In the present paper a method to determine the probability distribution of the maximum value process (MVP) of a Markov process is proposed. In this method, an augmented vector process of a physical process and its MVP is constructed. The joint probability density function is then calculated by the path integral solution (PIS), and further the probability density function of the MVP can be obtained as the marginal probability density. A numerical example is shown to validate the proposed method.Financial supports from the National Natural Science Foundation of China (NSFC Grant Nos. 11672209, 51538010 and the National Distinguished Youth Fund of NSFC with Grant No.51725804), the NSFC-DFG joint program (11761131014) and the International Joint Research Program of Shanghai Municipal Government (Grant No. 18160712800) are highly appreciated