An explicit martingale version of the one-dimensional Brenier theorem

By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge-Kantorovich mass transport problem was introduced in [3, 24]. Further, by suitable adaptation of the notion of cyclical monotonicity, [4] obtained an extension of the one-dimensional Brenier’s theorem to the present martingale version. In this paper, we complement the previous work by extending the so-called Spence-Mirrlees condition to the case of martingale optimal transport. Under some technical conditions on the starting and the target measures, we provide an explicit characterization of the corresponding optimal martingale transference plans both for the lower and upper bounds. These explicit extremal probability measures coincide with the unique left and right monotone martingale transference plans introduced in [4]. Our approach relies on the (weak) duality result stated in [3], and provides, as a by-product, an explicit expression for the corresponding optimal semistatic hedging strategies. We finally provide an extension to the multiple marginals case.