When modelling financial time series, the main difficulty consists in finding a model that captures the so-called stylized facts. These are statistical regularities, such as lep- tocurticity, volatility clustering or strong autocorrelations for absolute and squared re- turns, which are common to most financial series. The most popular way to take such characteristics into account is formed by models of generalized autoregressive conditional heteroscedasticity (GARCH). These models, however, cannot explain the empirically often observed strong dependence in volatility. In 1996, Baillie, Bollerslev and Mikkelsen therefore introduced the fractionally integrated GARCH (FIGARCH) model. While the existence of a strictly stationary solution is ensured, its ability to model long range de- pendence in volatility is controversial. In the literature there exist serveral approaches to define a continuous-time analogue to the discrete GARCH process. The continuous-time GARCH (COGARCH) model of Klüppelberg, Lindner and Maller stands out as it directly generalizes the essential features of its discrete time analogue. In this thesis we present two approaches to incorporate long range dependence into the volatility process of the COGARCH(1,1). The first one is based on Molchan-Golosov fractional Levy processes (FLP), while for the second we use a modification of the Mandelbrot-van-Ness FLP.     «

When modelling financial time series, the main difficulty consists in finding a model that captures the so-called stylized facts. These are statistical regularities, such as lep- tocurticity, volatility clustering or strong autocorrelations for absolute and squared re- turns, which are common to most financial series. The most popular way to take such characteristics into account is formed by models of generalized autoregressive conditional heteroscedasticity (GARCH). These models, however,...     »