| 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.
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. |
