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

Abstract

The evolution of the probability distributions of Japan and US major market indices, NIKKEI 225 and NASDAQ composite index, and JPY/DF,M 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 Δx(Δ(t)) distribution functions for different time lags Δ(t). From the probability distribution functions at variousΔ(t),the Fokker-Planck equation for p(Δx(t), Δ(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 driftD (1) and diffusionD (2)coefficients. The Markov nature of the indices and exchange rates is shown and an apparent difference in the NASDAQD (2) is pointed out.