  We consider robust Markov Decision Processes with Borel state and action
spaces, unbounded cost and finite time horizon. Our formulation leads to a
Stackelberg game against nature. Under integrability, continuity and
compactness assumptions we derive a robust cost iteration for a fixed policy of
the decision maker and a value iteration for the robust optimization problem.
Moreover, we show the existence of deterministic optimal policies for both
players. This is in contrast to classical zero-sum games. In case the state
space is the real line we show under some convexity assumptions that the
interchange of supremum and infimum is possible with the help of Sion's minimax
Theorem. Further, we consider the problem with special ambiguity sets. In
particular we are able to derive some cases where the robust optimization
problem coincides with the minimization of a coherent risk measure. In the
final section we discuss two applications: A robust LQ problem and a robust
problem for managing regenerative energy.
