  Let X be some homogeneous additive functional of a skew Bessel process Y. In
this note, we compute the asymptotics of the first passage time of X to some
fixed level b, and study the position of Y when X exits a bounded interval [a,
b]. As a by-product, we obtain the probability that X reaches the level b
before the level a. Our results extend some previous works on additive
functionals of Brownian motion by Isozaki and Kotani for the persistence
problem, and by Lachal for the exit time problem.
