2018/08/03 by Benjamin Jourdain, Jourdain, Benjamin, William Margheriti +1 · 1 citation
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.1808.01390
arxiv created 2019/03/07 · arxiv updated 2019/03/08
In this paper, we exhibit a new family of martingale couplings between two one-dimensional probability measures μ and ν in the convex order. This family is parametrised by two dimensional probability measures on the unit square with respective marginal densities proportional to the positive and negative parts of the difference between the quantile functions of μ and ν. It contains the inverse transform martingale coupling which is explicit in terms of the associated cumulative distribution functions. The integral of \vert x-y\vert with respect to each of these couplings is smaller than twice the W1 distance between μ and ν. When μ and ν are in the decreasing (resp. increasing) convex order, the construction is generalised to exhibit super (resp. sub) martingale couplings.