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On first exit times and their means for Brownian bridges

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

Abstract

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.

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Corresponding author

* Postal address: Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35 (MaD), FI-40014, Finland.
** Email address: christel.geiss@jyu.fi
*** Email address: antti.k.luoto@jyu.fi
**** Postal address: Faculty of Science and Engineering, Åbo Akademi University, Domkyrkotorget 3, 20500 Åbo, Finland. Email address: paavo.salminen@abo.fi

References

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On first exit times and their means for Brownian bridges

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

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