On the Bellman equation in recursive stochastic dynamic programming with the CES aggregator

In this paper we investigate discrete-time infinite horizon Markov decision processes (dynamic programming models) with recursive utilities defined by the classical \emph{CES} aggregator and the certainty equivalent operator being the negative of the entropic risk measure. Under mild conditions on the primitive data we prove existence and uniqueness of a solution to the Bellman equation. Moreover, we show that for any given stationary policy, the Koopmans equation has a unique solution. These two facts imply that a measurable selector of maxima in the Bellman equation is an optimal stationary policy. The methods of our proofs are applied to study solutions to the Bellman equation for other certainty equivalents, including risk-neutral case.