Recurrence and parameter estimation for degenerate diffusions with internal variables and randomly perturbed time-inhomogeneous deterministic input
Date issued
Authors
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
License
Abstract
Taking a multidimensional time-homogeneous dynamical system and adding a randomly perturbed time-dependent deterministic signal to some of its components gives rise to a high-dimensional system of stochastic differential equations which is driven by possibly very low-dimensional noise. Equations of this type are commonly used in biology for modelling neurons or in statistical mechanics for certain Hamiltonian systems.
This thesis is focused on studying two general properties of such a system. In the first part, we use methods from stability theory and control theory as well as Hörmander's condition in order to provide conditions that are sufficient for the corresponding stochastic process to be positive recurrent in the sense of Harris. Harris recurrence gives rise to Limit Theorems for a large class of functionals of the process and can thus be the foundation for applications in asymptotic statistics.
In the second part, considering a statistical model associated to a parametrised class of smooth signals, we exploit Harris recurrence in order to prove Local Asymptotic Normality in the sense of LeCam for the estimation of these parameters under continuous observation of certain components of the process.