Deterministic and stochastic influences on Japan and US stock and foreign exchange markets. A Fokker-Planck approach

The evolution of the probability distributions of Japan and US major market indices, NIKKEI 225 and NASDAQ composite index, and JPY /DEM and DEM/USD currency exchange rates is described by means of the Fokker-Planck equation (FPE). In order to distinguish and quantify the deterministic and random influences on these financial time series we perform a statistical analysis of their increments ∆x(∆(t)) distribution functions for different time lags ∆(t). From the probability distribution functions at various ∆(t), the Fokker-Planck equation for p(∆x(t),∆(t)) is explicitly derived. It is written in terms of a drift and a diffusion coefficient. The Kramers-Moyal coefficients, are estimated and found to have a simple analytical form, thus leading to a simple physical interpretation for both drift D and diffusion D coefficients. The Markov nature of the indices and exchange rates is shown and an apparent difference in the NASDAQ D is pointed out.