On misspecification in regularity and properties of MLE for some stochastic processes

This is a review of some recent results on parameter estimation by the continuous time observations for two models of observations. The first
one is the so-called signal in white Gaussian noise, and the second is inhomogeneous Poisson process. The main question in all statements is: what are the properties of the MLE if there is a misspecification in the regularity conditions. We consider three types of regularity: smooth signals, signals with cusp-type singularity and discontinuous signals (change-point). We suppose that the statistician assumes one type of regularity/singularity, but the real observations contain signals with different type of singularity/regularity. For example, the theoretical (assumed) model has a discontinuous signal, but the real observed (close) signal has cusp-type singularity. We describe the asymptotic behavior of the MLE in such situations.