Applications of Ornstein-Uhlenbeck Type Stochastic Differential Equations
In this dissertation, we show with plausible arguments that the Stochastic Differential Equations (SDEs) arising on the superposition and coupling system of independent Ornstein-Uhlenbeck process is a new method available in modern literature that takes the properties and behavior of the data into consideration when performing the statistical analysis of the time series. The time series to be analyzed is thought of as a source of fluctuations, and thus we need a model that takes this behavior into consideration when performing such analysis. Most of the standard methods fail to take into account the physical behavior of the time series, and some of the models are not completely stochastic. Thus in an attempt to overcome the modeling problems associated with the memory-less property models used in the traditional methods, we propose a continuous-time stationary and non-negative stochastic differential equation that is useful for describing a unique type of dependence in a sequence of events. The Ornstein-Uhlenbeck type SDE offers plenty of analytic flexibility which is not available to more standard models such as the geometric Gaussian Ornstein-Uhlenbeck processes. Moreover, the SDE provides a class of continuous time processes capable of exhibiting long memory behavior. The presence of long memory suggests that current information is highly correlated with past information at different levels. This facilitates prediction. The proposed SDE is applied to two different sets of real data; financial and geophysical time series. In the analysis of the time series, we show that the SDE makes new properties and estimate parameters that are useful for making inferences and predicting these types of events.
Tweneboah, Osei Kofi, "Applications of Ornstein-Uhlenbeck Type Stochastic Differential Equations" (2020). ETD Collection for University of Texas, El Paso. AAI27956843.