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Maximum Likelihood Estimators for Markov Switching Autoregressive Processes with ARCH Component
(2009)

We consider a mixture of AR-ARCH models where the switching between the basic states of the observed time series is controlled by a hidden Markov chain. Under simple conditions, we prove consistency and asymptotic normality of the maximum likelihood parameter estimates combining general results on asymptotics of Douc et al (2004) and of geometric ergodicity of Franke et al (2007).

We study nonlinear finite element discretizations for the density gradient equation in the quantum drift diffusion model. Especially, we give a finite element description of the so--called nonlinear scheme introduced by {it Ancona}. We prove the existence of discrete solutions and provide a consistency and convergence analysis, which yields the optimal order of convergence for both discretizations. The performance of both schemes is compared numerically, especially with respect to the influence of approximate vacuum boundary conditions.

Based on general partitions of unity and standard numerical flux functions, a class of mesh-free methods for conservation laws is derived. A Lax-Wendroff type consistency analysis is carried out for the general case of moving partition functions. The analysis leads to a set of conditions which are checked for the finite volume particle method FVPM. As a by-product, classical finite volume schemes are recovered in the approach for special choices of the partition of unity.

To prove difficult theorems in a mathematical field requires substantial know-ledge of that field. In this paper a frame-based knowledge representation formalismis presented, which supports a conceptual representation and to a large extent guar-antees the consistency of the built-up knowledge bases. We define a semantics ofthe representation by giving a translation into the underlaying logic.