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Let X be some homogeneous additive functional of a skew Bessel process Y. Inthis note, we compute the asymptotics of the first passage time of X to somefixed 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 bbefore the level a. Our results extend some previous works on additivefunctionals of Brownian motion by Isozaki and Kotani for the persistenceproblem, and by Lachal for the exit time problem.

Persistence and exit times for some additive functionals of skew Bessel processes