A Note on Positive Dynamic Programming

This note considers total reward Markov decision processes with countable state space. For these models it is well known that in the positive case, i.e. the immediate reward function is nonnegative, without further conditions (1) the value iteration holds and (2) there exist pointwise good stationary strategies. Here we show that (1) remains true if the nonnegativity of the immediate reward function is replaced by the nonnegativity of the value function and that (2) remains true if there exists a strategy with nonnegative total rewards.