A remark on the Lagrangian formulation of optimal transport with a non-convex cost

We study the Lagrangian formulation of a class of the Monge-Kantorovich optimal transportation problem. It can be considered a stochastic optimal transportation problem for absolutely continuous stochastic processes. A cost function and stochastic processes under consideration is not convex and have essentially bounded time derivatives almost surely, respectively. This paper is a continuation of the second author’s master thesis.