A Mass Transport Approach to Maximization of Expectation of Some Functions of the Final Value and the Running Maximum

Nikolay Lysenko

Published 2014-10-20Version 1

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 functions depending on the terminal value and the running maximum. In the present paper one can find an extension of this result that relates to another class of bivariate functions. Some functions that make stochastic processes other than the Azema-Yor embedding optimal, are also studied. The suggested approach is quite straightforward modulo Monge-Kantorovich theory basics and theorems about a joint distribution of the final value and the maximum. Keywords: Azema-Yor embedding, Monge-Kantorovich problem, supermodular functions, terminal value of a martingale.