On the global convergence of relative value iteration for infinite-horizon risk-sensitive control of diffusions

In Arapostathis and Borkar (2020), a multiplicative relative value iteration algorithm (RVI) for infinite-horizon risk-sensitive control of diffusions in R^d is studied. Assuming that there exists a control for which the diffusion is positive recurrent, the authors have established that the multiplicative value iteration (VI) algorithm converges to the solution of the multiplicative (risk-sensitive) HJB equation starting from an initial condition within the neighborhood of the solution (local convergence). Under a blanket (uniform) exponential ergodicity assumption, the authors have also shown that the RVI algorithm converges to the solution of the multiplicative HJB equation starting from any positive initial condition (global convergence). In this paper, we revisit this problem without assuming the blanket (uniform) condition. We instead assume a near-monotone running cost, and in addition, a structural assumption relating the running cost function to the solution of the multiplicative HJB equation. We show that this structural assumption implies the existence of a control under which the ground state diffusion is exponentially ergodic. More importantly, a global convergence result of the multiplicative VI/RVI algorithms is established; thus, extending upon the results in Arapostathis and Borkar (2020).