Optimal Control of Diffusion Processes with Terminal Constraint in Law

TL;DR

This paper develops a framework for stochastic optimal control problems with terminal distribution constraints, deriving necessary conditions through coupled PDEs using convex duality.

Contribution

It introduces a novel approach to handle terminal distribution constraints in stochastic control via coupled PDEs and convex duality techniques.

Findings

Established necessary optimality conditions involving PDE systems.

Connected control problems with terminal distribution constraints to convex duality.

Provided a mathematical foundation for future numerical methods.

Abstract

Stochastic optimal control problems with constraints on the probability distribution of the final output are considered. Necessary conditions for optimality in the form of a coupled system of partial differential equations involving a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation are proved using convex duality techniques.