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On first exit times and their means for Brownian bridges

Part of: Mathematical finance Markov processes

Published online by Cambridge University Press:  01 October 2019

Christel Geiss ,
Antti Luoto  and
Paavo Salminen
Christel Geiss*
Affiliation:
University of Jyväskylä
Antti Luoto*
Affiliation:
University of Jyväskylä
Paavo Salminen*
Affiliation:
Åbo Akademi University
*
* Postal address: Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35 (MaD), FI-40014, Finland.
* Postal address: Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35 (MaD), FI-40014, Finland.
**** Postal address: Faculty of Science and Engineering, Åbo Akademi University, Domkyrkotorget 3, 20500 Åbo, Finland. Email address: paavo.salminen@abo.fi
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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.

Keywords

Bessel processBlack–Scholes modelBrownian motiondiffusion processh-transformlast exit timePoisson summation formula

MSC classification

Primary: 60J65: Brownian motion 60J60: Diffusion processes
Secondary: 91G60: Numerical methods (including Monte Carlo methods)
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 3 , September 2019 , pp. 701 - 722
Copyright
© Applied Probability Trust 2019 

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