Dynamic Programming Principle for Stochastic Control Problems on Riemannian Manifolds

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.