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In this paper the fault detection (FD) task in stochastic continuous-time dynamical systems is addressed. A new family of FD approaches is presented, which is based on the application of hypothesis testing on continuous-time estimators. The given FD schemes are widely analyzed in the framework of their characteristics, such as fault detectability, false alarms and missed detection. A collection of sufficient detectability conditions are given for a class of faults (referred here as generic), characterizing the faults which can be detected with certain formalized guarantee by the given FD schemes, and providing also an upper bound for the detection time in a probabilistic sense. The application and comparative performance of these FD approaches is illustrated for different faults in a simulation example.