On the first-passage time of integrated Brownian motion

Christian H. Hesse

International Journal of Stochastic Analysis, 2005, vol. 2005, 1-10

Abstract:

Let ( B t ; t ≥ 0 ) be a Brownian motion process starting from B 0 = ν and define X ν ( t ) = ∫ 0 t B s d s . For a ≥ 0 , set τ a , ν : = inf { t : X ν ( t ) = a } (with inf φ = ∞ ). We study the conditional moments of τ a , ν given τ a , 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 ( τ a , ν | τ a , 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 .

Date: 2005
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:960738

DOI: 10.1155/JAMSA.2005.237

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