Degree Type


Date of Award


Degree Name

Doctor of Philosophy



First Advisor

Alicia Carriquiry

Second Advisor

Wolfgang Kliemann


Data collected over time may exhibit some type of memory structure, such as a short or long term memory. Two commonly used indicators of memory are the Hurst exponent and the self-similarity index. We investigate the relationship between the Hurst exponent and the self-similarity index and show that the Hurst exponent is an estimator of the self-similarity index in some time series such as fractional Brownian motion. For time series with constant self-similarity index, we compare the statistical properties of various estimators of the self-similarity index via simulation for a range of nominal H-values between 0 and 1. We also employ windowing techniques to study the over-time behavior of the memory structure in a subset of the S&P500 series.

Further, we incorporate the memory indicators into dynamical models. In particular, and due to their popularity in terms of use, we look at two continuous-timed dynamical systems - the Log Ornstein-Uhlenbeck (LogOU) and the Cox-Ingersoll-Ross (CIR) models and investigate how to extend them by substituting the standard Brownian motion driver for a fractional driver in order to allow more flexibility in their memory structures. From the point of view of Young's integrals we confirm the well-definedness of the two new models by noticing that the smoothness of the CIR and fractional OU solutions is similar to the smoothness of their random drivers. We also explore the memory structures underlying these two updated models, and develop related results through analytical and numerical approaches. Finally, we discuss how to estimate the memory indicators and other model parameters simultaneously in the two model systems within a Bayesian framework.

Copyright Owner

Wen Li



Date Available


File Format


File Size

123 pages