With an ever-increasing deployment density of probe and fixed sensors, massive vehicular trajectory data is available and show a promising foundation to improve the observability of dynamic traffic demand pattern. However, due to technical and privacy issues, the raw trajectories are not always complete and the paths and destinations between discontinuous trajectory nodes are usually missing. This paper proposes a probabilistic method to jointly reconstruct the missing driving path and destination location of vehicles with incomplete trajectory data. One problem-specific HMM-structured model incorporating spatial and temporal analysis (ST-HMM) is constructed to define the matching probability between observed data and possible movement. Two algorithms, namely candidate set generation and best-match search algorithms, are developed to seek the most possible one as matching result. It can implement end-to-end processing from incomplete trajectory data to complete and connective paths and destinations for the target vehicle. The proposed method is tested based on field-test data and city-wide road network. Compared with two benchmark methods, the proposed method improved the matching accuracy in terms of both path identification and destination inference. Additionally, sensitivity analyses on the size of training dataset and candidate set were performed. We believe that experiment results of these sensitivity analyses can help to provide guidance on data sensing and candidate generation.