Power law distributions and dynamic behaviour of stock markets

A simple agent model is introduced by analogy with the mean field approach to the Ising model for a magnetic system. Our model is characterised by a g eneralised Langevin equation phi over dot = F (phi)+ G (phi)(n) over cap (t ) where (n) over cap (t) is the usual Gaussian white noise, i.e.: <<(<eta>) over cap>(t)<(<eta>)over cap>(t')> = 2D delta (t - t') and <<(<eta>)over ca p>(t)> = 0. Both the associated Fokker Planck equation and the long time pr obability distribution function can be obtained analytically. A steady stat e solution may be expressed as P (phi) = 1/zexp{-Psi (phi) - lnG(phi)} wher e Psi(phi) = -1/D integral (phi) F/(G)(2)d phi and Z is a normalization fac tor. This is explored for the simple case where F (phi) = J phi + b phi (2) - c phi (3) and fluctuations characterised by the amplitude G (phi) = phi + epsilon when it readily yields for phi much greater than epsilon, a distr ibution function with power law tails, viz: P(phi) = 1/Z \ phi \ (1-j/D) ex p{(2b phi - c phi (2)) / D}. The parameter c ensures convergence of the dis tribution function for large values of,-. It might be loosely associated wi th the activity of so-called value traders. The parameter J may be associat ed with the activity of noise traders. Output for the associated time serie s show all tile characteristics of familiar financial time series providing J < 0 and D approximate to \J \.