We use Markov risk measures to formulate a risk averse version of a total cost problem on a controlled Markov process in infinite horizon. The one step costs are in $L^1$ but not necessarily bounded. We derive the conditions for the existence of the optimal strategies and present the robust dynamic programming equations. We illustrate our results in an optimal investment problem on infinite horizon.

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View Optimal Control of MDP's with Unbounded Cost on Infinite Horizon