Journal of Applied Mathematics and Stochastic Analysis
Volume 13 (2000), Issue 2, Pages 99-124
doi:10.1155/S1048953300000137
The moments of the area under reflected Brownian bridge
conditional on its local time at zero
University of Illinois, Department of Mathematics, 1409 West Green Street, Urbana 61801, IL, USA
Received 1 May 1999; Revised 1 August 1999
Copyright © 2000 Frank B. Knight. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
This paper develops a recursion formula for the conditional moments of
the area under the absolute value of Brownian bridge given the local time
at 0. The method of power series leads to a Hermite equation for the generating function of the coefficients which is solved in terms of the parabolic
cylinder functions. By integrating out the local time variable, this leads to
an integral expression for the joint moments of the areas under the positive and negative parts of the Brownian bridge.