From the Monge-Ampere equation to Stochastic Optimal Control
A Monge-Ampere equation is a nonlinear second-order partial differential equation involving the determinant of the Hessian matrix of the unknown function. It arises in many fields, such as Optimal Transport. Under appropriate conditions, the Monge-Ampere equation has an equivalent Hamilton-Jacobi-Bellman (HJB) equation formulation in the classical sense. In this talk, we will look at a model Monge-Ampere equation and its corresponding HJB equation and identify which Stochastic Optimal Control (SOC) problem underlies the HJB equation. We will discuss the proof of the existence of weak solutions to the SOC problem and showcase attempts at solving it numerically using Reinforcement Learning. Finally, we will look at some more complicated examples of the Monge-Ampere equation and discuss the challenges of applying the same methodology.