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On the maximum of the generalized Brownian bridge

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Abstract

We present some extensions of the distributions of the maximum of the Brownian bridge in [0,t] when the conditioning event is placed at a future timeu>t or at an intermediate timeu<t. The standard distributions of Brownian motion and Brownian bridge are obtained as limiting cases. These results permit us to derive also the distribution of the first-passage time of the Brownian bridge. Similar generalizations are carried out for the Brownian bridge with drift μ; in this case, it is shown that the maximal distribution is independent of μ (whenu≥t). Finally, the case of the two-sided maximal distribution of Brownian motion in [0,t], conditioned onB(u)=η (for bothu>t andu<t), is considered.

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Authors
  1. L. Beghin
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  2. E. Orsingher
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Additional information

Dip. di Statistica, Probabilità e Stat. Applicate, Università di Roma “La Sapienza,” Piazzale Aldo Moros, 00185 Roma, Italy. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp. 200–213, April–June, 1999.

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Beghin, L., Orsingher, E. On the maximum of the generalized Brownian bridge. Lith Math J 39, 157–167 (1999). https://doi.org/10.1007/BF02469280

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  • DOI: https://doi.org/10.1007/BF02469280

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