New approaches for modeling and estimation of discrete and continuous time stationary processes

Stationary processes form an important class of stochastic processes that has been extensively studied in the literature, and widely applied in many fields of science. Applications include modeling and forecasting various real-life phenomena such as stock market behavior, sales of a company, natural disasters and velocity of a Brownian particle under the influence of friction, to mention a few. In this dissertation, we consider new methods for modeling and estimation of discrete and continuous time stationary processes. We characterize discrete and continuous time strictly stationary processes by AR(1) and Langevin equations, respectively. From these characterizations, we derive quadratic (matrix) equations for the corresponding model parameter (matrix) in terms of autocovariance of the stationary process. Based on the equations, we construct an estimator for the model parameter. Furthermore, we show that the estimator inherits consistency and the rate of convergence from the chosen autocovariance estimators. Moreover, its limiting distribution is given by a linear function of the limiting distribution of the autocovariance estimators. In addition, we apply the presented general theory in modeling and estimationof a generalization of the ARCH model with stationary liquidity.