This paper is concerned with the solution of higher order moment stability region for Markov jump systems with respect to cases of both known and unknown transition probabilities. By exploring the cumulant generating function, the original stochastic system with Markov jumping modes is transformed to a deterministic system. Then based on the estimation of matrix eigenvalues, the explicit solution for higher order moment stability region is expressed in terms of transition probabilities, operation modes, state transition matrix and its dimension, and the moment order. Several examples are presented to illustrate the effectiveness of the proposed method and its associated algorithm. (C) 2018 Elsevier Ltd. All rights reserved.