Embedding in Brownian motion with drift and the Azéma–Yor construction

We consider the embedding of a probability distribution in Brownian motion with drift. We first give a sufficient condition on the target measure, under which a variant of the Azéma–Yor (1979a, Séminaire de Probabilités XIII, Lecture Notes in Mathematics, Vol. 721, Springer, Berlin, pp. 90–115) construction for this problem works. A necessary and sufficient condition for embeddability by means of some stopping time, not necessarily finite, is also provided. This latter condition is then analyzed in some detail.