probability product kernels the advantages of discriminative learning algorithms and kernel machines are combined with generative modeling using a novel kernel between distributions in the probability product kernel data points in the input space are mapped to distributions over the sample space and a general inner product is then evaluated as the integral of the product of pairs of distributions the kernel is straightforward to evaluate for all exponential family models such as multinomials and gaussians and yields interesting nonlinear kernels furthermore the kernel is computable in closed form for latent distributions such as mixture models hidden markov models and linear dynamical systems for intractable models such as switching linear dynamical systems structured mean field approximations can be brought to bear on the kernel evaluation for general distributions even if an analytic expression for the kernel is not feasible we show a straightforward sampling method to evaluate it thus the kernel permits discriminative learning methods including support vector machines to exploit the properties metrics and invariances of the generative models we infer from each datum experiments are shown using multinomial models for text hidden markov models for biological data sets and linear dynamical systems for time series data

partially implicit schemes for the numerical solutions of some non linear differential equations as numerical methods are one of the main tools used to discover the non linear behaviour characteristics of continuous dynamical systems a study on the use of numerical schemes which delineate the true physics from numerical artefact is desirable as a step in that direction partially implicit rational schemes based on first order methods and a second order method are derived and analysed from the point of view of dynamical systems for a scalar autonomous ordinary differential equation the second order method is developed by taking a linear combination of first order methods though implicit in nature the methods are applied explicitly the approach adopted is extended to obtain numerical schemes for the solution of a class of reaction diffusion partial differential equations the methods are tested on differential equations with cubic and quadratic reaction terms it is found that for certain sets of initial conditions the first order partially implicit schemes converge monotonically to the correct fixed points for all time steps unlike the euler method and the second order method in other words for the same steady state solution the associated domain of attraction of the discrete problem and its continuous counterpart are exactly the same however for other initial conditions it is found that the domains of attraction of the second order method mimic more closely the domains of attraction of the continuous differential equations the comparative study of the various explicit runge kutta methods of orders 2 6 shows that it is the discretized form with its specific algebraic form recursive relation which are the source of spurious steady state solutions and numerical chaos from these methods it is found however that the use in general of fifth order runge kutta methods is seen to be preferred to other runge kutta methods as they have an extended stability range in particular the fifth order lawson method which has a far superior range of stability than the other runge kutta methods

generalized correlation function definition properties and application to blind equalization with an abundance of tools based on kernel methods and information theoretic learning a void still exists in incorporating both the time structure and the statistical distribution of the time series in the same functional measure in this paper a new generalized correlation measure is developed that includes the information of both the distribution and that of the time structure of a stochastic process it is shown how this measure can be interpreted from a kernel method as well as from an information theoretic learning points of view demonstrating some relevant properties to underscore the effectiveness of the new measure a simple blind equalization problem is considered using a coded signal

an error entropy minimization algorithm for supervised training of nonlinear adaptive systems the paper investigates error entropy minimization in adaptive systems training we prove the equivalence between minimization of error s renyi 1970 entropy of order spl alpha and minimization of a csiszar 1981 distance measure between the densities of desired and system outputs a nonparametric estimator for renyi s entropy is presented and it is shown that the global minimum of this estimator is the same as the actual entropy the performance of the error entropy minimization criterion is compared with mean square error minimization in the short term prediction of a chaotic time series and in nonlinear system identification

adaptive learning in complex reproducing kernel hilbert spaces employing wirtinger s subgradients this paper presents a wide framework for non linear online supervised learning tasks in the context of complex valued signal processing the complex input data are mapped into a complex reproducing kernel hilbert space rkhs where the learning phase is taking place both pure complex kernels and real kernels via the complexification trick can be employed moreover any convex continuous and not necessarily differentiable function can be used to measure the loss between the output of the specific system and the desired response the only requirement is the subgradient of the adopted loss function to be available in an analytic form in order to derive analytically the subgradients the principles of the recently developed wirtinger s calculus in complex rkhs are exploited furthermore both linear and widely linear in rkhs estimation filters are considered to cope with the problem of increasing memory requirements which is present in almost all online schemes in rkhs the sparsification scheme based on projection onto closed balls has been adopted we demonstrate the effectiveness of the proposed framework in a non linear channel identification task a non linear channel equalization problem and a quadrature phase shift keying equalization scheme using both circular and non circular synthetic signal sources

stochastic blind equalization based on pdf fitting using parzen estimator this work presents a new blind equalization approach that aims to force the probability density function pdf at the equalizer output to match the known constellation pdf quadratic distance between pdf s is used as the cost function to be minimized the proposed method relies on the parzen window method to estimate the data pdf and is implemented by a stochastic gradient descent algorithm the kernel size of the parzen estimator allows a dual mode switch or a soft switch between blind and decision directed equalization the proposed method converges faster than the constant modulus algorithm cma working at the symbol rate with a similar computational burden and reduces the residual error of the cma in multilevel modulations at the same time a comparison with the most common blind techniques is presented

hitting time distributions in financial markets we analyze the hitting time distributions of stock price returns in different time windows characterized by different levels of noise present in the market the study has been performed on two sets of data from us markets the first one is composed by daily price of 1071 stocks trade for the 12 year period 1987 1998 the second one is composed by high frequency data for 100 stocks for the 4 year period 1995 1998 we compare the probability distribution obtained by our empirical analysis with those obtained from different models for stock market evolution specifically by focusing on the statistical properties of the hitting times to reach a barrier or a given threshold we compare the probability density function pdf of three models namely the geometric brownian motion the garch model and the heston model with that obtained from real market data we will present also some results of a generalized heston model r 2007 elsevier b v all rights reserved

comparison between the probability distribution of returns in the heston model and empirical data for stock indexes abstract we compare the probability distribution of returns for the three major stock market indexes nasdaq s p500 and dow jones with an analytical formula recently derived by dr gulescu and yakovenko for the heston model with stochastic variance for the period of 1982 1999 we find a very good agreement between the theory and the data for a wide range of time lags from 1 to 250 days on the other hand deviations start to appear when the data for 2000 2002 are included we interpret this as a statistical evidence of the major change in the market from a positive growth rate in 1980s and 1990s to a negative rate in 2000s

mean escape time in a system with stochastic volatility we study the mean escape time in a market model with stochastic volatility the process followed by the volatility is the cox ingersoll and ross process which is widely used to model stock price fluctuations the market model can be considered as a generalization of the heston model where the geometric brownian motion is replaced by a random walk in the presence of a cubic nonlinearity we investigate the statistical properties of the escape time of the returns from a given interval as a function of the three parameters of the model we find that the noise can have a stabilizing effect on the system as long as the global noise is not too high with respect to the effective potential barrier experienced by a fictitious brownian particle we compare the probability density function of the return escape times of the model with those obtained from real market data we find that they fit very well

correntropy properties and applications in non gaussian signal processing the optimality of second order statistics depends heavily on the assumption of gaussianity in this paper we elucidate further the probabilistic and geometric meaning of the recently defined correntropy function as a localized similarity measure a close relationship between correntropy and m estimation is established connections and differences between correntropy and kernel methods are presented as such correntropy has vastly different properties compared with second order statistics that can be very useful in non gaussian signal processing especially in the impulsive noise environment examples are presented to illustrate the technique

non extensive behavior of a stock market index at microscopic time scales abstract this paper presents an empirical investigation of the intraday brazilian stock market price fluctuations considering q gaussian distributions that emerge from a non extensive statistical mechanics our results show that when price returns are measured over intervals less than one hour the empirical distributions are well fitted by q gaussians with exponential damped tails scaling behavior is also observed for these microscopic time intervals we find that the time evolution of the return distributions is according to a super diffusive q gaussian stationary process within a nonlinear fokker planck equation this regime breaks down due to the exponential fall off of the tails which in turn governs the transient dynamics to the long term macroscopic gaussian regime this exponentially damped non extensive modeling provides a new framework to investigate the dynamics of other stock markets intraday price fluctuations

phase distribution and phase correlation of financial time series the scaling phase distribution and phase correlation of financial time series are investigated based on the dow jones industry average and nasdaq 10 min intraday data for a period from 1 aug 1997 to 31 dec 2003 the returns of the two indices are shown to have nice scaling behaviors and belong to stable distributions according to the criterion of l evy s ensuremath alpha stable distribution condition an approach catching characteristic features of financial time series based on the concept of instantaneous phase is further proposed to study the phase distribution and correlation analysis of the phase distribution concludes that return time series fall into a class which is different from other nonstationary time series the correlation between returns of the two indices probed by the distribution of phase difference indicates that there was a remarkable change of trading activities after the event of the 9 11 attack and this change persisted in later trading activities

statistical distribution and time correlation of stock returns runs abstract in this paper we focus on the statistical features and time correlation of runs which is defined as a sequence of consecutive gain loss rise fall stock returns by studying daily data of the dow jones industrial average djia we get the following points firstly the distribution of length and magnitude of stock returns runs both follow an exponential law secondly runs length do lack significant time correlation while runs magnitude exhibit a slow decay of time correlation with long persistence up to several months which implies existence of volatility clustering we expect the above properties may add new members to the family of stylized facts about stock returns

stochastic volatility models and kelvin waves we use stochastic volatility models to describe the evolution of an asset price its instantaneous volatility and its realized volatility in particular we concentrate on the stein and stein model ssm 1991 for the stochastic asset volatility and the heston model hm 1993 for the stochastic asset variance by construction the volatility is not sign definite in ssm and is non negative in hm it is well known that both models produce closed form expressions for the prices of vanilla option via the lewis lipton formula however the numerical pricing of exotic options by means of the finite difference and monte carlo methods is much more complex for hm than for ssm until now this complexity was considered to be an acceptable price to pay for ensuring that the asset volatility is non negative we argue that having negative stochastic volatility is a psychological rather than financial or mathematical problem and advocate using ssm rather than hm in most applications we extend ssm by adding volatility jumps and obtain a closed form expression for the density of the asset price and its realized volatility we also show that the current method of choice for solving pricing problems with stochastic volatility via the affine ansatz for the fourier transformed density function can be traced back to the kelvin method designed in the 19th century for studying wave motion problems arising in fluid dynamics

explicit finite difference methods for non linear dynamic systems froude s pendulum abstract numerical schemes based on first order methods and a second order method are developed and analysed for solving the non linear froude s pendulum system the second order method is developed by taking a linear combination of three first order methods though implicit in nature with the resulting improvements in stability the methods are applied explicitly these methods are extended to a more general second order ordinary differential equation stability of the numerical methods is approached from the point of view of dynamical systems it is found that the methods are stable under the same conditions as those required by the linearized schemes in the neighbourhood of constant stable fixed points of the underlying initial value problem for the second order method this result is found to hold for any chosen value of the step length a major advantage of using the second order method is the absence of solutions to the discrete finite difference equation that do not correspond to the continuous dynamical system this contrasts starkly with the runge kutta methods which are found to produce solutions that converge to a false asymptote or wrong solutions in a deceptively smooth manner if the step length is not rightly chosen a selection of examples are carried out on the non linear froude s pendulum system to illustrate the differing predictions obtained by the methods

boltzmann distribution and market temperature abstract we show that the minute fluctuations of s p 500 and nasdaq 100 indices show boltzmann statistics over a wide range of positive as well as negative returns thus allowing us to define a market temperature for either sign with increasing time the sharp boltzmann peak broadens into a gaussian whose volatility measured in 1 min is related to the temperature t by t 2 plots over the years 1990 2006 show that the arrival of the 2000 crash was preceded by an increase in market temperature suggesting that this increase can be used as a warning signal for crashes

faithful representation of separable distributions a geometric approach to data representation incorporating information theoretic ideas is presented the task of finding a faithful representation where the input distribution is evenly partitioned into regions of equal mass is addressed for input consisting of mixtures of statistically independent sources we treat independent component analysis ica as a computational geometry problem first we consider the separation of sources with sharply peaked distribution functions where the ica problem becomes that of finding high density directions in the input distribution second we consider the more general problem for arbitrary input distributions where ica is transformed into the task of finding an aligned equipartition by modifying the kohonen self organized feature maps we arrive at neural networks with local interactions that optimize coding while simultaneously performing source separation the local nature of our approach results in networks with nonlinear ica capabilities

dynamic behaviors and measurements of financial market crash rate abstract this paper proposes the conditional crash rate ccr to study the degree of financial market crisis and evaluate the risk of financial market crash based on maximum drawdown and escape rate it is found that the ccr of stock price in financial market is equivalent to the escape rate of the drawdown time series corresponding to price series based on the mean escape rate the analytical solution of ccr of financial market of stock price described by black scholes model is obtained for the ccr empirical researches show that the theoretical results of the heston model are in good agreement with the results of the real financial data of the sp ii we can find critical phenomena resonance and inverse resonance behaviors in the function of ccr vs mean reversion of volatility ii the parameter variation induces the appearance of noise enhancement stability in the function of ccr vs amplitude of volatility fluctuation

empirical distributions of chinese stock returns at different microscopic timescales abstract we study the distributions of event time returns and clock time returns at different microscopic timescales using ultra high frequency data extracted from the limit order books of 23 stocks traded in the chinese stock market in 2003 we find that the returns at the one trade timescale obey the inverse cubic law for larger timescales 2 32 trades and 1 5 min the returns follow the student distribution with power law tails with the decrease in timescale the tail becomes fatter which is consistent with the variational theory in turbulence

a unified framework for quadratic measures of independence this paper proposes a unified framework for several available measures of independence by generalizing the concept of information theoretic learning itl the key component of itl is the use of inner product between two density functions as a measure of similarity between two random variables we show that by generalizing the inner product using a symmetric strictly positive definite kernel and by choosing appropriate kernels it is possible to reproduce a number of popular measures of independence this unified framework also allows the design of new strictly positive definite kernels and corresponding measures of independence following this framework we explore a new measure of independence and apply it in the context of linear independent component analysis ica an attractive property of the proposed method is that it does not involve any free parameter and we demonstrate that it performs equally well compared to the existing methods for ica

continuum versus discrete networks graph laplacians and reproducing kernel hilbert spaces abstract motivated by applications to machine learning we construct a reversible and irreducible markov chain whose state space is a certain collection of measurable sets of a chosen l c h space x we study the resulting network connected undirected graph including transience royden and riesz decompositions and kernel factorization we describe a construction for hilbert spaces of signed measures which comes equipped with a new notion of reproducing kernels and there is a unique solution to a regularized optimization problem involving the approximation of l 2 functions by functions of finite energy the latter has applications to machine learning for markov random fields for example

the time delay restraining the herd behavior with bayesian approach abstract we explore the herd behavior of stock prices influenced by the time delay in a finance system with the delayed heston model on the basis of a bayesian approach for delayed heston model that we proposed and parameter bayesian estimation we simulate the absolute deviation between mean residence time of positive and negative returns to characterize the herd behavior in the plots of absolute deviation against the mean reversion long run variance of volatility or cross correlation between two wiener processes of stock price and volatility the results demonstrate an increasing phenomenon of herd behavior with increasing delay time in the plot of absolute deviation against the amplitude of volatility fluctuations the result indicates an optimal delay time matching minimum herd behavior i e the time delay restrains the herd behavior also the optimal mean reversion long run variance of volatility and cross correlation concerning minimum herd behavior can be observed

properties of a simple bilinear stochastic model estimation and predictability we analyze the properties of arguably the simplest bilinear stochastic multiplicative process proposed as a model of financial returns and of other complex systems combining both nonlinearity and multiplicative noise by construction it has no linear predictability zero two point correlation but a certain nonlinear predictability non zero three point correlation it can thus be considered as a paradigm for testing the existence of a possible nonlinear predictability in a given time series we present a rather exhaustive study of the process including its ability to produce fat tailed distributions from gaussian innovations the unstable characteristics of the inversion of the key nonlinear parameters and of the two initial conditions necessary for the implementation of a prediction scheme and an analysis of the associated super exponential sensitivity of the inversion of the innovations in the presence of a large impulse our study emphasizes the conditions under which a degree of predictability can be achieved and describe a number of different attempts which overall illuminates the properties of the process in conclusion notwithstanding its remarkable simplicity the bilinear stochastic process exhibits remarkably rich and complex behavior which makes it a serious candidate for the modeling of financial time series among others

statistical properties dynamic conditional correlation and scaling analysis evidence from dow jones and nasdaq high frequency data abstract this paper investigates statistical properties of high frequency intraday stock returns across various frequencies both time series and panel data are utilized to explore the properties of probability distribution dynamic conditional correlations and scaling analysis in dow jones industrial average djia and nasdaq intraday returns across 10 min 30 min 60 min 120 min and 390 min frequencies the evidence shows that both returns and volatility standard deviation increase with the increasing scaling from 10 min to 390 min intervals by fitting an ar 1 garch 1 1 model to intraday data we find that ar 1 coefficients are negative for djia returns and positive for nasdaq exhibiting a positive and negative feedback strategy in djia and nasdaq respectively the evidence also shows that these coefficients are statistically significant for either including or excluding opening returns for the 10 min and 30 min frequencies by examining the dynamic conditional correlation between the djia and the nasdaq across different frequencies a positive correlation ranging from 0 6 to 0 8 was found in addition the variance of the dynamic correlation coefficients is decreasing and appears to be stable for the 2001 2003 period finally both returns on djia and nasdaq satisfy the stable levy distributions implying that both markets can converge to equilibrium by self governing mechanism after shocks results of this work provide relevant implications for investors and policy makers

kernel based nonlinear blind source separation we propose ktdsep a kernel based algorithm for nonlinear blind source separation bss it combines complementary research fields kernel feature spaces and bss using temporal information this yields an efficient algorithm for nonlinear bss with invertible nonlinearity key assumptions are that the kernel feature space is chosen rich enough to approximate the nonlinearity and that signals of interest contain temporal information both assumptions are fulfilled for a wide set of real world applications the algorithm works as follows first the data are implicitly mapped to a high possibly infinite dimensional kernel feature space in practice however the data form a smaller submanifold in feature space even smaller than the number of training data points a fact that has already been used by for example reduced set techniques for support vector machines we propose to adapt to this effective dimension as a preprocessing step and to construct an orthonormal basis of this submanifold the latter dimension reduction step is essential for making the subsequent application of bss methods computationally and numerically tractable in the reduced space we use a bss algorithm that is based on second order temporal decorrelation finally we propose a selection procedure to obtain the original sources from the extracted nonlinear components automatically experiments demonstrate the excellent performance and efficiency of our ktdsep algorithm for several problems of nonlinear bss and for more than two sources

on the valuation of variance swaps with stochastic volatility abstract this paper is an extension to a recent paper by zhu and lian 2011 1 in which a closed form exact solution was presented for the price of variance swaps with a particular definition of the realized variance here we further demonstrate that our approach is quite versatile and can be used for other definitions of the realized variance as well in particular we present a closed form formula for the price of a variance swap with the realized variance in the payoff function being defined as a logarithmic return of the underlying asset at some pre specified discretely sampling points the simple formula presented here is a result of successfully finding an exact solution of the partial differential equation pde system based on the heston 1993 s 2 two factor stochastic volatility model a distinguishable feature of this new solution is that the computational time involved in pricing variance swaps with discretely sampling time has been substantially improved

the augmented complex kernel lms recently a unified framework for adaptive kernel based signal processing of complex data was presented by the authors which besides offering techniques to map the input data to complex reproducing kernel hilbert spaces developed a suitable wirtinger like calculus for general hilbert spaces in this short paper the extended wirtinger s calculus is adopted to derive complex kernel based widely linear estimation filters suitable for applications involving noncircular data furthermore we illuminate several important characteristics of the widely linear filters we show that although in many cases the gains from adopting widely linear estimation filters as alternatives to ordinary linear ones are rudimentary for the case of kernel based widely linear filters significant performance improvements can be obtained

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

analytical representation of stock and stock indexes returns non gaussian random walks with various jump laws abstract empirical data for stock and stock indexes returns that is available for international markets as well as for the russian stock market is introduced and discussed random walk process with a specific law of an elementary independent increment jump in some random walk space is proposed for a proper basic description an analytical representation of a random walk distribution law is derived from a general conception of stochastic elementary jumps this representation covers classic s chandrasekhar s results for gauss random walks with limited jumps and jumps having distributions with all moments as well as p levy flights with jump distributions without moments starting from the second one finally it describes truncated levy random walks for jumps having distributions with at least the second moment the natural generalization of truncated levy walks is done the random walk process is compared with known empirical data for the above discussed return fluctuations of international assets and stock indexes there is a very good agreement of the elaborated theory with empirical data

exponential distribution of financial returns at mesoscopic time lags a new stylized fact we study the probability distribution of stock returns at mesoscopic time lags return horizons ranging from about an hour to about a month while at shorter microscopic time lags the distribution has power law tails for mesoscopic times the bulk of the distribution more than 99 of the probability follows an exponential law the slope of the exponential function is determined by the variance of returns which increases proportionally to the time lag at longer times the exponential law continuously evolves into gaussian distribution the exponential to gaussian crossover is well described by the analytical solution of the heston model with stochastic volatility r 2004 elsevier b v all rights reserved

extraction of signals with specific temporal structure using kernel methods this work derives and evaluates a method for blind source extraction bse in a reproducing kernel hilbert space rkhs framework the a priori information about the autocorrelation function of the signal under study is translated in a linear transformation of the gram matrix of the transformed data in hilbert space our method proved to be more robust than methods presented in the literature of bse with respect to ambiguities in the available a priori information of the signal to be extracted the approach here introduced can also be seen as a generalization of kernel principal component analysis kpca to analyze autocorrelation matrices at specific time lags

microscopic origin of non gaussian distributions of financial returns abstract in this paper we study the possible microscopic origin of heavy tailed probability density distributions for the price variation of financial instruments we extend the standard log normal process to include another random component in the so called stochastic volatility models we study these models under an assumption akin to the born oppenheimer approximation in which the volatility has already relaxed to its equilibrium distribution and acts as a background to the evolution of the price process in this approximation we show that all models of stochastic volatility should exhibit a scaling relation in the time lag of zero drift modified log returns we verify that the dow jones industrial average index indeed follows this scaling we then focus on two popular stochastic volatility models the heston and hull white models in particular we show that in the hull white model the resulting probability distribution of log returns in this approximation corresponds to the tsallis t student distribution the tsallis parameters are given in terms of the microscopic stochastic volatility model finally we show that the log returns for 30 years dow jones index data is well fitted by a tsallis distribution obtaining the relevant parameters

model for non gaussian intraday stock returns stock prices are known to exhibit non gaussian dynamics and there is much interest in understanding the origin of this behavior here we present a model that explains the shape and scaling of the distribution of intraday stock price fluctuations called intraday returns and verify the model using a large database for several stocks traded on the london stock exchange we provide evidence that the return distribution for these stocks is non gaussian and similar in shape and that the distribution appears stable over intraday time scales we explain these results by assuming the volatility of returns is constant intraday but varies over longer periods such that its inverse square follows a gamma distribution this produces returns that are student distributed for intraday time scales the predicted results show excellent agreement with the data for all stocks in our study and over all regions of the return distribution

investigation of non gaussian effects in the brazilian option market abstract an empirical study of the brazilian option market is presented in light of three option pricing models namely the black scholes model the exponential model and a model based on a power law distribution the so called q gaussian distribution or tsallis distribution it is found that the q gaussian model performs better than the black scholes model in about one third of the option chains analyzed but among these cases the exponential model performs better than the q gaussian model in 75 of the time the superiority of the exponential model over the q gaussian model is particularly impressive for options close to the expiration date where its success rate rises above ninety percent

non poisson intermittent events in price formation in a ising spin model of market abstract the formation of price in a financial market is modelled as a chain of ising spin with three fundamental figures of trading we investigate the time behaviour of the model and we compare the results with the real euro usd change rate by using the test of local poisson hypothesis we show that this minimal model leads to clustering and declustering in the volatility signal typical of the real market data

pricing forward start variance swaps with stochastic volatility abstract in this paper a general approach is presented to price forward start variance swaps with discrete sampling times based on the heston 1993 s two factor stochastic volatility model using this approach we work out two analytical closed form formulae for the price of forward start variance swap with the realized variance being defined by the actual return realized variance and the log return realized variance respectively the main features of this new approach and the developed formulae as a result include 1 treating the pricing problem of variance swaps with different definitions of discretely sampled realized variance in a highly unified way 2 easily obtaining analytical closed form solutions for forward start variance swaps with two popularly used definitions of discretely sampled realized variance 3 enabling the investigation of some important properties of the forward start variance swaps utilizing the elegant and simple form of the obtained solutions with these advantages we believe that the approach can be applied to price variance swaps based on other stochastic volatility models as well

stochastic volatility of financial markets as the fluctuating rate of trading an empirical study we present an empirical study of the subordination hypothesis for a stochastic time series of a stock price the fluctuating rate of trading is identified with the stochastic variance of the stock price as in the continuous time random walk ctrw framework the probability distribution of the stock price changes log returns for a given number of trades n is found to be approximately gaussian the probability distribution of n for a given time interval dt is nonpoissonian and has an exponential tail for large n and a sharp cutoff for small n combining these two distributions produces a non trivial distribution of log returns for a given time interval dt which has exponential tails and a gaussian central part in agreement with empirical observations r 2007 elsevier b v all rights reserved

power law and multiscaling properties of the chinese stock market abstract we investigate the cumulative probability density function pdf and the multiscaling properties of the returns in the chinese stock market by using returns data adjusted for thin trading we find that the distribution has power law tails at shorter microscopic timescales or lags however the distribution follows an exponential law for longer timescales furthermore we investigate the long range correlation and multifractality of the returns in the chinese stock market by the dfa and mfdfa methods we find that all the scaling exponents are between 0 5 and 1 by dfa method which exhibits the long range power law correlations in the chinese stock market moreover we find by mfdfa method that the generalized hurst exponents h q are not constants which shows the multifractality in the chinese stock market we also find that the correlation of shenzhen stock market is stronger than that of shanghai stock market

chaos in numerical analysis of ordinary differential equations abstract the discretisation of the ordinary nonlinear differential equation d y d t y 1 y by the entral difference scheme is studied for fixed mesh size in the usual numerical computation this method produces some ghost solution for the long range calculation regarding this discretisation as a dynamical system in r2 these pathological behaviors are shown to be a kind of chaos in the dynamical system for any mesh size moreover some combination of the central difference scheme and the euler s scheme is studied for the above equation it gives some motivation for henon s model the usual discretisation of a second order differential equation are studied also it gives some chaotic behaviors numerically which is similar to the behavior of the orbits of the system of differential equations proposed by henon heiles

an infinite dimensional geometric structure on the space of all the probability measures equivalent to a given one let m be the set of all probability densities equivalent to a given reference probability measure this set is thought of as the maximal regular i e with strictly positive densities dominated statistical model for each f m we define 1 a banach space l f with unit ball v f and 2 a mapping s f from a subset u f of m onto v f in such a way that the system s f u f f m is an affine atlas on m moreover each parametric exponential model dominated by is a finite dimensional affine submanifold and each parametric statistical model dominated by with a suitable regularity is a submanifold the global geometric framework given by the manifold structure adds some insight to the so called geometric theory of statistical models in particular the present paper gives some of the developments connected with the fisher information metrics rao and the hilbert bundle introduced by amari

waiting time distributions of the volatility in the italian mib30 index clustering or poisson functions abstract we investigate the time behaviour of the italian mib30 stock index collected every minute during two months in the period from may 17 2006 up to july 24 2006 we find short range correlations in the price returns and on the contrary a long persistent time lag and slow decay in the autocorrelation functions of volatility besides we find that the probability density functions pdfs of returns show fat tails which are well fit by the log normal model of castaing b castaing y gagne e j hopfinger physica d 46 1990 177 and a convergence toward a normal distribution for large time scales we also find that the pdfs of volatility for short time horizons fit better with a log normal distribution than with a gaussian most of these features characterize the indexes and stocks of the largest american european and asian markets we also investigate the distribution of stochastic separation between isolated strong events in the volatility signal this is interesting because this gives us a deeper understanding about the price formation process by using a test for the occurrence of local poisson hypothesis we show that the process we examined strongly departs from a poisson statistics the origin of this failure stemming from the presence of temporal clustering and of a certain amount of memory

the kernel least mean square algorithm the combination of the famed kernel trick and the least mean square lms algorithm provides an interesting sample by sample update for an adaptive filter in reproducing kernel hilbert spaces rkhs which is named in this paper the klms unlike the accepted view in kernel methods this paper shows that in the finite training data case the klms algorithm is well posed in rkhs without the addition of an extra regularization term to penalize solution norms as was suggested by kivinen kivinen smola and williamson ldquoonline learning with kernels rdquo ieee transactions on signal processing vol 52 no 8 pp 2165 2176 aug 2004 and smale smale and yao ldquoonline learning algorithms rdquo foundations in computational mathematics vol 6 no 2 pp 145 176 2006 this result is the main contribution of the paper and enhances the present understanding of the lms algorithm with a machine learning perspective the effect of the klms step size is also studied from the viewpoint of regularization two experiments are presented to support our conclusion that with finite data the klms algorithm can be readily used in high dimensional spaces and particularly in rkhs to derive nonlinear stable algorithms with comparable performance to batch regularized solutions

an ideal gas approach to classify countries using financial indices abstract traditionally countries development is classified based on several features that can be related to economic and social factors however this classification task is costly due to the difficulty of obtaining those features we propose a method to classify countries based on financial indices using an ideal gas model the probability density function pdf of the return series of the financial indices is used to characterize the fluctuation of a market based on the pdf the volatility and the b coefficient which describe the behavior of world markets are estimated the evaluation procedure uses 34 indices from developed and developing countries the results show that the proposed method is an accurate fast and low cost computational alternative to the classifications provided by traditional organizations

nonextensive statistical features of the polish stock market fluctuations the statistics of return distributions on various time scales constitutes one of the most informative characteristics of the financial dynamics here we present a systematic study of such characteristics for the polish stock market index wig20 over the period 04 01 1999 31 10 2005 for the time lags ranging from 1min1min up to 1 h this market is commonly classified as emerging still on the shortest time scales studied we find that the tails of the return distributions are consistent with the inverse cubic power law as identified previously for majority of the mature markets within the time scales studied a quick and considerable departure from this law towards a gaussian can however be traced interestingly all the forms of the distributions observed can be comprised by the single q gaussians which provide a satisfactory and at the same time compact representation of the distribution of return fluctuations over all magnitudes of their variation the corresponding nonextensivity parameter q was found to systematically decrease when increasing the time scales the temporal correlations quantified here in terms of multifractality provide further arguments in favor of nonextensivity

kernel independent component analysis we present a class of algorithms for independent component analysis ica which use contrast functions based on canonical correlations in a reproducing kernel hilbert space on the one hand we show that our contrast functions are related to mutual information and have desirable mathematical properties as measures of statistical dependence on the other hand building on recent developments in kernel methods we show that these criteria and their derivatives can be computed efficiently minimizing these criteria leads to flexible and robust algorithms for ica we illustrate with simulations involving a wide variety of source distributions showing that our algorithms outperform many of the presently known algorithms

a reproducing kernel hilbert space framework for information theoretic learning this paper provides a functional analysis perspective of information theoretic learning itl by defining bottom up a reproducing kernel hilbert space rkhs uniquely determined by the symmetric nonnegative definite kernel function known as the cross information potential cip the cip as an integral of the product of two probability density functions characterizes similarity between two stochastic functions we prove the existence of a one to one congruence mapping between the itl rkhs and the hilbert space spanned by square integrable probability density functions therefore all the statistical descriptors in the original information theoretic learning formulation can be rewritten as algebraic computations on deterministic functional vectors in the itl rkhs instead of limiting the functional view to the estimators as is commonly done in kernel methods a connection between the itl rkhs and kernel approaches interested in quantifying the statistics of the projected data is also established

pricing european options with a log student s t distribution a gosset formula abstract the distributions of returns for stocks are not well described by a normal probability density function pdf student s t distributions which have fat tails are known to fit the distributions of the returns we present pricing of european call or put options using a log student s t distribution which we call a gosset approach in honour of w s gosset the author behind the nom de plume student the approach that we present can be used to price european options using other distributions and yields the black scholes formula for returns described by a normal pdf

measures of entropy from data using infinitely divisible kernels information theory provides principled ways to analyze different inference and learning problems such as hypothesis testing clustering dimensionality reduction classification and so forth however the use of information theoretic quantities as test statistics that is as quantities obtained from empirical data poses a challenging estimation problem that often leads to strong simplifications such as gaussian models or the use of plug in density estimators that are restricted to certain representation of the data in this paper a framework to nonparametrically obtain measures of entropy directly from data using operators in reproducing kernel hilbert spaces defined by infinitely divisible kernels is presented the entropy functionals which bear resemblance with quantum entropies are defined on positive definite matrices and satisfy similar axioms to those of renyi s definition of entropy convergence of the proposed estimators follows from concentration results on the difference between the ordered spectrum of the gram matrices and the integral operators associated to the population quantities in this way capitalizing on both the axiomatic definition of entropy and on the representation power of positive definite kernels the proposed measure of entropy avoids the estimation of the probability distribution underlying the data moreover estimators of kernel based conditional entropy and mutual information are also defined numerical experiments on independence tests compare favorably with state of the art

the roles of mean residence time on herd behavior in a financial market abstract we investigate the herd behavior of stock prices in a finance system with the heston model based on parameter estimation of the heston model obtained by minimizing the mean square deviation between the theoretical and empirical return distributions we simulate mean residence time of positive return mrtpr plots of mrtpr against the amplitude or mean reversion of volatility demonstrate a phenomenon of herd behavior for a positive cross correlation between noise sources of the heston model also for a negative cross correlation a phenomenon of herd behavior is observed in plots of mrtpr against the long run variance by increasing amplitude or mean reversion of volatility

fast and robust fixed point algorithms for independent component analysis independent component analysis ica is a statistical method for transforming an observed multidimensional random vector into components that are statistically as independent from each other as possible we use a combination of two different approaches for linear ica comon s information theoretic approach and the projection pursuit approach using maximum entropy approximations of differential entropy we introduce a family of new contrast functions for ica these contrast functions enable both the estimation of the whole decomposition by minimizing mutual information and estimation of individual independent components as projection pursuit directions the statistical properties of the estimators based on such contrast functions are analyzed under the assumption of the linear mixture model and it is shown how to choose contrast functions that are robust and or of minimum variance finally we introduce simple fixed point algorithms for practical optimization of the contrast functions

chaos free numerical solutions of reaction diffusion equations two numerical methods which do not bring contrived chaos into the solution are proposed for the solution of the riccati logistic equation though implicit in nature with the resulting improvements in stability the methods are applied explicitly when extended to the numerical solution of fisher s equation in which the quadratic polynomial representing the derivative in the riccati equation appears as the reaction term the solution is found by solving a linear system of algebraic equations at each time step as opposed to solving a nonlinear system which frequently happens when solving nonlinear partial differential equations the approaches adopted are extended to an ordinary differential equation in which the derivative is expressed as a cubic polynomial in the dependent variable the solution of this initial value problem is not available in closed form for finite values of the independent variable t under the conditions stated numerical solutions are seen to converge to the correct steady state solution a nonlinear partial differential equation which governs the conduction of electrical impulses along a nerve axon and which has the aforementioned cubic polynomial as its reaction term is solved by applying the numerical methods developed for solving the ordinary differential equation the solution to this nonlinear reaction diffusion equation is determined by solving a linear algebraic system at each time step

multiple time scales and the empirical models for stochastic volatility abstract the most common stochastic volatility models such as the ornstein uhlenbeck ou the heston the exponential ou expou and hull white models define volatility as a markovian process in this work we check the applicability of the markovian approximation at separate times scales and will try to answer the question which of the stochastic volatility models indicated above is the most realistic to this end we consider the volatility at both short a few days and long a few months time scales as a markovian process and estimate for it the coefficients of the kramers moyal expansion using the data for dow jones index it has been found that the empirical data allow to take only the first two coefficients of expansion to be non zero that define form of the volatility stochastic differential equation of ito it proved to be that for the long time scale the empirical data support the expou model at the short time scale the empirical model coincides with expou model for the small volatility quantities only

finite difference schemes having the correct linear stability properties for all finite step sizes iii abstract we construct a class of finite difference schemes for two coupled first order ordinary differential equations such that the difference equations have the correct linear stability properties for all finite values of the step size the differential equations considered are assumed to have only a single real fixed point a major consequence of such schemes is the absence of elementary numerical instabilities illustrations of such schemes are given for the duffing and van der pol equations

central difference scheme and chaos abstract the existence of chaotic orbits is proved for area preserving diffeomorphisms of the plane defined by polynomials of degree two derived from the discretized logistic equation

turbulence in multistep methods for initial value problems solving a scalar initial value problem with a numerical k step scheme amounts to iterating a k dimensional discrete dynamical system in this paper we investigate regular and chaotic behaviour of some dynamical systems associated with adams bashforth formulas applied to a simple nonlinear test equation

a contribution to the systematics of stochastic volatility models abstract we systematically compare several classes of stochastic volatility models of stock market fluctuations we show that the long time return distribution is either gaussian or develops a power law tail while the short time return distribution has generically a stretched exponential form but can also assume an algebraic decay in the family of models which we call garch type the intermediate regime is found in the exponential ornstein uhlenbeck process we also calculate the decay of the autocorrelation function of volatility

a blind source separation technique using second order statistics separation of sources consists of recovering a set of signals of which only instantaneous linear mixtures are observed in many situations no a priori information on the mixing matrix is available the linear mixture should be blindly processed this typically occurs in narrowband array processing applications when the array manifold is unknown or distorted this paper introduces a new source separation technique exploiting the time coherence of the source signals in contrast with other previously reported techniques the proposed approach relies only on stationary second order statistics that are based on a joint diagonalization of a set of covariance matrices asymptotic performance analysis of this method is carried out some numerical simulations are provided to illustrate the effectiveness of the proposed method

scaling of the distribution of fluctuations of financial market indices up to more than one month we find that the distributions for dt 4d 1560 min are consistent with a power law asymptotic behavior characterized by an exponent a 3 well outside the stable levy regime 0 a 2 to test the robustness of the s p result we perform a parallel analysis on two other financial market indices database iv contains 3560 daily records of the nikkei index for the 14 year period 1984 1997 and database v contains 4649 daily records of the hang seng index for the 18 year period 1980 1997 we find estimates ofa consistent with those describing the distribution of s p 500 daily returns one possible reason for the scaling of these distributions is the long persistence of the autocorrelation function of the volatility for time scales longer than dt 3 4 d our results are consistent with a slow convergence to gaussian behavior s1063 651x 99 11211 x pacs number s 05 40 fb 05 45 tp 89 90 1n

treatment of kurtosis in financial markets abstract since mandelbrot 1963 2 highlighted the fact that data on the yield of financial assets exhibit leptokurtosis different distributions have been presented as alternatives to the normal distribution so far little consideration has been given to the capacity that these distributions have to recover the kurtosis of the sample data our work aims to present distributions which given the broad range of their kurtosis have the capacity to perform adjustment on many occasions where other distributions fail while also being capable of recovering the peakedness of the empirical data another key characteristic of these distributions is that they are defined within a bounded domain in the same way as the sample data an empirical application of these distributions is presented within the financial field by using daily returns

microscopic spin model for the dynamics of the return distribution of the korean stock market index abstract in this paper we studied the dynamics of the log return distribution of the korean composition stock price index kospi from 1992 to 2004 based on the microscopic spin model we found that while the index during the late 1990s showed a power law distribution the distribution in the early 2000s was exponential this change in distribution shape was caused by the duration and velocity among other parameters of the information that flowed into the market

roles of capital flow on the stability of a market system abstract the roles of capital flow in an ensemble composed of sub markets are investigated a modified heston model and recycled noises are employed to describe the dynamics of stock price and capital flow in the ensemble respectively the mean escape times of two sub markets with a cubic nonlinearity are calculated by using numerical simulation the results evidence that i there is a worst delay time or rate of capital inflow concerning the minimal stability of stock price and an optimal delay time or rate of capital outflow concerning the maximal stability of stock price when 0 denotes strength of correlation between two wiener processes of the stock price and the volatility ii when 0 the stability of stock price is maximally enhanced by an optimal delay time or rate of capital inflow and reduced by a worst rate of capital outflow but monotonously strengthened by delay time of capital outflow

application of heston model and its solution to german dax data abstract we compare two well known examples of stochastic volatility models the heston model and the hull white model we derive the stationary probability density distribution of the variance in addition we apply this stationary solution to the probability density distribution of the logarithmic returns by using the conditional probability density distribution furthermore we compare the received solutions of the logarithmic returns with empirical high frequency data of dax and its stocks

information theoretic clustering clustering is an important topic in pattern recognition since only the structure of the data dictates the grouping unsupervised learning information theory is an obvious criteria to establish the clustering rule the paper describes a novel valley seeking clustering algorithm using an information theoretic measure to estimate the cost of partitioning the data set the information theoretic criteria developed here evolved from a renyi entropy estimator a renyi 1960 that was proposed recently and has been successfully applied to other machine learning applications j c principe et al 2000 an improved version of the k change algorithm is used in optimization because of the stepwise nature of the cost function and existence of local minima even when applied to nonlinearly separable data the new algorithm performs well and was able to find nonlinear boundaries between clusters the algorithm is also applied to the segmentation of magnetic resonance imaging data mri with very promising results

input space versus feature space in kernel based methods this paper collects some ideas targeted at advancing our understanding of the feature spaces associated with support vector sv kernel functions we first discuss the geometry of feature space in particular we review what is known about the shape of the image of input space under the feature space map and how this influences the capacity of sv methods following this we describe how the metric governing the intrinsic geometry of the mapped surface can be computed in terms of the kernel using the example of the class of inhomogeneous polynomial kernels which are often used in sv pattern recognition we then discuss the connection between feature space and input space by dealing with the question of how one can given some vector in feature space find a preimage exact or approximate in input space we describe algorithms to tackle this issue and show their utility in two applications of kernel methods first we use it to reduce the computational complexity of sv decision functions second we combine it with the kernel pca algorithm thereby constructing a nonlinear statistical denoising technique which is shown to perform well on real world data

truncated levy walks and an emerging market economic index abstract in this paper we perform a statistical analysis of the major stock index in latin america the sao paulo stock exchange index in brazil ibovespa database contains daily records for the 15 year period 1986 2000 we find that the time evolution of the index of share prices is well described by an exponentially truncated levy flight etlf characterized by a levy exponent 1 6 1 7 and a cutoff exponent 1 7 the etlf statistics accounts for the observed short term large fluctuations of the financial data time series and describes the long term convergence to the gaussian regime we derive the characteristic crossover time scale n c dependence on and according to this model as well as the volatility dependence on and n c we find an uncorrelated behaviour of the historical data and n c 20 trading days which are in numerical agreement with the analytical results this dynamic model provides a framework within which it is possible to develop an efficient risk management and option pricing practice for emerging economies

multikernel least mean square algorithm the multikernel least mean square algorithm is introduced for adaptive estimation of vector valued nonlinear and nonstationary signals this is achieved by mapping the multivariate input data to a hilbert space of time varying vector valued functions whose inner products kernels are combined in an online fashion the proposed algorithm is equipped with novel adaptive sparsification criteria ensuring a finite dictionary and is computationally efficient and suitable for nonstationary environments we also show the ability of the proposed vector valued reproducing kernel hilbert space to serve as a feature space for the class of multikernel least squares algorithms the benefits of adaptive multikernel mk estimation algorithms are illuminated in the nonlinear multivariate adaptive prediction setting simulations on nonlinear inertial body sensor signals and nonstationary real world wind signals of low medium and high dynamic regimes support the approach