Runtime monitoring has been proposed as an alternative to formal verification of cyber-physical systems. A monitor processes the sequence of outputs generated by the system and computes the likelihood of violation of a given property. In turn, it decides whether to raise an alarm. It has been shown that for systems that are monitorable, so-called threshold monitors can be constructed that achieve desired levels of accuracy. In this work we propose decision-theoretic monitors as an alternative to threshold monitors. These monitors are obtained by assigning rewards and penalties for making the correct or wrong decisions. Specifically, we define the monitor using POMDPs (Partially Observable Markov Decision Processes). We present the solution of the monitoring POMDP using a variation of POMCP (Partially Observable Monte-Carlo Planning), and demonstrate its performance on the problem of monitoring a mobile robot system in real-time.