Abstract : We consider the problem of detecting a change in the drift cfficient of a diffusion process (disorder problem), using the generalized likelihood ratio test (GLRT). Assuming that a change has occured, the asymptotic behaviour (consistency, asymptotic probability distribution) of the maximum likelihood estimate of the change time has already been investigated in the small noise asymptotics. The purpose of this paper is to study the asymptotics of the GLRT itself, i.e. the asymptotic behaviour of both the probability of false alarm and the probability of miss detection. We prove that these probabilities go to zero with exponential rate, provided a simple detectability assumption is satisfied by the limiting deterministic system. We also investigate the robustness of the GLRT with respect to model misspecification, wich is a very important property for practical implementation. Here, misspecification means that some wrong expressions are used for either the drift cfficient (before change), or the change cfficient. We obtain roughly the same behaviour for the error probabilities, as in the correctly specified case, although the detectability assumption will include the requirement that the change to be detected is larger in some sense than the misspecification error.
