On Convergence and Rate of Convergence of Policy Improvement Algorithms

In this paper, we provide a simple proof from scratch for the convergence of the Policy Improvement Algorithm(PIA) for a continuous time entropy-regularized stochastic control problem. Such convergence has been established by Huang-Wang-Zhou(2023) by using sophisticated PDE estimates for the iterative PDEs involved in the PIA. Our approach builds on some Feynman-Kac type probabilistic representation formulae for solutions of PDEs and their derivatives. Moreover, in the infinite horizon model with a large discount factor and in the finite horizon model, we obtain the exponential rate of convergence with similar arguments.