar. 62, 75101.
[16] Smirnov, N. V. (1939). On the estimation of the discrepancy between empirical curves of distribution for two independent samples. Bul. Math. de l’Univ. de Moscou 2, 314 (in Russian).
[17] Walsh, J. B. (2003). The rate of convergence of the binomial tree scheme. Finance Stoch . 7, 337361.


MSC classification

On first exit times and their means for Brownian bridges

  • Christel Geiss (a1), Antti Luoto (a1) and Paavo Salminen (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed