Variance-penalized Markov decision process

We consider a Markov decision process with both the expected limiting average, and the discounted total return criteria, appropriately modified to include a penalty for the variability in the stream of rewards. In both cases we formulate appropriate nonlinear programs in the space of state-action frequencies (averaged, or discounted) whose optimal solutions are shown to be related to the optimal policies in the corresponding “variance-penalized MDP.” The analysis of one of the discounted cases is facilitated by the introduction of a “Cartesian product of two independent MDPs.