Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

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

Auteur(s):

Code(s) de Classification MSC:

• 60G40 Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
• 60G44 Martingales with continuous parameter

Résumé: 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: $T_\mu=\inf\{t>0:F_t\geq \varphi^F_\mu(L_t)\}$. As an application, the Skorokhod embedding problem for a number of functionals $(F_t:t\geq 0)$, including the age (length) and the maximum (height) of excursions, is solved. Explicit formulae for the corresponding stopping times $T_\mu$, such that $F_{T_\mu}\sim\mu$, are given. It is shown that the function $\varphi_\mu^F$ is the same for the maximum and for the age, $\varphi_\mu=\psi_\mu^{-1}$, where $\psi_\mu(x)=\int_{[0,x]}\frac{y}{\overline{\mu}(y)}d\mu(y)$. The joint law of $(g_{T_\mu},T_\mu,L_{T_\mu})$, in the case of the age functional, is characterized. Examples for specific measures $\mu$ are discussed. Finally, a randomized solution to the embedding problem for Az\'ema martingale is deduced. Throughout the article, two possible approaches, using excursions and martingale theories, are presented in parallel.

Mots Clés: Skorokhod embedding problem ; age of Brownian excursions ; Azéma martingale ; functionals of Brownian excursions

Date: 2003-06-24

Prépublication numéro: PMA-833