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Introduction to Complex Systems

Physicists have long investigated examples of complex systems such as fluids and materials exhibiting special properties such as magnetism. Here the atoms themselves are the interacting entities that exchange energy and momentum and the approaches used to study such complex systems are rooted in methods of thermodynamics, statistical physics and stochastic processes.

In recent years it has become clear that the tools of theoretical and computational physics can also be used to study complex systems outside of the physical domain. In particular problems found in economics and the social sciences are now being studied using these methods. This may seem surprising but physics is the oldest of the quantitative sciences and one should not really be too surprised to find that the methods developed to tackle quantitative issues can be applied to other sciences that are of more recent provenance. A nice expose of these matters for the layperson is to be found in the book by Ball.

At TCD we have been especially interested in the nature and origin of income and wealth distributions, where the interacting entities might be thought of as 'people' competing for work or exchanging money via a trading process. These ideas can also be used to study the emergence or otherwise of consensus in groups. This latter topic has obvious applications to voting and political processes; it is also responsible for the emergence of cultural attributes of entire societies.

Applications to finance are also of interest. In the past we have studied the statistical characteristics of financial time series from both the UK and Ireland spanning the 19th century to recent times. More recently we have begun to examine the structure and dynamics of multi dimensional time series for asset portfolios. Minimal spanning trees and random matrix theory methods are being used to assess the nature of the underlying correlations.

Fluctuations of financial time series are now well known to exhibit 'fat tails' that are not accounted for by assuming Gaussian processes. We have been developing new analytic methods that allow us to assess the value of multidimensional Levy processes for the computation of option prices. Results obtained so far would seem to offer potential for the calculation of option prices over a wide range of exercise prices and maturity times.