Asymptotic inference for semimartingale models with singular parameter points
The author studies the asymptotic inference for stochastic processes using semimartingale models. In particular he discusses singularities in the parameter space. The paper clarifies some of the reasons for singular points for differentiable experiments in this context. The basic setup is the following: the law of the semimartingale depends on a \(k\)-dimensional Euclidean parameter \(\vartheta\) in such a way that the triplet of the local characteristics, which depends on the unknown parameter \(\vartheta\), is differentiable, and so the score martingale exists. Using the predictable bracket of the score martingale the author derives a local quadratic approximation for the likelihood. With the help of this approximation he then studies singularity points in the parameter space. He then gives a criterion for the quadratic approximation to be mixed normal (and then there is no singularity at this point). This criterion is also based on the bracket of the score martingale. Next he studies the limiting distributions of certain estimators. The paper ends with examples dealing with continuous diffusion processes, diffusions with jumps and point processes. Since the author works with quasi-left continuous semimartingales, the results obtained cannot be applied to discrete time models.
scientific article; zbMATH DE number 605310
scientific article