On the Joint Law of the L1 and L2 Norms of a 3-Dimensional Bessel Bridge

We give an analytical expression for the joint Laplace transform of the L1 and L2 norms of a 3-dimensional Bessel bridge. We derive the results by using merely probabilistic arguments. More precisely we show that the law of this functional is closely connected with the one of the first passage time of an Ornstein– Uhlenbeck process. The motivation for studying this problem are multiple; as an instance, they include the computation of the density of the first passage time of Brownian motion over some moving boundaries such as the square root and the quadratic ones. Key words: Bessel bridges, Ornstein–Uhlenbeck process, Williams’ time-reversal theorem, Feynman–Kac formula, Cylinder parabolic function, Boundary crossing