【正文】 procedure reproduce the conditional probability densities, as the central quantity for a prehensive multiscale characterisation. 16 Banking crisis and financial structure: A survivaltime analysis Ay?e Y. Evrensel Department of Economics and Finance, Southern Illinois University Edwardsville, Edwardsville, IL 620261102, United States Received 20 October 2020。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。y2, τ2。y N, τN)= p(y1, τ1│y2)……p(y N1, τN1│yN, τN) yN, τN)＝ p(y1, τ1│y2) (3) Consequently, p(y1, τ1。y3, τ3。 . . . ?！?Fokker–Planck equation 11 One of the outstanding features of the plexity of financial markets is that very often financial quantities display nonGaussian statistics often denoted as heavy tailed or intermittent statistics . To characterize the fluctuations of a financial time series x(t), most monly 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 mon 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 analysis and reconstruction of the processes for a given data window (time period). The analysis presented is mainly based on Bayer data for the time span of 1993–2020. The financial data sets were provided by the Karlsruher Kapitalmarkt Datenbank (KKMDB) . 2. Smallscale analysis One remarkable feature of financial data is the fact that the probability density functions (pdfs) are not Gaussian, but exhibit heavy tailed shapes. Another remarkable feature is the change of the shape with the size of the scale variable τ. To analyse the changing statistics of the pdfs with the scale t a nonparametric approach is chosen. The distance between the pdf p(y(τ)) on a ti