THE 1ST HITTING TIME FOR THE INTEGRATED BROWNIAN-MOTION

Let B(t), t greater-than-or-equal-to 0, be the standard Brownian motion in R. Define X(t) = integral 0t B(s)ds, U(t) = (X(t) + x + ty, B(t) + y), (x, y) is-an-element-of R2, and tau-a = inf {t > 0: U(t) is-an-element-of {a} x R}. In this note we compute explicitely the joint distribution of tau-a and U-tau-a. We also indicate a simple proof of a recent result of Lefebvre with some improvement.