An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale

A general methodology allowing to solve the Skorokhod stopping problem for positive functionals of Brownian excursions, with the help of Brownian local time, is developed. The stopping times we consider have the following form: Tμ=inf{t>0: Ft⩾ϕμF(Lt)}. As an application, the Skorokhod embedding problem for a number of functionals (Ft: t⩾0), including the age (length) and the maximum (height) of excursions, is solved. Explicit formulae for the corresponding stopping times Tμ, such that FTμ∼μ, are given. It is shown that the function ϕμF is the same for the maximum and for the age, ϕμ=ψμ−1, where ψμ(x)=∫[0, x](y/μ̄(y)) dμ(y). The joint law of (gTμ,Tμ,LTμ), in the case of the age functional, is characterized. Examples for specific measures μ are discussed. Finally, a randomized solution to the embedding problem for Azéma martingale is deduced. Throughout the article, two possible approaches, using excursions and martingale theories, are presented in parallel.