Maximization of functionals depending on the terminal value and the running maximum of a martingale: a mass transport approach

It is known that the Azéma-Yor solution to the Skorokhod embedding problem maximizes the law of the running maximum of a uniformly integrable martingale with a given terminal value distribution. Recently this optimality property has been generalized to expectations of certain bivariate cost functions depending on the terminal value and the running maximum. In this paper we give an extension of this result to another class of functions. In particular, we study a class of cost functions for which the corresponding optimal embeddings are not Azéma-Yor. The suggested approach is quite straightforward modulo basic facts of the Monge-Kantorovich mass transportation theory. Loosely speaking, the joint distribution of the running maximum and the terminal value in the Azéma-Yor embedding is concentrated on the graph of a monotone function, and we show that this fact follows from the cyclical monotonicity criterion for solutions to the Monge-Kantorovich problem.