A statistical analysis of the stochastic dynamics in financial and geomorphological systems using Artificial Intelligence and Probability Theory

This thesis is an investigation into the statistical properties of stochastic systems, primarily using financial data series and then applying the methods and analysis developed from the financial data to a geomorphological setting. This thesis starts by investigating the dynamics of the higher order statistical and standardised moments for financial time series when the time series are truncated to different lengths. We compare these dynamics to a Generalised Autoregressive Conditional Heteroskedasticity (GARCH) model, to elucidate the extent a GARCH model can be used to fit the empirical data within these time windows. When we seek to use a Gaussian conditional distribution to describe the distribution of price change, we cannot fit the empirical data using the higher order standardised moments of the time series. Nevertheless, when we instead use a double Gaussian conditional distribution, we are able to fit empirical data using the GARCH-double-normal model. However, with such a conditional distribution, the fitting becomes time dependent. The parameters of the distribution that can fit the time windows changes with the length of the time window. As a result, we are able to extract a signal of the 2008 banking crisis, and the 2020 global COVID-19 pandemic from the GARCH parameter's, α0, value for different financial securities. [...]