The Study On Week Convergences Of GJR-GARCH Model And Sheaf Theory

In the study of dynamic properties in the economic and financial phenomenon, the research of risk or instability plays a very important role. Discrete time autoregressive conditional heteroskedasticity models(ARCH model) are effective means as to these research. They are very important in the practical application. Generalized ARCH models(GARC H model) are easily used to do forecasting, estimation and testing about time series. While stochastic differential equations and continuous time diffusion processes occupy an important position in the theoretical research. Theoretical results about them have been very rich. Sheaf cohomology theory synthesizes various cohomology classes from algebra, topology and geometry background. It is an effective means as to the research of geometric objects. Smooth manifold is a kind of topological spaces with good topological and geometrical properties. When the sheaf cohomology theory is applied to smooth manifold, the results can get better. Thus, this paper mainly studies GJR-GARCH model and its limitation as diffusion processes and sheaf version of generalized Mayer-Vietoris sequence.Chapter 1 mainly introduces the research background of weakly converging theory of discrete time models and sheaf version, analyses and summarizes the research status of them at home and abroad. Besides, the main contents of this paper is gived out.Chapter 2 mainly introduces some necessary basics knowledge, including t he basic definitions and related knowledge in courses of probability and measure, stochastic process and algebraic topology. It lays a foundation for theoretical research and practical application in subsequent chapters.Chapter 3 discusses the theory of discrete time GJR-GARC H model weak ly converging to continuous time diffusion process through a concrete GARCH model: the GJR-GARCH model. Then a bridge is built between the GARCH model and the diffusion process. When doing estimation and testing, GARCH models can be used as a diffusion process approximation. On the contrary, when diffusion process models are as GARCH models approximation, their plentiful theoretical results may be applied to GARCH models.Chapter 4 investigates the sheaves of germs of differential forms on a smooth manifold are soft. To an open covering of a smooth manifold, the generalized Mayer- vietoris sequence and Mayer-vietoris theorem can be obtained from the sheaf theoretic viewpoint, with the aid of Cech-de Rham complex.Chapter 5 sums up all achievements of the paper and put forward the follow-up study.