Asymptotic Normality in Partially Observed Diffusions with Small Noise: Application to FDI

The problem of residual evaluation for fault detection in partially observed diffusions is investigated, using the local asymptotic approach, under the small noise asymptotics. The score function (i.e. the gradient of the log–likelihood function) evaluated at the nominal value of the parameter, and suitably normalized, is used as residual. It is proved that this residual is asymptotically Gaussian, with mean zero under the null hypothesis, with a different mean (depending linearly on the parameter change) and the same covariance matrix under the contiguous alternative hypothesis. This result relies on the local asymptotic normality (LAN) property for the family of probability distributions of the observation process, which is also proved.