Distribution functions, extremal limits and optimal transport

2015/02/24 by Maria Rita Iacò, Iacò, Maria Rita, Stefan Thonhauser +3
Mathematics · #FOS: Mathematics #Optimization and Control (math.OC) #math.OC

paper · pdf · doi:10.48550/arxiv.1502.06839

arxiv created 2015/02/24 · arxiv updated 2015/02/25

Abstract

Encouraged by the study of extremal limits for sums of the form lim_N→∞(1 )/(N)∑_n=1^N c(x_n,y_n) with uniformly distributed sequences \x_n\, \y_n\ the following extremal problem is of interest max_γ∫_[0,1]²c(x,y)γ(dx,dy), for probability measures γ on the unit square with uniform marginals, i.e., measures whose distribution function is a copula. The aim of this article is to relate this problem to combinatorial optimization and to the theory of optimal transport. Using different characterizations of maximizing γ's one can give alternative proofs of some results from the field of uniform distribution theory and beyond that treat additional questions. Finally, some applications to mathematical finance are addressed.

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