The VaR at Risk.

I show that the structure of the firm is not neutral in respect to regulatory capital budgeted under rules which are based on the Value-at-Risk. Indeed, when a holding company has the liberty to divide its risk into as many subsidiaries as needed, and when the subsidiaries are subject to capital requirements according to the Value-at-Risk budgeting rule, then there is an optimal way to divide risk which is such that the total amount of capital to be budgeted by the shareholder is zero. This result may lead to regulatory arbitrage by some firms.Value-at-Risk;

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

The var at risk

I show that the structure of the firm is not neutral in respect to regulatory capital budgeted under rules which are based on the Value-at-Risk.value-at-risk

Matching with Trade-offs: Revealed Preferences over Competiting Characteristics

We investigate in this paper the theory and econometrics of optimal matchings with competing criteria. The surplus from a marriage match, for instance, may depend both on the incomes and on the educations of the partners, as well as on characteristics that the analyst does not observe. The social optimum must therefore trade off matching on incomes and matching on educations. Given a exible specification of the surplus function, we characterize under mild assumptions the properties of the set of feasible matchings and of the socially optimal matching. Then we show how data on the covariation of the types of the partners in observed matches can be used to estimate the parameters that define social preferences over matches. We provide both nonparametric and parametric procedures that are very easy to use in applications.matching, marriage, assignment.

The var at risk

I show that the structure of the firm is not neutral in respect to regulatory capital budgeted under rules which are based on the Value-at-Risk

On the representation of the nested logit model

We give a two-line proof of a long-standing conjecture of Ben-Akiva and Lerman (1985) regarding the random utility representation of the nested logit model, thus providing a renewed and straightforward textbook treatment of that model. As an application, we provide a closed-form formula for the correlation between two Fr\'echet random variables coupled by a Gumbel copula

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