Sample Complexity for Markov Decision Processes and Stochastic Optimal Control with Static Risk Measures

We present an elementary state augmentation method for a class of static risk measure applied to the total cost for both Markov decision processes and stochastic optimal control, such that dynamic programming equations can be derived on the augmented space. Through this we discuss the sample complexities of these two problems for both finite-horizon and infinite-horizon settings. We demonstrate the application of the proposed approach through studying distributionally robust functional generated by $\phi$-divergences including conditional value-at-risk.