A POTENTIAL-BASED CONSTRUCTION OF THE INCREASING SUPERMARTINGALE COUPLING

The increasing supermartingale coupling, introduced by Nutz and Stebegg (Ann. Probab. 46 (2018) 3351-3398) is an extreme point of the set of "supermartingale" couplings between two real probability measures in convex-decreasing order. In the present paper we provide an explicit construction of a triple of functions, on the graph of which the increasing super -martingale coupling concentrates. In particular, we show that the increasing supermartingale coupling can be identified with the left-curtain martingale coupling and the antitone coupling to the left and to the right of a uniquely determined regime-switching point, respectively.Our construction is based on the concept of the shadow measure. We show how to determine the potential of the shadow measure associated to a supermartingale, extending the recent results of Beiglbock et al. (Electron. Commun. Probab. 27 (2022) 1-12) obtained in the martingale setting.