This paper deals with the first passage optimality and variance minimisation problems of discrete-time Markov decision processes (MDPs) with varying discount factors and unbounded rewards/costs. First, under suitable conditions slightly weaker than those in the previous literature on the standard (infinite horizon) discounted MDPs, we establish the existence and characterisation of the first passage expected-optimal stationary policies. Second, to further distinguish the expected-optimal stationary policies, we introduce the variance minimisation problem, prove that it is equivalent to a new first passage optimality problem of MDPs, and, thus, show the existence of a variance-optimal policy that minimises the variance over the set of all first passage expected-optimal stationary policies. Finally, we use a computable
example to illustrate our main results and also to show the difference between the first passage optimality here and the standard discount optimality of MDPs in the previous literature.
