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It is known that the Azema-Yor solution to the Skorokhod embedding problemmaximizes the law of the running maximum of an uniformly integrable martingalewith given terminal value distribution. Recently this optimality property hasbeen generalized to expectations of certain bivariate cost functions dependingon the terminal value and the running maximum.  In this paper we give an extension of this result to another class offunctions. In particular, we study a class of cost functions with the propertythat the corresponding optimal embeddings are not Azema-Yor. The suggestedapproach is quite straightforward modulo basic facts of the Monge-Kantorovichmass transportation theory. Loosely speaking, the joint distribution of therunning maximum and the terminal value in the Azema-Yor embedding isconcentrated on the graph of a monotone function, and we show that this factfollows from the cyclical monotonicity criterion for solutions to theMonge-Kantorovich problem.  Keywords: Skorokhod problem, Azema-Yor embedding, Monge-Kantorovich problem,optimal transport, supermodular functions, running maximum and the terminalvalue of a martingale.

Maximization of Functionals Depending on the Terminal Value and the Running Maximum of a Martingale: A Mass Transport Approach