In this paper the problem of active fault detection for a non-linear discrete-time stochastic system over an infinite time horizon is considered. The faults in the system are modeled by switching between fault-free and finitely many faulty models. The optimal active fault detector is derived by reformulating the original problem and solving corresponding Bellman's functional equation. A suboptimal solution is based on application of the value iteration or the policy iteration algorithms and a representation of the continuous hyper-state space by a uniform grid. A comparison of both algorithms for solving Bellman's functional equation is illustrated in a numerical example.