Risk Measurement under Dependence Structure Ambiguity

Abstract

In this thesis, we work on a generalization of the entropy regularized optimal transport problem, with the objective function being (spectral) risk measures. We accomplish three goals: to present the corresponding dual problem and prove Kantorovich duality, to prove stability of the optimal value under the weak convergence of marginals, the reference measure and the regularization threshold, and to explore an efficient numerical algorithm for a solution of the optimization problem.

The analogue of the Kantorovich duality is proved using techniques from convex analysis. Stability and convergence of approximating optimization problems are studied using the techniques of Gamma convergence, combined with recent results on shadow couplings. For the numerical solution of the optimization problem, a variation on Sinkhorn’s algorithm is developed, which improves on a naive linear programming implementation significantly, in terms of both running time and storage requirements.