Rates of convergence of Markov chain approximation for controlled regime-switching diffusions with stopping times

This work develops rates of convergence of Markov chain approximation methods for controlled switching diffusions, where the cost function is over an infinite horizon with stopping times and without discount. The discrete events are modeled by continuous-time Markov chains to delineate random environment and other random factors that cannot be represented by diffusion processes. The paper presents a first attempt using probabilistic approach for studying rates of convergence. In contrast to the significant developments in the literature using partial differential equation (PDE) methods for approximation of controlled diffusions, there appear to be yet any PDE results to date for rates of convergence of numerical solutions for controlled switching diffusions, to the best of our knowledge. Moreover, in the literature, to prove the convergence using Markov chain approximation methods for control problems involving cost functions with stopping (even for uncontrolled diffusion without switching), an added assumption was used to avoid the so-called tangency problem. In this paper, by modifying the value function, it is demonstrated that the anticipated tangency problem will not arise in the sense of convergence in probability and convergence in L¹.