On the First-passage Time of Integrated Brownian Motion

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