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 $JPY\mathrm{/}DEM$ and $DEM\mathrm{/}USD$ 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 $\mathrm{\Delta }x(\mathrm{\Delta }(t))$ distribution functions for different time lags $\mathrm{\Delta }(t)$. From the probability distribution functions at various $\mathrm{\Delta }(t)$, the Fokker-Planck equation for $p(\mathrm{\Delta }x(t),\mathrm{\Delta }(t))$ 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 $D^{(1)}$ and diffusion $D^{(2)}$ coefficients. The Markov nature of the indices and exchange rates is shown and an apparent difference in the NASDAQ $D^{(2)}$ is pointed out.