new results on composite control of singularly perturbed uncertain linear systems abstract we consider the problem of stabilizing uncertain singularly perturbed linear time varying systems whose fast dynamics can be unstable we propose a class of nonlinear composite controllers which assure global uniform ultimate boundedness of the trajectories of the closed loop system provided the singular perturbation parameter is sufficiently small these controllers consist of a fast controller which stabilizes the fast dynamics and a slow controller which yields the desired stability properties for the slow dynamics we consider the structure of the fast controller to be simpler than structures previously proposed in the literature to obtain these controllers we first develop some new results for singularly perturbed systems under output feedback in particular it is shown that the matching assumption which deals with the manner in which the uncertainties enter the system and is made on the reduced order system is invariant under linear feedback of the fast variable

kramers like escape driven by fractional gaussian noise we investigate the escape from a potential well of a test particle driven by fractional gaussian noise with hurst exponent 0 h 1 from a numerical analysis we demonstrate the exponential distribution of escape times from the well and analyze in detail the dependence of the mean escape time on the hurst exponent h and the particle diffusivity d we observe different behavior for the subdiffusive antipersistent and superdiffusive persistent domains in particular we find that the escape becomes increasingly faster for decreasing values of h consistent with previous findings on the first passage behavior approximate analytical calculations are shown to support the numerically observed dependencies

correlation measure to detect time series distances whence economy globalization abstract an instantaneous time series distance is defined through the equal time correlation coefficient the idea is applied to the gross domestic product gdp yearly increments of 21 rich countries between 1950 and 2005 in order to test the process of economic globalisation some data discussion is first presented to decide what eks gk or derived gdp series should be studied distances are then calculated from the correlation coefficient values between pairs of series the role of time averaging of the distances over finite size windows is discussed three network structures are next constructed based on the hierarchy of distances it is shown that the mean distance between the most developed countries on several networks actually decreases in time which we consider as a proof of globalization an empirical law is found for the evolution after 1990 similar to that found in flux creep the optimal observation time window size is found 15 years

nonlinear decoupling via feedback a differential geometric approach the paper deals with the nonlinear decoupling and noninteracting control problems a complete solution to those problems is made possible via a suitable nonlinear generalization of several powerful geometric concepts already introduced in studying linear multivariable control systems the paper also includes algorithms concerned with the actual construction of the appropriate control laws

first passage time distribution for variable diffusion processes first passage time distribution which presents the likelihood of a stock reaching a pre specified price at a given time is useful in establishing the value of financial instruments and in designing trading strategies first passage time distribution for wiener processes has a single peak while that for stocks exhibits a notable second peak within a trading day this feature has only been discussed sporadically often dismissed as due to insufficient incorrect data or circumvented by conversion to tick time and to the best of our knowledge has not been explained in terms of the underlying stochastic process it was shown previously that intra day variations in the market can be modeled by a stochastic process containing two variable diffusion processes hua et al in physica a 419 221 233 2015 we show here that the first passage time distribution of this two stage variable diffusion model does exhibit a behavior similar to the empirical observation in addition we find that an extended model incorporating overnight price fluctuations exhibits intra and inter day behavior similar to those of empirical first passage time distributions

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

information theoretical analysis of resting state eeg microstate sequences non markovianity non stationarity and periodicities we present an information theoretical analysis of temporal dependencies in eeg microstate sequences during wakeful rest we interpret microstate sequences as discrete stochastic processes where each state corresponds to a representative scalp potential topography testing low order markovianity of these discrete sequences directly we find that none of the recordings fulfils the markov property of order 0 1 or 2 further analyses show that the microstate transition matrix is non stationary over time in 80 window size 10 s 60 window size 20 s and 44 window size 40 s of the subjects and that transition matrices are asymmetric in 14 20 70 subjects to assess temporal dependencies globally the time lagged mutual information function autoinformation function of each sequence is compared to the first order markov model defined by the classical transition matrix approach the autoinformation function for the markovian case is derived analytically and numerically for experimental data we find non markovian behaviour in the range of the main eeg frequency bands where distinct periodicities related to the subject s eeg frequency spectrum appear in particular the microstate clustering algorithm induces frequency doubling with respect to the eeg power spectral density while the tail of the autoinformation function asymptotically reaches the first order markov confidence interval for time lags above 1000 ms in summary our results show that resting state microstate sequences are non markovian processes which inherit periodicities from the underlying eeg dynamics our results interpolate between two diverging models of microstate dynamics memoryless markov models on one side and long range correlated models on the other microstate sequences display more complex temporal dependencies than captured by the transition matrix approach in the range of the main eeg frequency bands but show finite memory content in the long run

burst and inter burst duration statistics as empirical test of long range memory in the financial markets abstract we address the problem of long range memory in the financial markets there are two conceptually different ways to reproduce power law decay of auto correlation function using fractional brownian motion as well as non linear stochastic differential equations in this contribution we address this problem by analyzing empirical return and trading activity time series from the forex from the empirical time series we obtain probability density functions of burst and inter burst duration our analysis reveals that the power law exponents of the obtained probability density functions are close to 3 2 which is a characteristic feature of the one dimensional stochastic processes this is in a good agreement with earlier proposed model of absolute return based on the non linear stochastic differential equations derived from the agent based herding model

feedback control of two time scale nonlinear systems this article concerns a class of two time scale single input single output nonlinear systems for such systems combination of singular perturbation and geometric methods is employed to synthesize well conditioned static state feedback laws that induce a well characterized input output behaviour and guarantee stability of the closed loop system the developed control laws are applied to two nonlinear chemical processes with time scale multiplicity and their performance is evaluated through simulations

a generalized kolmogorov sinai like entropy under markov shifts in symbolic dynamics abstract in this paper a generalized kolmogorov sinai like entropy ks q entropy in the sense of tsallis is proposed with a nonextensive parameter q under markov shifts which contains the classical kolmogorov sinai ks entropy and the renyi entropy as well as bernoulli shifts as special cases to verify the formula of this ks q entropy a one dimensional iterative system is chosen as an example of markov shifts and its ks q entropy is evaluated by a new refinement method of symbolic dynamics called symbolic refinement which differs from the conventional numerical method the numerical results show that this ks q entropy is monotonically decreasing as q increases

singular perturbations and input to state stability this paper establishes a type of total stability for the input to state stability property with respect to singular perturbations in particular if the boundary layer system is uniformly globally asymptotically stable and the reduced system is input to state stable with respect to disturbances then these properties continue to hold up to an arbitrarily small offset for initial conditions disturbances and their derivatives in an arbitrarily large compact set as long as the singular perturbation parameter is sufficiently small

analytical and empirical fluctuation functions of the eeg microstate random walk short range vs long range correlations we analyze temporal autocorrelations and the scaling behaviour of eeg microstate sequences during wakeful rest we use the recently introduced random walk approach and compute its fluctuation function analytically under the null hypothesis of a short range correlated first order markov process the empirical fluctuation function and the hurst parameter h as a surrogate parameter of long range correlations are computed from 32 resting state eeg recordings and for a set of first order markov surrogate data sets with equilibrium distribution and transition matrices identical to the empirical data in order to distinguish short range correlations h 0 5 from previously reported long range correlations h 0 5 statistically confidence intervals for h and the fluctuation functions are constructed under the null hypothesis comparing three different estimation methods for h we find that only one data set consistently shows h 0 5 compatible with long range correlations whereas the majority of experimental data sets cannot be consistently distinguished from markovian scaling behaviour our analysis suggests that the scaling behaviour of resting state eeg microstate sequences though markedly different from uncorrelated zero order markov processes can often not be distinguished from a short range correlated first order markov process our results do not prove the microstate process to be markovian but challenge the approach to parametrize resting state eeg by single parameter models

a theory of fluctuations in stock prices abstract the distribution of price returns is studied for a class of market models with markovian dynamics the models have a non constant diffusion coefficient that depends on the value of the return an analytical expression for the distribution of returns is obtained and shown to match the results of computer simulations for two simple cases those two cases are shown to have exponential and fat tailed power law decaying distributions respectively

martingales nonstationary increments and the efficient market hypothesis abstract we discuss the deep connection between nonstationary increments martingales and the efficient market hypothesis for stochastic processes x t with arbitrary diffusion coefficients d x t we explain why a test for a martingale is generally a test for uncorrelated increments we explain why martingales look markovian at the level of both simple averages and 2 point correlations but while a markovian market has no memory to exploit and cannot be beaten systematically a martingale admits memory that might be exploitable in higher order correlations we also use the analysis of this paper to correct a misstatement of the fair game condition in terms of serial correlations in fama s paper on the emh we emphasize that the use of the log increment as a variable in data analysis generates spurious fat tails and spurious hurst exponents

martingales detrending data and the efficient market hypothesis we discuss martingales detrending data and the efficient market hypothesis for stochastic processes x t with arbitrary diffusion coefficients d x t beginning with x independent drift coefficients r t we show that martingale stochastic processes generate uncorrelated generally nonstationary increments generally a test for a martingale is therefore a test for uncorrelated increments a detrended process with an x dependent drift coefficient is generally not a martingale and so we extend our analysis to include the class of x t dependent drift coefficients of interest in finance we explain why martingales look markovian at the level of both simple averages and 2 point correlations and while a markovian market has no memory to exploit and presumably cannot be beaten systematically it has never been shown that martingale memory cannot be exploited in 3 point or higher correlations to beat the market we generalize our markov scaling solutions presented earlier and also generalize the martingale formulation of the efficient market hypothesis emh to include x t dependent drift in log returns we also use the analysis of this paper to correct a misstatement of the fair game condition in terms of serial correlations in fama s paper on the emh we end with a discussion of levy scharacterization of brownian motion and prove that an arbitrary martingale is topologically inequivalent to a wiener process

non crossing partitions of type e e r abstract we investigate a new lattice of generalised non crossing partitions constructed using the geometry of the complex reflection group g e e r for the particular case e 2 resp r 2 our lattice coincides with the lattice of simple elements for the type d n resp i 2 e dual braid monoid using this lattice we construct a garside structure for the braid group b e e r as a corollary one may solve the word and conjugacy problems in this group

direct approach to almost disturbance and almost input output decoupling a direct solution to the almost input output and disturbance decoupling problem is given the approach is based on a new canonical form for linear systems which is obtained in a fully constructive way together with some elementary approximation results the solution is given in the form of an explicit feedback law parameterized by a scalar e by letting e go to zero the accuracy of the resulting approximate solution can be improved to any desired degree

feedforward and feedback linearization of non linear systems with disturbances the problem of transforming non linear systems with input disturbances to their linear equivalents is investigated a characterization of the class of non linear systems with disturbances that are equivalent to controllable quasi linear systems with disturbances z az bv z d is given the necessary transformations state space coordinate changes and control input space coordinate changes by non linear feedforward and feedback are presented and their implications for control system synthesis are discussed finally a numerical example is given

mutual information identifies spurious hurst phenomena in resting state eeg and fmri data long range memory in time series is often quantified by the hurst exponent h a measure of the signal s variance across several time scales we analyze neurophysiological time series from electroencephalography eeg and functional magnetic resonance imaging fmri resting state experiments with two standard hurst exponent estimators and with the time lagged mutual information function applied to discretized versions of the signals a confidence interval for the mutual information function is obtained from surrogate markov processes with equilibrium distribution and transition matrix identical to the underlying signal for eeg signals we construct an additional mutual information confidence interval from a short range correlated tenth order autoregressive model we reproduce the previously described hurst phenomenon h 0 5 in the analytical amplitude of alpha frequency band oscillations in eeg microstate sequences and in fmri signals but we show that the hurst phenomenon occurs without long range memory in the information theoretical sense we find that the mutual information function of neurophysiological data behaves differently from fractional gaussian noise fgn for which the hurst phenomenon is a sufficient condition to prove long range memory two other well characterized short range correlated stochastic processes ornstein uhlenbeck cox ingersoll ross also yield h 0 5 whereas their mutual information functions lie within the markovian confidence intervals similar to neural signals in these processes which do not have long range memory by construction a spurious hurst phenomenon occurs due to slow relaxation times and heteroscedasticity time varying conditional variance in summary we find that mutual information correctly distinguishes long range from short range dependence in the theoretical and experimental cases discussed our results also suggest that the stationary fgn process is not sufficient to describe neural data which seem to belong to a more general class of stochastic processes in which multiscale variance effects produce hurst phenomena without long range dependence in our experimental data the hurst phenomenon and long range memory appear as different system properties that should be estimated and interpreted independently

dynamic compensation of measurable disturbances in non linear multivariable systems this paper concerns multivariable nonlinear systems with measurable disturbance inputs for such systems we address the problem of eliminating the effect of the disturbance inputs on the outputs and achieving a well characterized linear input output behaviour in the closed loop system necessary and sufficient conditions are derived under which dynamic feedforward static state feedback laws solve the above problem and explicit synthesis formulas are developed

a comment on the paper stochastic feedback nonlinear families of markov processes and nonlinear fokker planck equations by t d frank abstract the purpose of this comment is to correct mistaken assumptions and claims made in the paper stochastic feedback nonlinear families of markov processes and nonlinear fokker planck equations by t d frank t d frank stochastic feedback non linear families of markov processes and nonlinear fokker planck equations physica a 331 2004 391 our comment centers on the claims of a non linear markov process and a non linear fokker planck equation first memory in transition densities is misidentified as a markov process second the paper assumes that one can derive a fokker planck equation from a chapman kolmogorov equation but no proof was offered that a chapman kolmogorov equation exists for the memory dependent processes considered a non linear markov process is claimed on the basis of a non linear diffusion pde for a 1 point probability density we show that regardless of which initial value problem one may solve for the 1 point density the resulting stochastic process defined necessarily by the conditional probabilities the transition probabilities is either an ordinary linearly generated markovian one or else is a linearly generated non markovian process with memory we provide explicit examples of diffusion coefficients that reflect both the markovian and the memory dependent cases so there is neither a non linear markov process nor a non linear fokker planck equation for a conditional probability density the confusion rampant in the literature arises in part from labeling a non linear diffusion equation for a 1 point probability density as non linear fokker planck whereas neither a 1 point density nor an equation of motion for a 1 point density can define a stochastic process in a closely related context we point out that borland misidentified a translation invariant 1 point probability density derived from a non linear diffusion equation as a conditional probability density finally in the appendix a we present the theory of fokker planck pdes and chapman kolmogorov equations for stochastic processes with finite memory

selfdecomposability and selfsimilarity a concise primer abstract we summarize the relations among three classes of laws infinitely divisible selfdecomposable and stable first we look at them as the solutions of the central limit problem then their role is scrutinized in relation to the levy and the additive processes with an emphasis on stationarity and selfsimilarity finally we analyze the ornstein uhlenbeck processes driven by levy noises and their selfdecomposable stationary distributions and we end with a few particular examples

anomalous scaling of stochastic processes and the moses effect the state of a stochastic process evolving over a time t is typically assumed to lie on a normal distribution whose width scales like t frac 1 2 however processes in which the probability distribution is not normal and the scaling exponent differs from frac 1 2 are known the search for possible origins of such anomalous scaling and approaches to quantify them are the motivations for the work reported here in processes with stationary increments where the stochastic process is time independent autocorrelations between increments and infinite variance of increments can cause anomalous scaling these sources have been referred to as the joseph effect and the noah effect respectively if the increments are nonstationary then scaling of increments with t can also lead to anomalous scaling a mechanism we refer to as the moses effect scaling exponents quantifying the three effects are defined and related to the hurst exponent that characterizes the overall scaling of the stochastic process methods of time series analysis that enable accurate independent measurement of each exponent are presented simple stochastic processes are used to illustrate each effect intraday financial time series data are analyzed revealing that their anomalous scaling is due only to the moses effect in the context of financial market data we reiterate that the joseph exponent not the hurst exponent is the appropriate measure to test the efficient market hypothesis

generating synthetic time series from bak sneppen co evolution model mixtures abstract the bak sneppen model of co evolution is used to derive synthetic time series with a priori specified fractal dimension or hurst exponent through a mixing of processes in various lattice dimensions both theoretical and numerical analyses concern the avalanches at the critical threshold and provide a model for time series reconstruction that can be tested as an alternative to the classical fractional brownian motion fbm because of differences in properties new results on critical threshold and avalanche structure are obtained up to euclidean dimension d 6 the method involves a lattice based structure and therefore is suitable for the application of parallel computing

martingale option pricing we show that our earlier generalization of the black scholes partial differential equation pde for variable diffusion coefficients is equivalent to a martingale in the risk neutral discounted stock price previously the equivalence of black scholes to a martingale was proven for the case of the gaussian returns model by harrison and kreps but we prove it for much a much larger class of returns models where the returns diffusion coefficient depends irreducibly on both returns x and time t that option prices blow up if fat tails in logarithmic returns x are included in market return is also proven

scaling behaviors in differently developed markets scaling properties of four di erent stock market indices are studied in terms of a generalized hurst exponent approach we nd that the deviations from pure brownian motion behavior are associated with the degrees of development of the markets and we observe strong di erentiations in the scaling properties of markets at di erent development stage c

compensation of measurable disturbances for two time scale nonlinear systems this paper deals with a class of two time scale nonlinear systems with time varying disturbances modeled within the framework of singular perturbations systems in both standard and nonstandard form are considered for these systems we synthesize well conditioned control laws that utilize feedback of the full state vector and feedforward compensation of disturbances to exponentially stabilize the fast dynamics and enforce a pre specified input output behavior independently of the disturbances in the closed loop slow subsystem singular perturbation methods are employed to establish that the discrepancy between the output of the closed loop full order system and the output of the closed loop slow subsystem is proportional to the value of the singular perturbation parameter the stability of the closed loop system is analyzed using lyapunov s direct method and precise conditions that guarantee boundedness of the trajectories for sufficiently small values of the singular perturbation parameter are derived the application of the developed methodology is illustrated through a catalytic continuous stirred tank reactor example modeled as a singularly perturbed system in nonstandard form

variable diffusion in stock market fluctuations abstract we analyze intraday fluctuations in several stock indices to investigate the underlying stochastic processes using techniques appropriate for processes with nonstationary increments the five most actively traded stocks each contains two time intervals during the day where the variance of increments can be fit by power law scaling in time the fluctuations in return within these intervals follow asymptotic bi exponential distributions the autocorrelation function for increments vanishes rapidly but decays slowly for absolute and squared increments based on these results we propose an intraday stochastic model with linear variable diffusion coefficient as a lowest order approximation to the real dynamics of financial markets and to test the effects of time averaging techniques typically used for financial time series analysis we find that our model replicates major stylized facts associated with empirical financial time series we also find that ensemble averaging techniques can be used to identify the underlying dynamics correctly whereas time averages fail in this task our work indicates that ensemble average approaches will yield new insight into the study of financial markets dynamics our proposed model also provides new insight into the modeling of financial markets dynamics in microscopic time scales

hurst exponents markov processes and fractional brownian motion there is much confusion in the literature over hurst exponents recently we took a step in the direction of eliminating some of the confusion one purpose of this

sliding mode control for singularly perturbed systems this paper addresses the design of sliding mode controllers for singularly perturbed systems the full order system is separated into slow and fast subsystems and a sliding mode controller is designed for each subsystem a composite of these controls is then implemented on the full order system it is shown that if specific reaching conditions are satisfied for the reduced order models and an additional condition is met then the reaching conditions are also satisfied for the full order model furthermore sufficient conditions are derived which guarantee that the approximation error between the controlled reduced order response and the controlled full order model response is small a numerical example is given which successfully demonstrates the techniques developed in this paper

fractional calculus and continuous time finance ii the waiting time distribution we complement the theory of tick by tick dynamics of financial markets based on a continuous time random walk ctrw model recently proposed by scalas et al 4 and we point out its consistency with the behaviour observed in the waiting time distribution for bund future prices traded at liffe london