Langevin processes, agent models and socio-economic systems
Peter Richmond, Department of Physics, Trinity College, Dublin 2, Ireland
Abstract
We review some approaches to the understanding of
fluctuations of financial asset prices. Our approach builds on the development
of a simple Langevin equation that characterises stochastic processes. This
provides a unifying approach that allows first a straightforward description of
the early approaches of Bachelier. We generalise the approach to stochastic
equations that model interacting agents. The agent models recently advocated
byMarsilli and Solomon are motivated. Using a simple change of variable, we show
that the peer pressure model of Marsilli and the wealth dynamics model of
Solomon are essentially equivalent. The methods are further shown to be
consistent with a global free energy functional that invokes an entropy term
based on the Boltzmann formula.
A more recent approach by Michael and Johnson maximised a
Tsallis entropy function subject to simple constraints. They obtain a
distribution function for financial returns that exhibits power law tails and
which can describe the distribution of returns not only over low but also high
frequencies (minute by minute) data for the Dow Jones index. We show how
this approach can be developed from an agent model where the simple Langevin
process is now conditioned by local rather than global noise. Such local noise
may of course be the origin of speculative frenzy or herding in the market
place. The approach yields a BBGKY type hierarchy of equations for the system
correlation functions. Of especial interest is that the results can be obtained
from a new free energy functional similar to that mentioned above except that a
Tsallis like entropy term replaces the Boltzmann entropic term. A mean field
approximation yields the results of Michael and Johnson.
Generalising the model to include many body interactions, we
show how personal income data for Brazil, the US, Germany and the UK, analysed
recently by Borgas can be understood, albeit qualitatively by this latter
approach.