Related papers: Duality for Borel measurable cost functions
We introduce and investigate properties of a variant of the semi-discrete optimal transport problem. In this problem, one is given an absolutely continuous source measure and cost function, along with a finite set which will be the support…
The Gromov--Wasserstein problem is a non-convex optimization problem over the polytope of transportation plans between two probability measures supported on two spaces, each equipped with a cost function evaluating similarities between…
It is known that the topology of a Polish group is uniquely determined by its Borel structure and group operations, but this does not give us a way to find the topology. In this article we expand on this theorem and give a criterion for a…
The classical problem of optimal transportation can be formulated as a linear optimization problem on a convex domain: among all joint measures with fixed marginals find the optimal one, where optimality is measured against a cost function.…
In this paper, we study the optimal transport problem induced by separable cost functions. In this framework, transportation can be expressed as the composition of two lower-dimensional movements. Through this reformulation, we prove that…
This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former,…
We study the notion of debiasability for cost functions arising in optimal transport. We call a symmetric cost function $c:\mathscr{X}\times\mathscr{X}\to\mathbb{R}\cup\{+\infty\}$ debiasable if it satisfies $c(x,y)\ge…
We consider the optimal mass transportation problem in $\RR^d$ with measurably parameterized marginals, for general cost functions and under conditions ensuring the existence of a unique optimal transport map. We prove a joint measurability…
We consider optimal transport problems where the cost for transporting a given probability measure $\mu_0$ to another one $\mu_1$ consists of two parts: the first one measures the transportation from $\mu_0$ to an intermediate (pivot)…
We pursue robust approach to pricing and hedging in mathematical finance. We consider a continuous time setting in which some underlying assets and options, with continuous paths, are available for dynamic trading and a further set of…
Many results in probability (most famously, Strassen's theorem on stochastic domination), characterize some relationship between probability distributions in terms of the existence of a particular structured coupling between them. Optimal…
Based on the multidimensional irreducible paving of De March & Touzi, we provide a multi-dimensional version of the quasi sure duality for the martingale optimal transport problem, thus extending the result of Beiglb\"ock, Nutz & Touzi.…
We study cyclically monotone transport plans between measures in $\mathrm{M}_0(\mathbb{R}^d)$, the class of Borel measures on $\mathbb{R}^d \setminus \{0\}$ that are finite on sets bounded away from the origin but may have infinite total…
For probability measures on a complete separable metric space, we present sufficient conditions for the existence of a solution to the Kantorovich transportation problem. We also obtain sufficient conditions (which sometimes also become…
This paper investigates causal optimal transportation problems, in the framework of two Polish spaces, both endowed with filtrations. Specific concretizations yield primal problems equivalent to several classical problems of stochastic…
We consider Kantorovich optimal transportation problem in the case where the cost function and marginal distributions continuously depend on a parameter with values in a metric space. We prove the existence of approximate optimal Monge…
These notes constitute a sort of Crash Course in Optimal Transport Theory. The different features of the problem of Monge-Kantorovitch are treated, starting from convex duality issues. The main properties of space of probability measures…
The Monge-Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes containing…
We shall prove new contraction properties of general transportation costs along nonnegative measure-valued solutions to Fokker-Planck equations in $R^d$, when the drift is a monotone (or $\lambda$-monotone) operator. A new duality approach…
This note concerns the relationship between conditions on cost functions and domains and the convexity properties of potentials in optimal transportation and the continuity of the associated optimal mappings. In particular, we prove that if…