International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 2005 / Article

Open Access

Volume 2005 |Article ID 960738 | https://doi.org/10.1155/JAMSA.2005.237

Christian H. Hesse, "On the first-passage time of integrated Brownian motion", International Journal of Stochastic Analysis, vol. 2005, Article ID 960738, 10 pages, 2005. https://doi.org/10.1155/JAMSA.2005.237

On the first-passage time of integrated Brownian motion

Received30 Jan 2004
Revised29 Sep 2004

Abstract

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.

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.


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