An explicit Skorokhod embedding for functionals of Markovian excursions

We develop an explicit non-randomized solution to the Skorokhod embedding problem in an abstract setup of signed functionals of Markovian excursions. Our setting allows to solve the Skorokhod embedding problem, in particular, for diffusions and their (signed, scaled) age processes, for Azema's martingale, for spectrally one-sided Levy processes and their reflected versions, for Bessel processes of dimension smaller than $2$, and for their age processes, as well as for the age process of excursions of Cox-Ingersoll-Ross processes. This work is a continuation and an important generalization of Obloj and Yor (SPA 110) [35]. Our methodology, following [35], is based on excursion theory and the solution to the Skorokhod embedding problem is described in terms of the Ito measure of the functional. We also derive an embedding for positive functionals and we correct a mistake in the formula in [35] for measures with atoms.