Robust risk management via multi-marginal optimal transport

TL;DR

This paper links risk management problems to multi-marginal optimal transport, providing explicit solutions in simple cases, conditions for solution structure in higher dimensions, and stability results under perturbations.

Contribution

It establishes a novel equivalence between spectral risk maximization and multi-marginal optimal transport, with explicit solutions and structural insights for various dimensions.

Findings

Explicit solutions for one-dimensional variables

Conditions for solution concentration and uniqueness in higher dimensions

Stability and Lipschitz dependence results

Abstract

We study the problem of maximizing a spectral risk measure of a given output function which depends on several underlying variables, whose individual distributions are known but whose joint distribution is not. We establish and exploit an equivalence between this problem and a multi-marginal optimal transport problem. We use this reformulation to establish explicit, closed form solutions when the underlying variables are one dimensional, for a large class of output functions. For higher dimensional underlying variables, we identify conditions on the output function and marginal distributions under which solutions concentrate on graphs over the first variable and are unique, and, for general output functions, we find upper bounds on the dimension of the support of the solution. We also establish a stability result on the maximal value and maximizing joint distributions when the output function, marginal distributions and spectral function are perturbed; in addition, when the variables one dimensional, we show that the optimal value exhibits Lipschitz dependence on the marginal distributions for a certain class of output functions. Finally, we show that the equivalence to a multi-marginal optimal transport problem extends to maximal correlation measures of multi-dimensional risks; in this setting, we again establish conditions under which the solution concentrates on a graph over the first marginal.