A new family of one dimensional martingale couplings
| Autor: | Benjamin Jourdain, William Margheriti |
|---|---|
| Přispěvatelé: | Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), Mathematical Risk Handling (MATHRISK), Université Paris-Est Marne-la-Vallée (UPEM)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria) |
| Rok vydání: | 2020 |
| Předmět: |
Statistics and Probability
Inverse Unit square 01 natural sciences Combinatorics 010104 statistics & probability martingale coupling 91G80 FOS: Mathematics Wasserstein distance [MATH]Mathematics [math] 0101 mathematics 60G42 Mathematics Probability measure Coupling martingale optimal transport martingale couplings Probability (math.PR) 010102 general mathematics Regular polygon convex order [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] 60E15 Statistics Probability and Uncertainty Positive and negative parts Martingale (probability theory) Mathematics - Probability Quantile |
| Zdroj: | Electron. J. Probab. Electronic Journal of Probability Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2020, 25 (136), pp.1-50. ⟨10.1214/20-EJP543⟩ Electronic Journal of Probability, 2020, 25 (136), pp.1-50. ⟨10.1214/20-EJP543⟩ |
| ISSN: | 1083-6489 |
| DOI: | 10.1214/20-ejp543 |
| Popis: | In this paper, we exhibit a new family of martingale couplings between two one-dimensional probability measures $\mu $ and $\nu $ 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 $\mu $ and $\nu $. It contains the inverse transform martingale coupling which is explicit in terms of the quantile functions of these marginal densities. The integral of $|x-y|$ with respect to each of these couplings is smaller than twice the $\mathcal {W}_{1}$ distance between $\mu $ and $\nu $. When the comonotonous coupling between $\mu $ and $\nu $ is given by a map $T$, the elements of the family minimise $\int _{\mathbb {R}}\vert y-T(x)\vert \,M(dx,dy)$ among all martingale couplings between $\mu $ and $\nu $. When $\mu $ and $\nu $ are in the decreasing (resp. increasing) convex order, the construction is generalised to exhibit super (resp. sub) martingale couplings. |
| Databáze: | OpenAIRE |
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