This dissertation contains three essays on distinguishing between structural breaks under long memory, testing for fractional cointegration relationship between the financial markets and developing optimal forecast methods under long memory in the presence of a discrete structural break. Chapter 1 introduces the concepts of long memory, fractional cointegration and briefly describes the rest of the chapters.Chapter 2 suggests a testing procedure to discriminate between stationarity, a break in the mean and a break in persistence in a time series that may exhibit long memory is introduced. The asymptotic properties of test statistics based on the CUSUM statistic are studied. In a Monte Carlo study we further analyze the finite sample properties of the procedure. An application to inflation rates shows the potential of our procedure for future research.Chapter 3 revisits the question whether volatilities of different markets and trading zones have a long-run equilibrium in the sense that they are fractionally cointegrated. We consider the U.S., Japanese and German stock, bond and foreign exchange markets to see whether there is fractional cointegration between the markets in one trading zone or for one market across trading zones. Also the other combinations of different markets in different trading zones are considered. Applying a purely semiparametric approach through the whole analysis showsfractional cointegration can only be found for a small minority of different cases. Investigating further we find that all volatility series show persistence breaks during the observation period which may be a reason for different findings in previous studies.Finally, we develop methods in Chapter 4 to obtain optimal forecast under long memory in the presence of a discrete structural break based on different weighting schemes for the observations. We observe significant changes in the forecasts when long-range dependence is taken into account. Using Monte Carlo simulations, we confirm that our methods substantially improve the forecasting performance under long memory. We further present an empirical application to inflation rates that emphasizes the importance of our methods.