Medium and Small Scale Analysis of Financial Data

A stochastic analysis of financial data is presented. In particular we investigate how the statistics of log returns change with different time delays $\tau$. The scale dependent behaviour of financial data can be divided into two regions. The first time-range, the small-timescale region (in the range of seconds) seems to be characterized by universal features. The second time-range, the medium-timescale range from several minutes upwards and can be characterized by a cascade process, which is given by a stochastic Markov process in the scale $\tau$. A corresponding Fokker-Planck equation can be extracted from given data and provides a non equilibrium thermodynamical description of the complexity of financial data.