Re-optimization technique is an efficient approach for solving the shortest path problem in dynamic deterministic networks, where link travel times are updated in real-time. However, existing re-optimization techniques, built on the assumption that link travel times are deterministic, cannot be used to solve reliable shortest path problems in real road networks with noticeable levels of travel time uncertainties. This study proposes a novel re-optimization technique, named reliable lifelong planning A* (RLPA*), for re-optimizing reliable shortest path finding results in dynamic stochastic networks, where link travel time distributions are updated in real-time. The proposed RLPA* technique can efficiently determine the optimal solution in dynamic stochastic networks by reusing path search results produced in the previous time instance. The proposed RLPA* technique is further utilized to solve the K reliable shortest paths problem, which is regarded as a series of reliable shortest path searches in a dynamic stochastic network. To validate the proposed algorithms, a comprehensive case study using real traffic data is conducted. The case study results demonstrated that the proposed algorithms significantly outperform the corresponding state-of-the-art algorithms on all testing networks.