Discounting long run average growth in stochastic dynamic programs

Finding solutions to the Bellman equation relies on restrictive boundedness assumptions. The literature on endogenous growth or business cycle models with unbounded random shocks provide with numerous examples of recursive programs in which returns are not bounded along feasible paths. In this paper we develop a method of proof that allows to account for models of this type. In applications our assumptions only imply that long run average (expected) growth is sufficiently discounted, in sharp contrast with classical assumptons either absolutely bounding growth or bounding each period (instead of long run) maximum (instead of average) growth. We discuss our work in relation to the literature and provide several examples.