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Journal of Applied Mathematics and Stochastic Analysis
Volume 2005 (2005), Issue 3, Pages 237-246

On the first-passage time of integrated Brownian motion

Institut für Stochastik und Anwendungen, Fachbereich Mathematik, Universität Stuttgart, Stuttgart 70569, Germany

Received 30 January 2004; Revised 29 September 2004

Copyright © 2005 Christian H. Hesse. 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.


Let (Bt;t0) be a Brownian motion process starting from B0=ν and define Xν(t)=0tBsds. For a0, set τa,ν:=inf{t:Xν(t)=a} (with inf φ=). We study the conditional moments of τa,ν given τa,ν<. Using martingale methods, stopping-time arguments, as well as the method of dominant balance, we obtain, in particular, an asymptotic expansion for the conditional mean E(τa,ν|τa,ν<) as ν. Through a series of simulations, it is shown that a truncation of this expansion after the first few terms provides an accurate approximation to the unknown true conditional mean even for small ν.