medium and small-scale analysis of financial data（中小规模的金融数据分析）-（外文翻译）.doc

Medium and small-scale analysis of financial data Abstract A stochastic analysis of financial data is presented. In particular we investigate how the statistics of log returns change with different time delays t. 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 characterised by universal features. The second time range, the medium-timescale range from several minutes upwards can be characterised by a cascade process, which is given by a stochastic Markov process in the scale τ. A corresponding Fokker–Planck equation can be extracted from given data and provides a non-equilibrium thermodynamical description of the complexity of financial data. Keywords: Econophysics; Financial markets; Stochastic processes; Fokker–Planck equation 1.Introduction One of the outstanding features of the complexity of financial markets is that very often financial quantities display non-Gaussian statistics often denoted as heavy tailed or intermittent statistics . To characterize the fluctuations of a financial time series x(t), most commonly quantities like returns, log returns or price increments are used. Here, we consider the statistics of the log return y(τ) over a certain timescale t, which is defined as y(τ)=log x(t+τ) - log x(t), (1) where x(t) denotes the price of the asset at time t. A common problem in the analysis of financial data is the question of stationarity for the discussed stochastic quantities. In particular we find in our analysis that the methods seem to be robust against nonstationarity effects. This may be due to the data selection. Note that the use of (conditional) returns of scale τ corresponds to a specific filtering of the data. Nevertheless the particular results change slightly for different data windows, indicating a possible influence of nonstationarity effects. In this paper we focus on the analy