Shadow couplings
Mathias Beiglböck, Nicolas Juillet · Transactions of the American Mathematical Society · 2021
A classical result of Strassen asserts that given probabilities μ , ν \mu , u on the real line which are in convex order, there exists a martingale coupling with these marginals, i.e. a random vector ( X 1 , X 2 ) (X_1,X_2) such that X 1 ∼ μ , X 2 ∼ ν X_1\sim \mu , X_2\sim u and E [ X 2 | X 1 ] = X 1 \mathbb {E}[X_2|X_1]=X_1 . Remarkably, it is a non-trivial problem to construct particular solutions to this problem. Based on the concept of shadow for measures in convex order, we introduce a family of such martingale couplings, each of which admits several characterizations in terms of optimality properties/geometry of the support set/representation through a Skorokhod embedding. As a particular element of this family we recover the (left-)curtain martingale transport, which has recently been studied (see Beiglböck, Henry-Labordère, and Touzi [Stochastic Process. Appl. 127 (2017), pp. 3005–3013]; Beiglböck and Juillet [Ann. Probab. 44 (2016), pp. 42–106]; Campi, Laachir, and Martini [Finance Stoch. 21 (2017), pp. 471–486; Henry-Labordère and Touzi [Finance Stoch. 20 (2016), pp. 635–668]) and which can be viewed as a martingale analogue of the classical monotone rearrangement. As another canonical element of this family we identify a martingale coupling that resembles the usual product coupling and appears as an optimizer in the general transport problem recently introduced by Gozlan et al. In addition, this coupling provides an explicit example of a Lipschitz kernel, shedding new light on Kellerer’s proof of the existence of Markov martingales with specified marginals.