In this article, the problem of the asynchronous fault detection (FD) observer design is discussed for 2-D Markov jump systems (MJSs) expressed by a Roesser model. In general, the FD observer cannot work synchronously with the system, that is, the mode of the observer varies with the mode of the system in line with some conditional transitional probabilities. For dealing with this difficult point, a hidden Markov model (HMM) is employed. Then, combining the H∞ attenuation index and H_ increscent index, a multiobjective solution to the FD problem is formed. In terms of linear matrix inequality technology, sufficient conditions are gained to guarantee the existence of the asynchronous FD. Simultaneously, an asynchronous FD algorithm is generated to acquire the optimal performance indices. Finally, a numerical example concerned with the Darboux equation is demonstrated to exhibit the soundness of the developed approach.