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

It is known that the Azema-Yor solution to the Skorokhod embedding problem maximizes the law of the running maximum of an uniformly integrable martingale with 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 with the property that the corresponding optimal embeddings are not Azema-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 Azema-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. Keywords: Skorokhod problem, Azema-Yor embedding, Monge-Kantorovich problem, optimal transport, supermodular functions, running maximum and the terminal value of a martingale.