Exponential convergence for a convexifying equation and a non-autonomous gradient flow for global minimization

We consider an evolution equation similar to that introduced by Vese and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time. We then introduce a non-autonomous gradient flow and prove that its trajectories all converge to minimizers of the convex envelope

Vector Quantile Regression: An Optimal Transport Approach

We propose a notion of conditional vector quantile function and a vector quantile regression. A \emph{conditional vector quantile function} (CVQF) of a random vector $Y$, taking values in $\mathbb{R}^d$ given covariates $Z=z$, taking values in $\mathbb{R}$, is a map $u \longmapsto Q_{Y\mid Z}(u,z)$, which is monotone, in the sense of being a gradient of a convex function, and such that given that vector $U$ follows a reference non-atomic distribution $F_U$, for instance uniform distribution on a unit cube in $\mathbb{R}^d$, the random vector $Q_{Y\mid Z}(U,z)$ has the distribution of $Y$ conditional on $Z=z$. Moreover, we have a strong representation, $Y = Q_{Y\mid Z}(U,Z)$ almost surely, for some version of $U$. The \emph{vector quantile regression} (VQR) is a linear model for CVQF of $Y$ given $Z$. Under correct specification, the notion produces strong representation, $Y=\beta \left(U\right) ^\top f(Z)$, for $f(Z)$ denoting a known set of transformations of $Z$, where $u \longmapsto \beta(u)^\top f(Z)$ is a monotone map, the gradient of a convex function, and the quantile regression coefficients $u \longmapsto \beta(u)$ have the interpretations analogous to that of the standard scalar quantile regression. As $f(Z)$ becomes a richer class of transformations of $Z$, the model becomes nonparametric, as in series modelling. A key property of VQR is the embedding of the classical Monge-Kantorovich's optimal transportation problem at its core as a special case. In the classical case, where $Y$ is scalar, VQR reduces to a version of the classical QR, and CVQF reduces to the scalar conditional quantile function. An application to multiple Engel curve estimation is considered

Pareto efficiency for the concave order and multivariate comonotonicity

In this paper, we focus on efficient risk-sharing rules for the concave dominance order. For a univariate risk, it follows from a comonotone dominance principle, due to Landsberger and Meilijson [25], that efficiency is characterized by a comonotonicity condition. The goal of this paper is to generalize the comonotone dominance principle as well as the equivalence between efficiency and comonotonicity to the multi-dimensional case. The multivariate setting is more involved (in particular because there is no immediate extension of the notion of comonotonicity) and we address it using techniques from convex duality and optimal transportation

Letter from Alfred Guillaume Gabriel, Count d\u27Orsay to an unidentified recipient

Letter from Alfred Guillaume Gabriel, Count d\u27Orsay to an unidentified individual, dated \u27Marcredi\u27.https://scholarworks.umt.edu/whicker/1021/thumbnail.jp

Lithograph of Thomas Carlyle by Alfred Guillaume Gabriel, Count d\u27Orsay

Lithograph of Thomas Carlyle by D\u27Orsay, dated May 1839.https://scholarworks.umt.edu/whicker/1022/thumbnail.jp

Pareto efficiency for the concave order and multivariate comonotonicity

This paper studies efficient risk-sharing rules for the concave dominance order. For a univariate risk, it follows from a comonotone dominance principle, due to Landsberger and Meilijson (1994), that efficiency is characterized by a comonotonicity condition. The goal of the paper is to generalize the comonotone dominance principle as well as the equivalence between efficiency and comonotonicity to the multidimensional case. The multivariate case is more involved (in particular because there is no immediate extension of the notion of comonotonicity), and it is addressed by using techniques from convex duality and optimal transportation

A Spitzer Search for Water in the Transiting Exoplanet HD189733b

We present Spitzer Space Telescope observations of the extrasolar planet HD189733b primary transit, obtained simultaneously at 3.6 and 5.8 microns with the Infrared Array Camera. The system parameters, including planetary radius, stellar radius, and impact parameter are derived from fits to the transit light curves at both wavelengths. We measure two consistent planet-to-star radius ratios, (Rp/Rs)[3.6$\mu$m] = 0.1560 +/- 0.0008(stat) +/- 0.0002(syst) and (Rp/Rs)[5.8$\mu$m] = 0.1541 +/- 0.0009(stat) +/- 0.0009(syst), which include both the random and systematic errors in the transit baseline. Although planet radii are determined at 1%-accuracy, if all uncertainties are taken into account the resulting error bars are still too large to allow for the detection of atmospheric constituants like water vapour. This illustrates the need to observe multiple transits with the longest possible out-of-transit baseline, in order to achieve the precision required by transmission spectroscopy of giant extrasolar planets.Comment: Accepted in The Astrophysical Journal Letter