Stochastic Orders and Constrained Transport – SOCOT

Our goal is to improve understanding and extend the range of applications of stochastic orders ubiquitous objects in probability, statistics and mathematics of decision-making in connection with the weak optimal transport theory, that is an important setting of the generalized transport theory and carries new opportunities for applications. We shall address emblematic conjectures and federate a wide range of expertise. By bringing together a wide range of expertise, we shall address fundamental issues and exploit our tools in new contexts. The project is structured around three axes and three poles. Axis I: Transport problems under stochastic constraints and applications. Axis II: Application to functional inequalities and convex geometry. Axis III: Application to data processing. The partner universities are the University of Upper Alsace (for pole 1, probability and actuarial sciences), the University Paris-Cité (pole 2, analysis, geometry and probability) and the University of Toulon (pole 3 specialized in optimization: analysis and data processing).

A stochastic ordering is a partial order relation between measures, mostly probability measures. This order is induced by some spaces A of test functions, mostly cones. The most used orderings are the increasing order and the convex order, for which A is the space A_1 of increasing functions or the space A_2 of convex functions, respectively. The scope of stochastic orders is in fact immense; they appear in operational research, game theory, mathematics of decision-making or statistics. They also hold a central place in Choquet’s theory and in potential theory. The link between weak transport problems and stochastic orders is made with the stochastic kernels (the conditional laws of the considered couplings). The later appear both in the Strassen-type theorems and in the formulation of weak costs.