A potential-based construction of the increasing supermartingale coupling

TL;DR

This paper presents an explicit potential-based construction of the increasing supermartingale coupling, linking it to the left-curtain martingale and antitone couplings through a regime-switching point, extending shadow measure concepts.

Contribution

It provides a novel explicit construction of the increasing supermartingale coupling using potential theory and shadow measures, connecting it with known couplings.

Findings

Explicit construction of the coupling via potential functions

Identification with left-curtain and antitone couplings

Extension of shadow measure potential results to supermartingales

Abstract

The increasing supermartingale coupling, introduced by Nutz and Stebegg (Canonical supermartingale couplings, Annals of Probability, 46(6):3351--3398, 2018) 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 supermartingale 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 Beiglb\"{o}ck et al. (The potential of the shadow measure, Electron. Commun. Probab., 27, paper no. 16, 1--12, 2022) obtained in the martingale setting.