  In this paper, we consider the control problem with the Average-Value-at-Risk
(AVaR) criteria of the possibly unbounded $L^{1}$-costs in infinite horizon on
a Markov Decision Process (MDP). With a suitable state aggregation and by
choosing a priori a global variable $s$ heuristically, we show that there exist
optimal policies for the infinite horizon problem. To our knowledge, this is
the first work of deriving dynamic programming equations with $L^1$-unbounded
costs via AVaR-operator.
