工商管理专业外文文献翻译--中小规模的金融数据分析(编辑修改稿)

工商管理专业外文文献翻译--中小规模的金融数据分析(编辑修改稿)内容摘要：

easure seems to be universally present. If a normalised Gaussian distribution is taken as a reference distribution, the fast deviation from the Gaussian shape in the smalltimescale regime bees evident. For larger timescales dK remains approximately constant, indicating a very slow change of the shape of the pdfs. 3. Medium scale analysis Next the behaviour for larger timescales (τ1 min) is discussed. We proceed with the idea of a cascade. it is possible to grasp the plexity of financial data by cascade processes running in the variable τ. In particular it has been shown that it is possible to estimate directly from given data a stochastic cascade process in the form of a Fokker– Planck equation. The underlying idea of this approach is to access statistics of all orders of the financial data by the general joint nscale probability densities p(y1, τ1。 y2, τ2。 …。 y N, τN). Here we use the shorthand notation y1=y(τ1) and take without loss of generality τiτi+1. The smaller log returns y(τi) are nested inside the larger log returns y(τi+1) with mon end point t. The joint pdfs can be expressed as well by the multiple conditional probability densities p(yi, ti│yi+1, ti+1。 . . .。 yN, tN). This very general nscale characterisation of a data set, which contains the general npoint statistics, can be simplified essentially if there is a stochastic process in t, which is a Markov process. This is the case if the conditional probability densities fulfil the following relations: 13 p(y1, τ1│y2, τ2。 y3, τ3。 . . .。 yN, τN)＝ p(y1, τ1│y2) (3) Consequently, p(y1, τ1。 …。 y N, τN)= p(y1, τ1│y2)……p(y N1, τN1│yN, τN)p(yN, τN) (4) holds. Eq. (4) indicates the importance of the conditional pdf for Markov processes. Knowledge of p(y, τ│y0, τ0) (for arbitrary scales τ and τ0 with ττ0) is sufficient to generate the entire statistics of the increment, encoded in the Npoint probability density p(y1, τ1。 y2, τ2。 …。 y N, τN). For Markov processes the conditional probability density satisfies a master equation, which can be put into the form of a Kramers– Moyal expansion for which the Kramers– Moyal coefficients D(K)(y, τ) are defined as the limit △τ→0 of the conditional moments M(K)(y, τ, △τ): ( ) ( )0( , ) li m ( , , )KKtD y t M y t t (5) () ( , , ) ( ) ( , | , )!KktM y t t y y p y t t y t d ykt       (6) For a general stochastic process, all Kramers– Moyal coefficients are different from zero. According to Pawula’ s theorem, however, the Kramers– Moyal expansion stops after the second term, provided that the fourth order coefficient D(4)(y, τ) vanishes. In that case, the Kramers– Moyal expansion reduces to a Fokker– Planck equation (also known as the backwards or second Kolmogorov equation): 2( 1 ) ( 2 )0 0 0 02( , | , ) ( , ) ( , ) ( , | , )tt p y t y t D y t D y t p y t y tyy        (7) D(1) is denoted as drift term, D(2) as diffusion term. The probability density p(y, τ) has to satisfy the same equation, as can be shown by a simple integration of Eq. (7). 4. Results for Bayer data The Kramers– Moyal coefficients were calculated according to Eqs. (5) and (6). The timescale was divided into halfopen intervals [1/2(τi1+τi),1/2(τi+τi+1)] assuming that the Kramers– Moyal coefficients are constant with respect to the timescaleτin each of these subintervals of the timescale. The smallest timescale 14 considered was 240 s and all larger scales were chosen such that τi＝ *τi+1. The Kramers– Moyal coefficients themselves were parameterised in the following form: D(1)＝α 0+α 1y (8) D(2)＝β 0+β 1y+β 2y2 (9) This result shows that the rich and plex structure of financial data, expressed by multiscale statistics, can be pinned down to coefficients with a relatively simple functional form. 5. Discussion The results indicate that for financial data there are two scale regimes. In the smallscale regime the shape of the pdfs changes very fast and a measure like the Kullback– Leibler entropy increases linearly. At timescales of a few seconds not all available information may be included in the price and processes necessary for price formation take place. Nevertheless this regime seems to exhibit a welldefined structure, expressed by the very simple functional form of the Kullback– Leibler entropy with respect to the timescale τ. The upper boundary in timescale for this regime seems to be very similar for different stocks. Based on a stochastic analysis we have shown that a second time range, the medium scale range exists, where multiscale joint probability densities can be expressed by a stochastic cascade process. Here, the information on the prehensive multiscale statistics can be expressed by simple conditioned probability densities. This simplification may be seen in analogy to the thermodynamical description of a gas by means of statistical mechanics. The prehensive statistical quantity for the gas is the joint nparticle probability density, which describes the location and the momentum of all the individual particles. One essential simplification for the kiic gas theory is the single particle approximation. The Boltzmann equation is an equation for the time evolution of the probability density p(x。 p。 t) in oneparticle phase space, where x and p are position and momentum, respectively. In analogy to this we have obtained for 15 the financial data a Fokker– Planck equation for the scale t evolution of conditional probabilities, p(yi, τi│yi+1, τi+1). In our cascade picture the conditional probabilities cannot be reduced further to single probability densities, p(yi, τi), without loss of information, as it is done for the kiic ga。