Orateur
Description
We study a Lagrangian particle discretization of a risk estimation problem formulated in the framework of optimal transport. Broadly speaking, the problem consists in determining the most pessimistic dependence structure of a finite number of risk factors. Given finitely many univariate probability measures representing risk factors, together with a real-valued loss function that quantifies the severity of each scenario, we seek the joint distribution that maximizes a prescribed spectral risk measure. More precisely, the objective function is not the mean of the loss functions under the joint law, but a weighted average of the induced loss distribution. The weights emphasize large losses, which makes the functional nonlinear in the joint law. Ennaji, Mérigot, Nenna, and Pass showed that this problem is equivalent to a standard multimarginal optimal transport problem with one additional marginal encoding the weights of the spectral risk measure. As is often the case in optimal transport, solutions are typically supported on sets of dimension much smaller than that of the ambient space, which renders Eulerian grid-based methods inherently inefficient. To discretize the problem, we therefore approximate the joint law by a measure supported on N equally weighted Dirac masses and optimize their locations, in a Lagrangian-like framework. The marginal constraints are relaxed through quadratic Wasserstein penalization terms, each corresponding to a one-dimensional semi-discrete optimal transport cost. We show the discretized problem converges to the original one as N goes to infinity, and we establish two different convergence rates. The first one is determined by the dimension of the support of an optimal solution, and requires only mild assumptions. On the other hand, assuming supermodularity of the loss function — a rather strong assumption — the convergence rate depends solely on the asymptotic uniform quantization errors of the marginal distributions. Numerical experiments illustrate and support our theoretical results.