Motivated by security applications, where agent intentions are unknown, actions may have stochastic outcomes, and an observer may have an obfuscated view due to low sensor resolution, we introduce partially-observable states and unobservable actions into a stochastic goal recognition design framework. The proposed model is accompanied by a method for calculating the expected maximal number of steps before the goal of an agent is revealed and a new sensor refinement modification that can be applied to enhance goal recognition. A preliminary empirical evaluation on a range of benchmark applications shows the effectiveness of our approach.