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)=∫0tBsds. 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.