on fitting the pareto levy distribution to stock market index data selecting a suitable cutoff value abstract the so called pareto levy or power law distribution has been successfully used as a model to describe probabilities associated to extreme variations of stock markets indexes worldwide the selection of the threshold parameter from empirical data and consequently the determination of the exponent of the distribution is often done using a simple graphical method based on a log log scale where a power law probability plot shows a straight line with slope equal to the exponent of the power law distribution this procedure can be considered subjective particularly with regard to the choice of the threshold or cutoff parameter in this work a more objective procedure based on a statistical measure of discrepancy between the empirical and the pareto levy distribution is presented the technique is illustrated for data sets from the new york stock exchange djia and the mexican stock market ipc

the αth moment stability for the stochastic pantograph equation in this paper we investigate the ath moment asymptotical stability of the analytic solution and the numerical methods for the stochastic pantograph equation by using the razumikhin technique especially the linear stochastic pantograph equations and the semi implicit euler method applying them are considered the convergence result of the semi implicit euler method is obtained the stability conditions of the analytic solution of those equations and the numerical method are given finally some experiments are given

dynamics of stock prices we show that the dynamics of stock prices can be accurately described as a continuous time random walk with a time dependent diffusion coefficient the time evolution of the diffusion coefficient can be derived from tick by tick databases provided the stock price is characterized in terms of a couple of values describing the best ask and the best bid we are then led to a finding and namely that the transition rate of the random walk process is different from the frequency of transactions our results allow us to obtain a fast and reliable determination of the diffusion coefficient and precisely confirm that fat tails in the distribution of price variations are due to volatility fluctuations

numerical solutions of stochastic differential delay equations under the generalized khasminskii type conditions the classical existence and uniqueness theorem of the solution to a stochastic differential delay equation sdde requires the local lipschitz condition and the linear growth condition see e g 11 12 and 20 the numerical solutions under these conditions have also been discussed intensively see e g 4 10 13 16 17 18 21 22 and 24 recently mao and rassias 14 and 15 established the generalized khasminskii type existence and uniqueness theorems for sddes where the linear growth condition is no longer imposed these generalized khasminskii type theorems cover a wide class of highly nonlinear sddes but these nonlinear sddes do not have explicit solutions whence numerical solutions are required in practice however there is so far little numerical theory on sddes under these generalized khasminskii type conditions the key aim of this paper is to close this gap

asymptotic stability in the pth moment for stochastic differential equations with levy noise abstract this paper is devoted to study a class of stochastic differential equations with levy noise in comparison to the standard gaussian noise levy noise is more versatile and interesting with a wider range of applications however levy noise makes the analysis more difficult owing to the discontinuity of its sample paths in this paper we attempt to overcome this difficulty we propose several sufficient conditions under which we investigate the long time behavior of the solution including the asymptotic stability in the pth moment and almost sure stability also we discuss two types of continuity of the solution continuous in probability and continuous in the pth moment finally we provide two examples to illustrate the effectiveness of the theoretical results

a new computational algorithm for solving periodic tri diagonal linear systems in this article we present a new efficient computational algorithm for solving periodic tri diagonal linear systems the implementation of the algorithm using computer algebra systems cas such as maple macsyma mathematica and matlab is straightforward an example is given in order to illustrate the algorithm

trend and fractality assessment of mexico s stock exchange the total value of domestic market capitalization of the mexican stock exchange was calculated at 520 billion of dollars by the end of november 2013 to manage this system and make optimum capital investments its dynamics needs to be predicted however randomness within the stock indexes makes forecasting a difficult task to address this issue in this work trends and fractality were studied using gnu r over the opening and closing prices indexes over the past 24 years returns kernel density estimation autocorrelation function and r s analysis and the hurst exponent were used in this research as a result it was found that the kernel estimation density and the autocorrelation function shown the presence of long range memory effects in a first approximation the returns of closing prices seems to behave according to a markovian random walk with a length of step size given by an stable random process for extreme values returns decay asymptotically as a power law with a characteristic exponent approximately equal to 2 6

almost sure stability of linear stochastic differential equations with jumps under the nondegenerate condition as in the diffusion case see 14 21 6 the linear stochastic jump diffusion process projected on the unit sphere is a strong feller process and has a unique invariant measure which is also ergodic using the relation between the transition probabilities of jump diffusions and the corresponding diffusions due to 22 the unique deterministic lyapunov exponent can be represented by the furstenberg khas minskii formula as an integral over the sphere or the projective space with respect to the ergodic invariant measure so that the almost sure asymptotic stability of linear stochastic systems with jumps depends on its sign the critical case of zero lyapunov exponent is discussed and a large deviations result for asymptotically stable systems is further investigated several examples are treated for illustration

a backward three term recurrence relation for vector valued continued fractions and its applications this paper presents a backward three term recurrence relation which is a solution of an open problem in graves morris numer math 42 1983 331 by means of this relation we succeed in establishing some conclusions about the two backward successive convergents of vector valued continued fractions moreover an expression for the vector valued continued fraction in series form is obtained and some theorems of the convergence of the vector valued continued fraction are proved by virtue of the recurrence relation

analytical inversion of general periodic tridiagonal matrices in this paper we present an analytical forms for the inversion of general periodic tridiagonal matrices and provide some very simple analytical forms which immediately lead to closed formulae for some special cases such as symmetric or perturbed toeplitz for both periodic and non periodic tridiagonal matrices an efficient computational algorithm for finding the inverse of any general periodic tridiagonal matrices from the analytical form is given it is suited for implementation using computer algebra systems such as maple matlab macsyma and mathematica an example is also given to illustrate the algorithm

the scaling properties of stock markets based on modified multiscale multifractal detrended fluctuation analysis abstract here we propose the new method dh mma based on multiscale multifractal detrended fluctuation analysis mma to investigate the scaling properties in stock markets it is demonstrated that our approach can provide a more stable and faithful description of the scaling properties in comprehensive range rather than fixing the window length and slide length it allows the assessment of more universal and subtle scaling characteristics we illustrate dh mma by selecting power law artificial data sets and six stock markets from us and china the us stocks exhibit very strong multifractality for positive values of q however the chinese stocks show stronger multifractality for negative q than positive q in general the us stock markets show similar behaviors but chinese stock markets display distinguishing characteristics

stock market dynamics abstract we elucidate on several empirical statistical observations of stock market returns moreover we find that these properties are recurrent and are also present in invariant measures of low dimensional dynamical systems thus we propose that the returns are modeled by the first poincare return time of a low dimensional chaotic trajectory this modeling which captures the recurrent properties of the return fluctuations is able to predict well the evolution of the observed statistical quantities in addition it explains the reason for which stocks present simultaneously dynamical properties and high uncertainties in our analysis we use data from the s p 500 index and the brazilian stock telebras

stochastic process with multiplicative structure for the dynamic behavior of the financial market abstract a stochastic model with multiplicative noise has been proposed as a mathematical model for the prices dynamics of the financial market we have presented a model which allows us to test within the same framework the comparative explanatory power of rational agents versus irrational agents with respect to facts of the financial market we calculate the long range memory of the model and studied the behavior of the long tail distribution of the cumulative distribution of probabilities for the model with additive and multiplicative noise

scale invariance and universality organizing principles in complex systems this paper is a brief summary of a talk that was designed to address the question of whether two of the pillars of the eld of phase transitions and critical phenomena scale invariance and universality can be useful in guiding research on a broad class of complex phenomena we shall see that while scale invariance has been tested for many years universality is relatively more rarely discussed in particular we shall develop a heuristic argument that serves to make more plausible the universality hypothesis in both thermal critical phenomena and percolation phenomena and suggest that this argument could be developed into a possible coherent approach to understanding the ubiquity of scale invariance and universality in a wide range of complex systems c 2000 elsevier science b v all rights reserved

almost sure exponential stability of numerical solutions for stochastic delay differential equations using techniques based on the continuous and discrete semimartingale convergence theorems this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations sddes the important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of sddes directly this is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the chebyshev inequality and the borel cantelli lemma

on the eigenvalues of some tridiagonal matrices a solution is given for a problem on eigenvalues of some symmetric tridiagonal matrices suggested by william trench the method presented can be generalizable to other problems

discrete time approximations of stochastic delay equations the milstein scheme in this paper we develop a strong milstein approximation scheme for solving stochastic delay differential equations sddes the scheme has convergence order 1 in order to establish the scheme we prove an infinite dimensional ito formula for tame functions acting on the segment process of the solution of an sdde it is interesting to note that the presence of the memory in the sdde requires the use of the malliavin calculus and the anticipating stochastic analysis of nualart and pardoux given the nonanticipating nature of the sdde the use of anticipating calculus methods in the context of strong approximation schemes appears to be novel

the inverse of a tridiagonal matrix in this paper explicit formulae for the elements of the inverse of a general tridiagonal matrix are presented by first extending results on the explicit solution of a second order linear homogeneous difference equation with variable coefficients to the nonhomogeneous case and then applying these extended results to a boundary value problem a formula for the characteristic polynomial is obtained in the process we also establish a connection between the matrix inverse and orthogonal polynomials in addition the case of a cyclic tridiagonal system is discussed

stability analysis of neutral stochastic differential delay equations driven by levy noises abstract this paper mainly analyzes the well posedness and the stability analysis for the global solution of neutral stochastic differential delay equations nsddes driven by levy noises by using an integral lemma and a lyapunov function approach the existence and uniqueness theorem is proved then by using the inequality technique and the stochastic analysis theory the exponential stability in pth p 2 moment of such equations is discussed by using another integral lemma and using the baralat lemma as well as the stochastic analysis the almost surely asymptotic stability is also studied finally one example is given to check the effectiveness of the findings derived

economic fluctuations and statistical physics quantifying extremely rare and less rare events in finance one challenge of economics is that the systems treated by these sciences have no perfect metronome in time and no perfect spatial architecture crystalline or otherwise nonetheless as if by magic out of nothing but randomness one finds remarkably fine tuned processes in time we present an overview of recent research joining practitioners of economic theory and statistical physics to try to better understand puzzles regarding economic fluctuations one of these puzzles is how to describe outliers phenomena that lie outside of patterns of statistical regularity we review evidence consistent with the possibility that such outliers may not exist this possibility is supported by recent analysis of databases containing information about each trade of every stock r 2007 elsevier b v all rights reserved

the existence and asymptotic estimations of solutions to stochastic pantograph equations with diffusion and levy jumps in this paper we consider a class of stochastic pantograph differential equations with levy jumps spdewljs by using the burkholder davis gundy inequality and the kunita s inequality we prove the existence and uniqueness of solutions to spdewljs whose coefficients satisfying the lipschitz conditions and the local lipschitz conditions meantime we establish the p th exponential estimations and almost surely asymptotic estimations of solutions to spdewljs

intermittency in an interacting generalization of the geometric brownian motion model we propose a minimal interacting generalization of the geometric brownian motion model which turns out to be formally equivalent to a model describing the dynamics of networks of analogue neurons for sufficiently strong interactions such systems may have many meta stable states transitions between meta stable states are associated with macroscopic reorganizations of the system which can be triggered by random external forcing such a system will exhibit intermittent dynamics within a large part of its parameter space we propose market dynamics as a possible application of this model in which case random external forcing would correspond to the arrival of important information the emergence of a model of interacting prices of the type considered here can be argued to follow naturally from a general argument based on integrating out all non price degrees of freedom from the dynamics of a hypothetical complete description of economic dependences

introduction to the numerical analysis of stochastic delay differential equations abstract we consider the problem of the numerical solution of stochastic delay differential equations of ito form d x t f x t x t d t g x t x t d w t t 0 t and x t t for t 0 with given f g wiener noise w and given 0 with a prescribed initial function we indicate the nature of the equations of interest and give a convergence proof for explicit single step methods some illustrative numerical examples using a strong euler maruyama scheme are provided

numerical solutions of neutral stochastic functional differential equations this paper examines the numerical solutions of neutral stochastic functional differential equations nsfdes d x t u x t f x t dt g x t dw t t geq 0 the key contribution is to establish the strong mean square convergence theory of the euler maruyama approximate solution under the local lipschitz condition the linear growth condition and contractive mapping these conditions are generally imposed to guarantee the existence and uniqueness of the true solution so the numerical results given here are obtained under quite general conditions although the way of analysis borrows from x mao lms j comput math 6 2003 pp 141 161 to cope with u x t several new techniques have been developed

existence of global solutions and invariant measures for stochastic differential equations driven by poisson type noise with non lipschitz coefficients the purpose of this paper is twofold firstly we investigate the problem of existence and uniqueness of solutions to stochastic differential equations with one sided dissipative drift driven by semi martingales secondly we investigate the problem of existence of an invariant measure for such equations when the coefficients are time independent

existence and uniqueness of the solutions and convergence of semi implicit euler methods for stochastic pantograph equations abstract in this paper the sufficient conditions of existence and uniqueness of the solutions for stochastic pantograph equation are given i e the local lipschitz condition and the linear growth condition under the lipschitz condition and the linear growth condition it is proved that the semi implicit euler method is convergence with strong order 1 2

numerical methods for nonlinear stochastic differential equations with jumps we present and analyse two implicit methods for ito stochastic differential equations sdes with poisson driven jumps the first method ssbe is a split step extension of the backward euler method the second method cssbe arises from the introduction of a compensated martingale form of the poisson process we show that both methods are amenable to rigorous analysis when a one sided lipschitz condition rather than a more restrictive global lipschitz condition holds for the drift our analysis covers strong convergence and nonlinear stability we prove that both methods give strong convergence when the drift coefficient is one sided lipschitz and the diffusion and jump coefficients are globally lipschitz on the way to proving these results we show that a compensated form of the euler maruyama method converges strongly when the sde coefficients satisfy a local lipschitz condition and the pth moment of the exact and numerical solution are bounded for some p 2 under our assumptions both ssbe and cssbe give well defined unique solutions for sufficiently small stepsizes and ssbe has the advantage that the restriction is independent of the jump intensity we also study the ability of the methods to reproduce exponential mean square stability in the case where the drift has a negative one sided lipschitz constant this work extends the deterministic nonlinear stability theory in numerical analysis we find that ssbe preserves stability under a stepsize constraint that is independent of the initial data cssbe satisfies an even stronger condition and gives a generalization of b stability finally we specialize to a linear test problem and show that cssbe has a natural extension of deterministic a stability the difference in stability properties of the ssbe and cssbe methods emphasizes that the addition of a jump term has a significant effect that cannot be deduced directly from the non jump literature

minority games with score dependent and agent dependent payoffs score dependent and agent dependent payoffs of the strategies are introduced into the standard minority game the intrinsic periodicity is consequently removed and the stylized facts arise such as long range volatility correlations and fat tails in the distribution of the returns the agent dependence of the payoffs is essential in producing the long range volatility correlations the new payoffs lead to a better performance in the dynamic behavior nonlocal in time and can coexist with the inactive strategy we also observe that the standard deviation ensuremath sigma 2 n is significantly reduced thus the efficiency of the system is distinctly improved based on this observation we give a qualitative explanation for the long range volatility correlations

estimates for the inverse of tridiagonal matrices arising in boundary value problems by considering tridiagonal matrices as three term recurrence relations with dirichlet boundary conditions one can formulate their inverses in terms of green s functions this analysis is applied to three point difference schemes for 1 d problems and five point difference schemes for 2 d problems we derive either an explicit inverse of the jacobian or a sharp estimate for both uniform and nonuniform grids

time evolution of stochastic processes with correlations in the variance stability in power law tails of distributions abstract we model the time series of the s p500 index by a combined process the ar garch process where ar denotes the autoregressive process which we use to account for the short range correlations in the index changes and garch denotes the generalized autoregressive conditional heteroskedastic process which takes into account the long range correlations in the variance we study the ar garch process with an initial distribution of truncated levy form we find that this process generates a new probability distribution with a crossover from a levy stable power law to a power law with an exponent outside the levy range beyond the truncation cutoff we analyze the sum of n variables of the ar garch process and find that due to the correlations the ar garch process generates a probability distribution which exhibits stable behavior in the tails for a broad range of values n a feature which is observed in the probability distribution of the s p500 index we find that this power law stability depends on the characteristic scale in the correlations we also find that inclusion of short range correlations through the ar process is needed to obtain convergence to a limiting gaussian distribution for large n as observed in the data

bin size independence in intra day seasonalities for relative prices abstract in this paper we perform a statistical analysis over the returns and relative prices of the cac 40 and the s p 500 with the purpose of analysing the intra day seasonalities of single and cross sectional stock dynamics in order to do that we characterized the dynamics of a stock or a set of stocks by the evolution of the moments of its returns and relative prices during a typical day we show that these intra day seasonalities are independent of the size of the bin and the index we consider but characteristic for each index for the case of the relative prices but not for the case of the returns finally we suggest how this bin size independence could be used in order to characterize atypical days for indexes and anomalous behaviours in stocks

autocorrelation type timescale and statistical property in financial time series abstract earlier studies have documented that three types of autocorrelations exist in financial time series sign volatility and return autocorrelation in this paper we examine how each type of the above autocorrelations affects the statistical properties of financial time series and its role in maintaining such statistical properties using three different shuffling series that correspondingly destroy each type of autocorrelation upon different timescales we find that 1 the statistical properties of the shuffling series significantly vary from the original ones 2 volatility and return autocorrelations show greater impacts than sign autocorrelation 3 the effects on the statistical properties are intensified as time scale expands 4 the nonlinear component of autocorrelation is the major drive of the effect

factorizations and representations of the backward second order linear recurrences we show the relationships between the determinants and permanents of certain tridiagonal matrices and the negatively subscripted terms of second order linear recurrences also considering how to the negatively subscripted terms of second order linear recurrences can be connected to chebyshev polynomials by determinants of these matrices we give factorizations and representations of these numbers

volatility cluster and herding stock markets can be characterized by fat tails in the volatility distribution clustering of volatilities and slow decay of their time correlations for an explanation models with several mechanisms and consequently many parameters as the lux marchesi model have been used we show that a simple herding model with only four parameters leads to a quantitative description of the data as a new type of data we describe the volatility cluster by the waiting time distribution which can be used successfully to distinguish between different models

an analytical approach explicit inverses of periodic tridiagonal matrices abstract we derive an explicit formula for the inverse of a general periodic tridiagonal matrix our approach is to derive its l u factorization using backward continued fractions bcf which are an essential tool in number theory we then use these formulae to construct an algorithm for inverting a general periodic tridiagonal matrix which we implement in maple 1 finally we present the results of testing the efficiency of our new algorithm against another published implementation and against the library procedures available within maple to invert a general matrix and to compute its determinant

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

pantograph stochastic differential equations driven by g brownian motion abstract in this work we discuss a kind of pantograph stochastic functional differential equations driven by g brownian motion g psdes in short when the coefficients satisfy local lipschitz and generalized lyapunov conditions we establish existence uniqueness asymptotic boundedness and exponential stability of the solution for g psdes

tail distribution of index fluctuations in world markets abstract we have investigated the tail distribution of the daily fluctuations in 202 different indices in the stock markets of 59 countries for the time span of the last 20 years power law log normal weibull exponential and power law with exponential cutoff distributions are considered as possible candidates for the tail distribution of the normalized returns it is found that the power exponent depends strongly on the choice of the tail threshold and a sizeable number of indices can be better fitted by a distribution function other than the power law at the region that has power law exponent of 3 also we have found that the power exponent is not an indicator of the maturity of the market

the eigenvalues of tridiagonal sign matrices are dense in the spectra of periodic tridiagonal sign operators abstract chandler wilde chonchaiya and lindner conjectured that the set of eigenvalues of finite tridiagonal sign matrices i e plus and minus ones on the first sub and superdiagonal zeroes everywhere else is dense in the set of spectra of periodic tridiagonal sign operators on the usual hilbert space of square summable bi infinite sequences we give a simple proof of this conjecture as a consequence we get that the set of eigenvalues of tridiagonal sign matrices is dense in the unit disk in fact a recent paper further improves this result showing that this set of eigenvalues is dense in an even larger set

robustness of exponential stability of stochastic differential delay equations regard the stochastic differential delay equation dx t a a t b b t spl tau x t spl tau dt g t x t x t spl tau dw t as the result of the effects of uncertainty stochastic perturbation and time lag to a linear ordinary differential equation x spl dot t a b x t assume the linear system is exponentially stable in this paper we characterize how much the uncertainty stochastic perturbation and time lag the linear system can bear such that the stochastic delay system remains exponentially stable the result can also be extended to nonlinear systems

convergence of numerical solutions to stochastic pantograph equations with markovian switching in this paper a class of stochastic pantograph equations with markovian switching is considered the main purpose is to investigate the convergence of the euler method of the equations it is proved that the euler approximation solution converge to the analytic solution in probability under weaker conditions an example is provided to illustrate our theory

factorization of joint multifractality abstract complex systems often exhibit multifractal characteristics in various forms the study of the joint fluctuation of multifractal objects referred to as joint multifractality is presented in this work we use the joint partition function approach c meneveau et al phys rev a 41 1990 894 to show that joint multifractality admits a factorization into a common factor related to the notion of relative multifractality studied by riedi and scheuring r h riedi i scheuring fractals 5 1997 153 and a remainder term related to the individual multifractality we demonstrated our ideas using binomial measures and applied to the fluctuation of financial data

new algorithms for solving periodic tridiagonal and periodic pentadiagonal linear systems abstract recently an efficient computational algorithm for solving periodic pentadiagonal linear systems has been proposed by karawia a a karawia a computational algorithm for solving periodic pentadiagonal linear systems appl math comput 174 2006 613 618 the algorithm is based on the lu factorization of the periodic pentadiagonal matrix in this paper new algorithms are presented for solving periodic pentadiagonal linear systems based on the use of any pentadiagonal linear solver in addition an efficient way of evaluating the determinant of a periodic pentadiagonal matrix is discussed the corresponding results in this paper can be readily obtained for solving periodic tridiagonal linear systems

explicit formula for the inverse of a tridiagonal matrix by backward continued fractions in this paper we consider a general tridiagonal matrix and give the explicit formula for the elements of its inverse for this purpose considering usual continued fraction we define backward continued fraction for a real number and give some basic results on backward continued fraction we give the relationships between the usual and backward continued fractions then we reobtain the lu factorization and determinant of a tridiagonal matrix furthermore we give an efficient and fast computing method to obtain the elements of the inverse of a tridiagonal matrix by backward continued fractions comparing the earlier result and our result on the elements of the inverse of a tridiagonal matrix it is seen that our method is more convenient efficient and fast