Strategies in POMDPs with Stage Duration

Partially observable Markov decision processes (POMDPs) with stage duration provide a framework for approximating continuous-time behavior by scaling transition probabilities with a stage duration parameter $h \in (0,1]$. While previous literature has primarily focused on the limit of the discounted value as the stage duration $h$ vanishes, this paper investigates the global behavior of the asymptotic value, $V(h)$, across varying stage durations. Our main result demonstrates that any strategy in a POMDP with stage duration $h$ can be mimicked in the base POMDP ($h=1$). Specifically, we provide an explicit construction showing that for any strategy in the POMDP with stage duration $h$, there exists a strategy in the base POMDP that secures the same asymptotic payoff. As a consequence of this theorem, we establish that the value function $V(h)$ is nondecreasing with respect to $h$, and that the continuous-time limit $\lim_{h \to 0} V(h)$ exists.