Dynamic Programming Principle for Stochastic Control Problems on Riemannian Manifolds

TL;DR

This paper extends the dynamic programming principle and Hamilton-Jacobi-Bellman theory to stochastic control problems on compact Riemannian manifolds, including existence and uniqueness of solutions.

Contribution

It introduces a framework for stochastic control on Riemannian manifolds, deriving the HJB equation and proving viscosity solution properties and verification theorems.

Findings

Established DPP for control on manifolds

Proved existence and uniqueness of viscosity solutions

Characterized optimal controls via verification theorem

Abstract

In this paper, we first establish the dynamic programming principle for stochastic optimal control problems defined on compact Riemannian manifolds without boundary. Subsequently, we derive the associated Hamilton-Jacobi-Bellman (HJB) equation for the value function. We then prove the existence, uniqueness of viscosity solutions to the HJB equation, along with their continuous dependence on initial data and model parameters. Finally, under appropriate regularity conditions on the value function, we establish a verification theorem that characterizes optimal controls.