Jacobi-like relative value iteration algorithms for ergodic risk-sensitive control of Markov chains

We propose a Jacobi-like relative value iteration (RVI) algorithm and a Gauss-Seidel-like implementation for the ergodic risk-sensitive control (ERSC) problem of a controlled discrete time Markov chain (DTMC) on a finite state space. Under the assumption that the DTMC is irreducible and recurrent under every stationary Markov policy, we prove that the iterates of the proposed RVI algorithms converge at a geometric rate. The main challenge stems from the multiplicative structure of the ERSC cost criterion and the associated Bellman-like operators, which prevents us from adapting the analogous global contraction and bi-Lipschitz continuity properties that underlie the proof of convergence in the average cost setting. We overcome this by establishing local contraction properties for the risk-sensitive Bellman-like operators and a local bi-Lipschitz continuity property for their fixed points, and use these properties to show the iterates converge geometrically. We conclude by implementing our proposed RVI algorithms on two examples: service effort control for a single-server queue of finite capacity, and maximizing the exit rate from a finite domain (on a graph).