A potential-based construction of the increasing supermartingale coupling

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 Beiglböck et al. (The potential of the shadow measure, arXiv preprint, 2020) obtained in the martingale setting. Contents 1. Introduction 2. Preliminaries 2.1. Measures and Convex order 2.2. Convex hull 3. The shadow measure and π I 3.1. The maximal element 3.2. The shadow measure 3.3. The increasing supermartingale coupling π I 3.4. Lifted supermartingale transport plans 4. The geometric construction of π I 4.1. The regime-switching point u * 4.2. Proof of Theorem 1.1 5. Appendix References

Supermartingale Brenier's Theorem with full-marginals constraint

“…• on (−∞, x 1 (t)]×R, Pµt,µt+ coincides with the left-curtain martingale coupling of µ t | (−∞,x 1 (t)] and S µt+ (µ t | (−∞,x 1 (t)] ); • the support of (µ t − µ t | (−∞,x 1 (t)] ) is strictly to the right of the support of (µ t+ − S µt+ (µ t | (−∞,x 1 (t)] )); • on (x 1 (t), ∞) × R, Pµt,µt+ coincides with the antitone coupling of (µ t − µ t | (−∞,x 1 (t)] ) and (µ t+ − S µt+ (µ t | (−∞,x 1 (t)] )); see Bayraktar et al [4] for details. (In fact, these properties hold for any measures µ ≤ cd ν, and not necessarily µ t ≤ cd µ t+ as in (14).)…”

“…In the martingale region, the two supporting maps T u and T d are characterized in Proposition B. 4.…”

“…Let (µ t ) t∈[0,1] be as in (14). Fix t ∈ [0, 1) and ∈ (0, 1 − t], and denote by Pµt,µt+ the increasing supermartingale coupling of µ t and µ t+ (see Nutz and Stebegg [57] and Bayraktar et al [4]).…”

“…More precisely, the limiting process, associated to the left-curtain coupling, is driven by downward jumps and upward drift, while in the right-curtain case it is the upward jumps and downward drift that drive the limiting process. On the other hand, in the supermartingale case, the increasing supermartingale transport plan and the decreasing supermartingale transport plan demonstrate different behaviour see Bayraktar et al [4], [5]. In the increasing case, the limiting process is a martingale in the interior of the support of marginals, and possesses strict supermartingale characteristics when it escapes to the upper boundary of the support (it jumps from the upper all the way down to the lower boundary).…”