Asymptotics for dynamical systems driven by jump noise

This dissertation studies asymptotic estimates for dynamical systems with jumps. We first focus on the parameter estimation problem for a linear partially observed system. A least-squares estimator for the intensity of a Poisson process is proposed, where the signal process is driven by the mixture of a Brownian motion and a Poisson precess and the observation is a diffusion process. Precisely, we verify the unbiasedness, consistency for the estimator of the intensity. Furthermore, the asymptotic distribution and convergence rate of the consistent estimator are studied as well as a statistics for statistical inference is constructed employing the central limit theorem, large and moderate deviation principles. The last part of this dissertation is concerned with large deviation principles for the optimal filtering of a general nonlinear model. First, the uniqueness of the solution of the Zakai and Kushner-Stratonovich equations are proved, by applying a pertinent transformation of the associated equations into SDEs in an appropriate Hilbert space. Taking into account the controlled analogue of Zakai and Kushner-Stratonovich equations, respectively, the large deviation principle follows by employing some qualitative properties of their solutions using weak convergence arguments.