On the first-passage time of integrated Brownian motion

Abstract

Let (Bt; t ≥ 0) be a Brownian motion process starting from B0 = ν and define Xν (t) = ∫0t Bs ds. For a ≥ 0, 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.