Optimal Transportation Problem by Stochastic Optimal Control

Abstract

We address an optimal mass transportation problem by means of optimal stochastic control. We consider a stochastic control problem which is a natural extension of the Monge–Kantorovich problem. Using a vanishing viscosity argument we provide a probabilistic proof of two fundamental results in mass transportation: the Kantorovich duality and the graph property for the support of an optimal measure for the Monge–Kantorovich problem. Our key tool is a stochastic duality result involving solutions of the Hamilton–Jacobi–Bellman PDE.