Continuous Policy and Value Iteration for Relaxed Stochastic Control Problems and Its Convergence

We introduce a continuous policy-value iteration algorithm where the approximations of the value function of a stochastic control problem and the optimal control are simultaneously updated through Langevin-type dynamics. This paper focuses on the entropy-regularized relaxed control problem with infinite horizon. We establish policy improvement and demonstrate convergence to the optimal control under the monotonicity condition of the Hamiltonian. By utilizing Langevin-type stochastic differential equations for continuous updates along the policy iteration direction, our approach enables the use of distribution sampling techniques in machine learning to optimize the value function and identify the optimal control simultaneously. Numerical experiments are presented for LQ model with a non-concave reward, and a non-LQ example with a skewed-Laplace optimal control.