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

TL;DR

This paper investigates a Lagrangian approach to a non-convex optimal transport problem involving stochastic processes with bounded derivatives, extending previous theoretical work.

Contribution

It introduces a Lagrangian formulation for a class of non-convex stochastic optimal transport problems with bounded derivatives.

Findings

Develops a Lagrangian framework for non-convex costs

Extends stochastic optimal transport theory to non-convex settings

Provides mathematical insights into bounded derivative processes

Abstract

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.