time vs ensemble averages for nonstationary time series abstract we analyze whether sliding window time averages applied to stationary increment processes converge to a limit in probability the question centers on averages correlations and densities constructed via time averages of the increment x t t x t t x t e g x t t ln p t t p t in finance and economics where p t is a price and the assumption is that the increment is distributed independently of t we apply tchebychev s theorem to the construction of statistical ensembles and then show that the convergence in probability condition is not satisfied when applied to time averages of functions of stationary increments we further show that tchebychev s theorem provides the basis for constructing approximate ensemble averages and densities from a single historic time series where as in fx markets the series shows a definite statistical periodicity the convergence condition is not satisfied strongly enough for densities and certain averages but is well satisfied by specific averages of direct interest rates of convergence cannot be established independently of specific models however our analysis shows how to decide which empirical averages to avoid and which ones to construct

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

scaling in currency exchange we study the scaling behavior in currency exchange rates our results suggest that they satisfy scaling with an exponent close to 0 5 but that it differs qualitatively from that of a simple random walk indeed price variations cannot be considered as independent variables and subtle correlations are present furthermore we introduce a novel statistical analysis for economic data which makes the physical properties of a signal more evident and eliminates the systematic effects of time periodicity

from dispersionless to soliton systems via weyl moyal like deformations the formalism of quantization deformation is reviewed and the weyl moyal like deformation is applied to systematic construction of the field and lattice integrable soliton systems from poisson algebras of dispersionless systems

weyl manifolds and deformation quantization this paper deals with non commutative objects based on the weyl algebra from the differential geometric point of view we propose to extend familiar notions from manifolds to non commutative objects in this algebraic approach the basic object is not a point of a space but certain sections of bundles

chapman kolmogorov equation for non markovian shift invariant measures we study the class c of probability measures invariant with respect to the shift transformation on k z where k is a finite set of integers which satisfies the chapman kolmogorov equation for a given stochastic matrix we construct a dense subset of measures in c distinct from the markov measure when is irreduable and aperiodic these measures are ergodic but not weary mixing we show that the set of measures with infinite memory is g dense in c and that the markov measure is the unique measure which maximizes the kolmogorov sinai k s entropy in c we give examples of ergodic measures in c with zero entropy

gauge theories in noncommutative homogeneous kahler manifolds we construct a gauge theory on a noncommutative homogeneous kahler manifold where we employ the deformation quantization with separation of variables for kahler manifolds formulated by karabegov a key point in this construction is to obtaining vector fields which act as inner derivations for the deformation quantization we give an explicit construction of this gauge theory on noncommutative cpn and noncommutative chn

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

formal series of generalized functions and their application to deformation quantization foundations of the formal series calculus in deformation quantization are discussed several classes of continuous linear functionals over algebras applied in classical and quantum physics are introduced the notion of non negativity in formal series calculus is proposed problems with defining quantum states over the set of formal series are analyzed foundations of the formal series calculus in deformation quantization are discussed several classes of continuous linear functionals over algebras applied in classical and quantum physics are introduced the notion of non negativity in formal series calculus is proposed problems with defining quantum states over the set of formal series are analyzed

on higher order relations in fedosov supermanifolds higher order relations existing in normal coordinates between affine extensions of the symplectic curvature tensor and basic objects for any fedosov supermanifolds are derived representation of these relations in general coordinates is discussed

tangent bundle geometry lagrangian dynamics various aspects of the differential geometry of the tangent bundle of a differentiable manifold are examined and the results applied to time independent lagrangian dynamics it is shown that a certain type 1 1 tensor field which is part of the intrinsic geometry of a tangent bundle being a tensorial equivalent of the projection map of tangent vectors plays a role in lagrangian theory scarcely less important than that of the canonical one form on a cotangent bundle in hamiltonian theory recent results in lagrangian theory are interpreted from this new viewpoint

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

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

formal symplectic groupoid of a deformation quantization we give a self contained algebraic description of a formal symplectic groupoid over a poisson manifold m to each natural star product on m we then associate a canonical formal symplectic groupoid over m finally we construct a unique formal symplectic groupoid with separation of variables over an arbitrary kahler poisson manifold

a formality theorem for hochschild chains abstract we prove tsygan s formality conjecture for hochschild chains of the algebra of functions on an arbitrary smooth manifold m using the fedosov resolutions proposed in math qa 0307212 and the formality quasi isomorphism for hochschild chains of r y 1 y d proposed in paper math qa 0010321 by shoikhet this result allows us to describe traces on the quantum algebra of functions on an arbitrary poisson manifold

fedosov products and quantum momentum maps the purpose of this paper is to study various aspects of star products on a symplectic manifold related to the fedosov method by introducing the notion of quantum exponential maps we give a characterization of fedosov connections as an application a geometric realization is obtained for the equivalence between an arbitrary product and a fedosov one every fedosov product is shown to be a vey product consequently we find that every product is equivalent to a vey product a classical result of lichnerowicz quantization of a hamiltonian g space and in particular quantum momentum maps are studied lagrangian submanifolds are also studied under a deformation quantization

hamiltonian systems on k cosymplectic manifolds the hamiltonian framework on symplectic and cosymplectic manifolds is extended in order to consider classical field theories to do this the notion of k cosymplectic manifold is introduced and a suitable hamiltonian formalism is developed so that the field equations for scalar and vector hamiltonian functions are derived

almost tangent and cotangent structures in the large we examine some global properties of integrable almost tangent and cotangent manifolds in particular we extend several results which essentially characterize tangent and cotangent bundles as respectively regular almost tangent and cotangent structures

markov processes hurst exponents and nonlinear diffusion equations with application to finance abstract we show by explicit closed form calculations that a hurst exponent h 1 2 does not necessarily imply long time correlations like those found in fractional brownian motion fbm we construct a large set of scaling solutions of fokker planck partial differential equations pdes where h 1 2 thus markov processes which by construction have no long time correlations can have h 1 2 if a markov process scales with hurst exponent h 1 2 then it simply means that the process has nonstationary increments for the scaling solutions we show how to reduce the calculation of the probability density to a single integration once the diffusion coefficient d x t is specified as an example we generate a class of student t like densities from the class of quadratic diffusion coefficients notably the tsallis density is one member of that large class the tsallis density is usually thought to result from a nonlinear diffusion equation but instead we explicitly show that it follows from a markov process generated by a linear fokker planck equation and therefore from a corresponding langevin equation having a tsallis density with h 1 2 therefore does not imply dynamics with correlated signals e g like those of fbm a short review of the requirements for fbm is given for clarity and we explain why the usual simple argument that h 1 2 implies correlations fails for markov processes with scaling solutions finally we discuss the question of scaling of the full green function g x t x t of the fokker planck pdes

some geometric structures associated with a k symplectic manifold a canonical connection is attached to any k symplectic manifold we study the properties of this connection and its geometric applications to k symplectic manifolds in particular we prove that under some natural assumptions any k symplectic manifold admits an ehresmann connection discuss some corollaries of this result and find vanishing theorems for characteristic classes on a k symplectic manifold

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 in the norwegian stock market the main objective of this paper is to investigate the validity of the much used assumptions that stock market returns follow a random walk and are normally distributed for this purpose the concepts of chaos theory and fractals are applied two independent models are used to examine price variations in the norwegian and us stock markets the first model used is the range over standard deviation or r s statistic which tests for persistence or antipersistence in the time series both the norwegian and us stock markets show significant persistence caused by long run memory components in the series in addition an average non periodic cycle of four years is found for the us stock market these results are not consistent with the random walk assumption the second model investigates the distributional scaling behaviour of the high frequency price variations in the norwegian stock market the results show a remarkable constant scaling behaviour between different time intervals this means that there is no intrinsic time scale for the dynamics of stock price variations the relationship can be expressed through a scaling exponent describing the development of the distributions as the time scale changes this description may be important when constructing or improving pricing models such that they coincide more closely with the observed market behaviour the empirical distributions of high frequency price variations for the norwegian stock market is then compared to the levy stable distribution with the relevant scaling exponent found by using the r s and distributional scaling analysis good agreement is found between the levy profile and the empirical distribution for price variations less than 6 standard deviations covering almost three orders of magnitude in the data for probabilities larger than 6 standard deviations there seem to be an exponential fall off from the levy profile in the tails which indicates that the second moment may be finite

indecomposable quadratic bundles of rank 4n over the real affine plane keywords real affine plane quadratic space extended from subring hermitian bundle non extended quadratic spaces indecomposable quadratic bundles indecomposable quadratic spaces reference cma article 1983 001doi 10 1007 bf02095998 record created on 2008 12 16 modified on 2016 08 08

the canonical connection of a bi lagrangian manifold we prove that the canonical connection of a bi lagrangian manifold introduced by hess is a levi civita connection showing that a bi lagrangian manifold i e a symplectic manifold endowed with two transversal lagrangian foliations is endowed with a canonical semi riemannian metric

quadratic spaces over laurent extensions of dedekind domains let abea dedekind domain in which 2 is invertible we show in this paper that any isotropic quadratic space over r t 7 1 is isometric to n ii q q hip with rank p n where if p denotes the hyperbolic space 1 some preliminary results in this section we prove some lemmas which will be used in the later sections we assume throughout the paper that in all the rings considered 2 is invertible if a q2 are quadratic spaces over r a j tq2 denotes the quadratic space a r r t t x l t a2 r r t t x over r t t x

twisted deformation quantization of algebraic varieties abstract let x be a smooth algebraic variety over a field k containing the real numbers we introduce the notion of twisted associative resp poisson deformation of the structure sheaf o x these are stack like versions of usual deformations we prove that there is a twisted quantization operation from twisted poisson deformations to twisted associative deformations which is canonical and bijective on gauge equivalence classes this result extends work of kontsevich and our own earlier work on deformation quantization of algebraic varieties

explicit formulas for noncommutative deformations of cpn and chn we give explicit expressions of a deformation quantization with separation of variables for cpn and chn this quantization method is one of the ways to perform a deformation quantization of kahler manifolds which is introduced by karabegov star products are obtained as explicit formulas in all order in the noncommutative parameter we also give the fock representations of the noncommutative cpn and chn

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

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

cotangent bundle quantization entangling of metric and magnetic field for manifolds of noncompact type endowed with an affine connection for example the levi civita connection and a closed 2 form magnetic field we define a hilbert algebra structure in the space and construct an irreducible representation of this algebra in this algebra is automatically extended to polynomial in momenta functions and distributions under some natural conditions this algebra is unique the non commutative product over is given by an explicit integral formula this product is exact not formal and is expressed in invariant geometrical terms our analysis reveals that this product has a front which is described in terms of geodesic triangles in the quantization of functions induces a family of symplectic reflections in and generates a magneto geodesic connection on this symplectic connection entangles on the phase space level the original affine structure on and the magnetic field in the classical approximation the 2 part of the quantum product contains the ricci curvature of and a magneto geodesic coupling tensor