Optimal Control for Absolutely Continuous Stochastic Processes and the Mass Transportation Problem

Abstract

We study the optimal control problem for $\mathbb{R}^d$-valued absolutely continuous stochastic processes with given marginal distributions at every time. When $d=1$, we show the existence and the uniqueness of a minimizer which is a function of a time and an initial point. When $d \gt 1$, we show that a minimizer exists and that minimizers satisfy the same ordinary differential equation.