The Azéma-Yor embedding in non-singular diffusions

Let (Xt)t[greater-or-equal, slanted]0 be a non-singular (not necessarily recurrent) diffusion on starting at zero, and let [nu] be a probability measure on Necessary and sufficient conditions are established for [nu] to admit the existence of a stopping time [tau]* of (Xt) solving the Skorokhod embedding problem, i.e. X[tau]* has the law [nu]. Furthermore, an explicit construction of [tau]* is carried out which reduces to the Azéma-Yor construction (Séminaire de Probabilités XIII, Lecture Notes in Mathematics, Vol. 721, Springer, Berlin, p. 90) when the process is a recurrent diffusion. In addition, this [tau]* is characterized uniquely to be a pointwise smallest possible embedding that stochastically maximizes (minimizes) the maximum (minimum) process of (Xt) up to the time of stopping.