Cross-Sectional and Temporal Dependence in Complex Data

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Understanding the temporal random nature of data sequences is central to many areas of research, be it academic or industry-based. The financial industry, for example, is particularly interested in inferring information from large data sets and thus discovering any patterns revealed by recognising a signal. There is an additional dimension: not only is advancement in the statistical analysis and the Mathematics of random time series needed (temporal analysis), but also their cross-sectional properties need to be better understood. That is, one would also like to know more about the dependence structures across several time series. This is a problem found in financial portfolio analysis, among others, where the price dynamics and the risk profile of a portfolio of financial assets may be determined by hundreds of time series. This research programme treats multivariate generalised diffusions and multivariate Lévy processes from the perspective of structural and driving noise-based dependence structures. Such an investigation includes the study of concordance measures induced by endogenous factors, observed in the market or an economy, which may influence the dynamics of price processes and hedging strategies central in risk management. Progress in these areas of Mathematics and Statistics could have an immediate impact on the analysis of market efficiency, the pricing and hedging of risks, model risk, and the wider concept of no-arbitrage in financial markets. Although these areas are of particular interest to the financial industry, it is a fact that these are important questions in the Mathematics of Data and Information, in general.