Understanding Extremes and Clustering in Chaotic Maps and Financial Returns Data

In this thesis we present a numerical and analytical study of modelling extremes in chaotic dynamical systems. We study a range of examples with different dependency structures, and different clustering characteristics. We compare our analysis to the extreme statistics observed for financial returns data, and hence consider the modelling potential of using chaotic systems for understanding financial returns. As part of the study we use the block maxima approach and the peak over threshold method to compute the distribution parameters that arise in the corresponding extreme value distributions. We compare these computations to the theoretical answers, and moreover we obtain error bounds on the rate of convergence of these schemes. In particular we investigate the optimal block size when applying the block maxima method. Since the time series of observations on a dynamical system have dependency we must therefore go beyond the classic approach of studying extremes for independent identically distributed random variables. This is the main purpose of our study. As part of this thesis, we also study clustering in financial returns, and again investigate the potential of using dynamical systems models. Moreover we can also compare numerical quantification of clustering with theoretical approaches. As further work, we measure the dependency structures in our models using a rescaled range analysis. We also make preliminary investigations into record statistics for dynamical systems models, and relate our findings to record statistics in financial data, and to other models (such as random walk models).