Strassen's classical martingale coupling theorem states that two real-valued random variables are ordered in the convex (resp. increasing convex) stochastic order if and only if they admit a martingale (resp. submartingale) coupling. By analyzing topological properties of spaces of probability measures equipped with a Wasserstein metric and applying a measurable selection theorem, we prove a conditional version of this result for real-valued random variables conditioned on a random element taking values in a general measurable space. We also provide an analogue of the conditional martingale coupling theorem in the language of probability kernels and illustrate how this result can be applied in the analysis of pseudo-marginal Markov chain Monte Carlo algorithms. We also illustrate how our results imply the existence of a measurable minimiser in the context of martingale optimal transport.
The main result of this article (Theorem ) extends Theorem to parametrised collections of ordered pairs of probability distributions, in contrast with ordered sequences as in @cite_3 @cite_24 . The proof of Theorem is based on measurability properties of related set-valued mappings and an application of a measurable selection theorem of Kuratowski and Ryll-Nardzewski @cite_1 . We are unaware of earlier results which would be directly applicable in this context. However, similar results related to martingale couplings have appeared recently in the context of optimal transport. Beiglboeck and Juillet @cite_9 consider the problem of finding an optimal transport plan under the constraint that the transport plan is a martingale. The work of Fontbona, Gu 'erin and M 'el 'eard @cite_26 has the most similarities with our developments. With the notation above, they consider finding a measurable optimal transport plan between @math and @math . The work of Hobson @cite_28 , brought to our attention by a referee, provides an explicit Skorokhod embedding of two univariate convex ordered distributions. This embedding could be used to prove our result in the scalar case.
@cite_26 @cite_28 @cite_9 @cite_1 @cite_3 @cite_24
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We consider the optimal mass transportation problem in @math with measurably parameterized marginals under conditions ensuring the existence of a unique optimal transport map. We prove a joint measurability result for this map, with respect to the space variable and to the parameter. The proof needs to establish the measurability of some set-valued mappings, related to the support of the optimal transference plans, which we use to perform a suitable discrete approximation procedure. A motivation is the construction of a strong coupling between orthogonal martingale measures. By this we mean that, given a martingale measure, we construct in the same probability space a second one with a specified covariance measure process. This is done by pushing forward the first martingale measure through a predictable version of the optimal transport map between the covariance measures. This coupling allows us to obtain quantitative estimates in terms of the Wasserstein distance between those covariance measures. The basic problem of optimal transportation consists in minimizing the expected costs E[c(X1,X2)] by varying the joint distribution (X1,X2) where the marginal distributions of the random variables X1 and X2 are fixed. Inspired by recent applications in mathematical finance and connections with the peacock problem, we study this problem under the additional condition that (Xi)i=1,2 is a martingale, that is, E[X2|X1]=X1. We establish a variational principle for this problem which enables us to determine optimal martingale transport plans for specific cost functions. In particular, we identify a martingale coupling that resembles the classic monotone quantile coupling in several respects. In analogy with the celebrated theorem of Brenier, the following behavior can be observed: If the initial distribution is continuous, then this “monotone martingale” is supported by the graphs of two functions T1,T2:R→R . We provide a framework to approximate the 2-Wasserstein distance and the optimal transport map, amenable to efficient training as well as statistical and geometric analysis. With the quadratic cost and considering the Kantorovich dual form of the optimal transportation problem, the Brenier theorem states that the optimal potential function is convex and the optimal transport map is the gradient of the optimal potential function. Using this geometric structure, we restrict the optimization problem to different parametrized classes of convex functions and pay special attention to the class of input-convex neural networks. We analyze the statistical generalization and the discriminative power of the resulting approximate metric, and we prove a restricted moment-matching property for the approximate optimal map. Finally, we discuss a numerical algorithm to solve the restricted optimization problem and provide numerical experiments to illustrate and compare the proposed approach with the established regularization-based approaches. We further discuss practical implications of our proposal in a modular and interpretable design for GANs which connects the generator training with discriminator computations to allow for learning an overall composite generator. We are interested in the computation of the transport map of an Optimal Transport problem. Most of the computational approaches of Optimal Transport use the Kantorovich relaxation of the problem to learn a probabilistic coupling @math but do not address the problem of learning the underlying transport map @math linked to the original Monge problem. Consequently, it lowers the potential usage of such methods in contexts where out-of-samples computations are mandatory. In this paper we propose a new way to jointly learn the coupling and an approximation of the transport map. We use a jointly convex formulation which can be efficiently optimized. Additionally, jointly learning the coupling and the transport map allows to smooth the result of the Optimal Transport and generalize it to out-of-samples examples. Empirically, we show the interest and the relevance of our method in two tasks: domain adaptation and image editing.