normal bases of ray class fields over imaginary quadratic fields we develop a criterion for a normal basis theorem 2 4 and prove that the singular values of certain siegel functions form normal bases of ray class fields over imaginary quadratic fields other than mathbb q sqrt 1 and mathbb q sqrt 3 theorem 4 2 this result would be an answer for the lang schertz conjecture on a ray class field with modulus generated by an integer 2 remark 4 3

interest rates in quantum finance the wilson expansion and hamiltonian interest rate instruments form a major component of the capital markets the libor market model lmm is the finance industry standard interest rate model for both libor and euribor which are the most important interest rates the quantum finance formulation of the libor market model is given in this paper and leads to a key generalization all the libors for different future times are imperfectly correlated a key difference between a forward interest rate model and the lmm lies in the fact that the lmm is calibrated directly from the observed market interest rates the short distance wilson expansion phys rev 179 1499 1969 of a gaussian quantum field is shown to provide the generalization of ito calculus in particular the wilson expansion of the gaussian quantum field a t x driving the libors yields a derivation of the libor drift term that incorporates imperfect correlations of the different libors the logarithm of libor phi t x is defined and provides an efficient and compact representation of the quantum field theory of the libor market model the lagrangian and feynman path integrals of the libor market model of interest rates are obtained as well as a derivation given by its hamiltonian the hamiltonian formulation of the martingale condition provides an exact solution for the nonlinear drift of the libor market model the quantum finance formulation of the lmm is shown to reduce to the industry standard bruce gatarek musiela jamshidian model when the forward interest rates are taken to be exactly correlated

the f q mock theta function conjecture and partition ranks in 1944 freeman dyson initiated the study of ranks of integer partitions here we solve the classical problem of obtaining formulas for ne n resp no n the number of partitions of n with even resp odd rank thanks to rademacher s celebrated formula for the partition function this problem is equivalent to that of obtaining a formula for the coefficients of the mock theta function f q a problem with its own long history dating to ramanujan s last letter to hardy little was known about this problem until dragonette in 1952 obtained asymptotic results in 1966 g e andrews refined dragonette s results and conjectured an exact formula for the coefficients of f q by constructing a weak maass poincare series whose holomorphic part is q 1f q24 we prove the andrews dragonette conjecture and as a consequence obtain the desired formulas for ne n and no n

a stochastic model for internal hiv dynamics in this paper we analyse a stochastic model representing hiv internal virus dynamics the stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modelling we show that the model established in this paper possesses non negative solutions as this is essential in any population dynamics model we also carry out analysis on the asymptotic behaviour of the model we approximate one of the variables by a mean reverting process and find out the mean and variance of this process numerical simulations conclude the paper

option pricing stochastic volatility singular dynamics and constrained path integrals abstract stochastic volatility models have been widely studied and used in the financial world the heston model heston 1993 7 is one of the best known models to deal with this issue these stochastic volatility models are characterized by the fact that they explicitly depend on a correlation parameter which relates the two brownian motions that drive the stochastic dynamics associated to the volatility and the underlying asset solutions to the heston model in the context of option pricing using a path integral approach are found in lemmens et al 2008 21 while in baaquie 2007 1997 12 13 propagators for different stochastic volatility models are constructed in all previous cases the propagator is not defined for extreme cases 1 it is therefore necessary to obtain a solution for these extreme cases and also to understand the origin of the divergence of the propagator in this paper we study in detail a general class of stochastic volatility models for extreme values 1 and show that in these two cases the associated classical dynamics corresponds to a system with second class constraints which must be dealt with using dirac s method for constrained systems dirac 1958 1967 22 23 in order to properly obtain the propagator in the form of a euclidean hamiltonian path integral henneaux and teitelboim 1992 25 after integrating over momenta one gets an euclidean lagrangian path integral without constraints which in the case of the heston model corresponds to a path integral of a repulsive radial harmonic oscillator in all the cases studied the price of the underlying asset is completely determined by one of the second class constraints in terms of volatility and plays no active role in the path integral

feynman perturbation expansion for the price of coupon bond options and swaptions in quantum finance ii empirical european options on coupon bonds are studied in a quantum field theory model of forward interest rates swaptions are briefly reviewed an approximation scheme for the coupon bond option price is developed based on the fact that the volatility of the forward interest rates is a small quantity the field theory for the forward interest rates is gaussian but when the payoff function for the coupon bond option is included it makes the field theory nonlocal and nonlinear a perturbation expansion using feynman diagrams gives a closed form approximation for the price of coupon bond option a special case of the approximate bond option is shown to yield the industry standard one factor hjm formula with exponential volatility

on twisted forms and relative algebraic k theory this paper introduces a new approach to the study of certain aspects of galois module theory by combining ideas arising from the study of the galois structure of torsors of finite group schemes with techniques coming from relative algebraic k theory

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

relative galois module structure of rings of integers and elliptic functions iii in this paper we consider the problem of describing the galois module structure of rings of integers of abelian extensions of a quadratic imaginary number field k in certain cases by use of elliptic functions we are able to exhibit specific generators of such rings of integers as galois modules the general problem of finding galois integral generators by means of evaluation of transcendental functions has been considered by a number of mathematicians our results represent the first major advance outside the cyclotomic case we recall that the principal cyclotomic results are the hilbert speiser theorem and more generally the following result of leopoldt cf le theorem 1 let e denote an abelian extension of q let f gal e q

stability of a stochastic sir system abstract we propose a stochastic sir model with or without distributed time delay and we study the stability of disease free equilibrium the numerical simulation of the stochastic sir model shows that the introduction of noise modifies the threshold of system for an epidemic to occur and the threshold stochastic value is found

on the time to extinction in recurrent epidemics summary an approximation is derived for the expected time to extinction in a stochastic model for recurrent epidemics numerical illustrations indicate that the approximation is crude but that it has the correct order of magnitude the quasi stationary distribution plays an important role in the derivation approximations for the critical community size and for the persistence threshold are derived comments are made on the classical study by bartlett 1956 1960

environmental variability in a stochastic epidemic model abstract in this paper we investigate the stochastic dynamics of a simple epidemic model incorporating the mean reverting ornstein uhlenbeck process analytically and numerically we define two threshold parameters the stochastic demographic reproduction number r d s and the stochastic basic reproduction number r 0 s to utilize in identifying the stochastic extinction and persistence of the disease we find that the stochastic disease dynamics can be determined by the environment fluctuations which measured by the intensity of volatility and the speed of reversion the larger intensity of volatility or the smaller speed of reversion can suppress the outbreak of the disease the smaller intensity of volatility or the the higher speed of reversion can enhance the outbreak of the disease furthermore via numerical simulations we find that the stochastic model has an endemic stationary distribution which leads to the stochastic persistence of the disease our results show that mean reverting process is a well established way of introducing stochastic environmental noise into biologically realistic population dynamic models

the sensitivity analysis of propagator for path independent quantum finance model abstract quantum finance successfully implements the imperfectly correlated fluctuation of forward interest rates at different maturities by replacing the wiener process with a two dimensional quantum field interest rate derivatives can be priced at a more realistic value under this new framework the quantum finance model requires three main ingredients for pricing the initial forward interest rates the volatility of forward interest rates and the correlation of forward interest rates at different maturities however the hedging strategy only focused on fluctuation of forward interest rates this hedging method is based on the assumption that the propagator the covariance of forward interest rates has an ergodic property since inserting the propagator is the main characteristic that distinguishes quantum finance from the libor market model lmm and the heath jarrow and morton hjm model understanding the impact of propagator dynamics on the price of interest rate derivatives is crucial this research is the first step in developing a hedge strategy with respect to the evolution of the propagator we analyze the dynamics of the propagator from libor futures data and the integrated propagator from zero coupon bond rate data then we study the sensitivity of the implied volatility of caplets and swaptions according to the three dominant dynamics of the propagator and the change of the zero coupon bond option price according to the two dominant dynamics of the integrated propagator

on arithmetic class invariants let f be a number field with ring of integers of and let s be a finite set of places of f assume that s contains the set s of archimedean places of f and write sf for the set of finite places contained in s let os or o when there is no danger of confusion denote the ring of sf integers of f write f for an algebraic closure of f let y be any scheme over spec o suppose thatg is a finite flat commutative group scheme over y of exponent n and let g denote the cartier dual of g let x y be ag torsor and write 0 g y for the trivialg torsor thenox is anog comodule and so it is also anogd module see 12 as an ogd module ox is locally free of rank one and it therefore gives a line bundle m over g set l m m 1 0 then the map h y g pic g l

a quantum anharmonic oscillator model for the stock market abstract a financially interpretable quantum model is proposed to study the probability distributions of the stock price return the dynamics of a quantum particle is considered an analog of the motion of stock price then the probability distributions of price return can be computed from the wave functions that evolve according to schrodinger equation instead of a harmonic oscillator in previous studies a quantum anharmonic oscillator is applied to the stock in liquid market the leptokurtic distributions of price return can be reproduced by our quantum model with the introduction of mixed state and multi potential the trend following dominant market in which the price return follows a bimodal distribution is discussed as a specific case of the illiquid market

hamiltonian and potentials in derivative pricing models exact results and lattice simulations the pricing of options warrants and other derivative securities is one of the great success of financial economics these financial products can be modeled and simulated using quantum mechanical instruments based on a hamiltonian formulation we show here some applications of these methods for various potentials which we have simulated via lattice langevin and monte carlo algorithms to the pricing of options we focus on barrier or path dependent options showing in some detail the computational strategies involved

the minimal length uncertainty and the quantum model for the stock market abstract we generalize the recently proposed quantum model for the stock market by zhang and huang to make it consistent with the discrete nature of the stock price in this formalism the price of the stock and its trend satisfy the generalized uncertainty relation and the corresponding generalized hamiltonian contains an additional term proportional to the fourth power of the trend we study a driven infinite quantum well where information as the external field periodically fluctuates and show that the presence of the minimal trading value of stocks results in a positive shift in the characteristic frequencies of the quantum system the connection between the information frequency and the transition probabilities is discussed finally

a stochastic differential equation sis epidemic model in this paper we extend the classical susceptible infected susceptible epidemic model from a deterministic framework to a stochastic one and formulate it as a stochastic differential equation sde for the number of infectious individuals i t we then prove that this sde has a unique global positive solution i t and establish conditions for extinction and persistence of i t we discuss perturbation by stochastic noise in the case of persistence we show the existence of a stationary distribution and derive expressions for its mean and variance the results are illustrated by computer simulations including two examples based on real life diseases

sensitivity analysis and feedback control of noise induced extinction for competition chemostat model with mutualism abstract we consider a stochastic chemostat model involving two obligate mutualistic species feeding on a limiting substrate it is shown that the spatial arrangement of the random states near the deterministic coexistence equilibrium is enlarged as noise intensity increases more precisely in this paper based on the technique of stochastic sensitivity functions a confidence ellipse of the random states is constructed and a critical value for the noise intensity is established it is shown that a new dynamic phenomenon of noise induced extinction can be observed when the noise intensity passes the critical value we then propose some feedback control strategies which can reduce the size of the confidence ellipse so that to prevent the noise induced extinction

path integral for equities dynamic correlation and empirical analysis abstract this paper develops a model to describe the unequal time correlation between rate of returns of different stocks a non trivial fourth order derivative lagrangian is defined to provide an unequal time propagator which can be fitted to the market data a calibration algorithm is designed to find the empirical parameters for this model and different de noising methods are used to capture the signals concealed in the rate of return the detailed results of this gaussian model show that the different stocks can have strong correlation and the empirical unequal time correlator can be described by the model s propagator this preliminary study provides a novel model for the correlator of different instruments at different times

simulation of coupon bond european and barrier options in quantum finance abstract coupon bond european and barrier options are studied in the framework of quantum finance the prices of european and barrier options are analyzed by generating sample values of the forward interest rates f t x using a two dimensional gaussian quantum field a t x the strong correlations of forward interest rates are described by the stiff propagator of the quantum field a t x using the cholesky decomposition a t x is expressed in terms of white noise the simulation results for european coupon bond and barrier options are compared with approximate formulas which are obtained as power series in the volatility of the forward interest rates the simulation shows that the simulated price deviates from the approximate value for large volatilities the numerical algorithm is flexible and can be used for pricing any kind of option it is shown that the three factor hjm model can be derived from the quantum finance formulation

price volume multifractal analysis and its application in chinese stock markets abstract an empirical research on chinese stock markets is conducted using statistical tools first the multifractality of stock price return series r i r i ln p t 1 ln p t and trading volume variation series v i v i ln v t 1 ln v t is confirmed using multifractal detrended fluctuation analysis furthermore a multifractal detrended cross correlation analysis between stock price return and trading volume variation in chinese stock markets is also conducted it is shown that the cross relationship between them is also found to be multifractal second the cross correlation between stock price p i and trading volume v i is empirically studied using cross correlation function and detrended cross correlation analysis it is found that both shanghai stock market and shenzhen stock market show pronounced long range cross correlations between stock price and trading volume third a composite index r based on price and trading volume is introduced compared with stock price return series r i and trading volume variation series v i r variation series not only remain the characteristics of original series but also demonstrate the relative correlation between stock price and trading volume finally we analyze the multifractal characteristics of r variation series before and after three financial events in china namely price limits reform of non tradable shares and financial crisis in 2008 in the whole period of sample to study the changes of stock market fluctuation and financial risk it is found that the empirical results verified the validity of r

a finite dimensional quantum model for the stock market abstract we present a finite dimensional version of the quantum model for the stock market proposed in c zhang and l huang a quantum model for the stock market physica a 389 2010 5769 our approach is an attempt to make this model consistent with the discrete nature of the stock price and is based on the mathematical formalism used in the case of the quantum systems with finite dimensional hilbert space the rate of return is a discrete variable corresponding to the coordinate in the case of quantum systems and the operator of the conjugate variable describing the trend of the stock return is defined in terms of the finite fourier transform the stock return in equilibrium is described by a finite gaussian function and the time evolution of the stock price directly related to the rate of return is obtained by numerically solving a schrodinger type equation

on some arithmetic properties of siegel functions we deal with several arithmetic properties of the siegel functions which are modular units by modifying the ideas in kubert and lang modular units grundlehren der mathematischen wissenschaften vol 244 spinger heidelberg 1981 we establish certain criterion for determining a product of siegel functions to be integral over mathbb z j we also find generators of the function fields mathcal k x 1 n by examining the orders of siegel functions at the cusps and apply them to evaluate the ramanujan s cubic continued fraction systematically furthermore we construct ray class invariants over imaginary quadratic fields in terms of singular values of j and siegel functions

quantum finance hamiltonian for coupon bond european and barrier options coupon bond european and barrier options are financial derivatives that can be analyzed in the hamiltonian formulation of quantum finance forward interest rates are modeled as a two dimensional quantum field theory and its hamiltonian and state space is defined european and barrier options are realized as transition amplitudes of the time integrated hamiltonian operator the double barrier option for a financial instrument is knocked out terminated with zero value if the price of the underlying instrument exceeds or falls below preset limits the barrier option is realized by imposing boundary conditions on the eigenfunctions of the forward interest rates hamiltonian the price of the european coupon bond option and the zero coupon bond barrier option are calculated it is shown that is general the constraint function for a coupon bond barrier option can to a good approximation be linearized a calculation using an overcomplete set of eigenfunctions yields an approximate price for the coupon bond barrier option which is given in the form of an integral of a factor that results from the barrier condition times another factor that arises from the payoff function

quantum spatial periodic harmonic model for daily price limited stock markets abstract we investigate the behaviors of stocks in daily price limited stock markets by purposing a quantum spatial periodic harmonic model the stock price is considered to be oscillating and damping in a quantum spatial periodic harmonic oscillator potential well a complicated non linear relation including inter band positive correlation and intra band negative correlation between the volatility and trading volume of a stock is numerically derived with the energy band structure of the model concerned the effectiveness of price limit is re examined with some observed characteristics of price limited stock markets in china studied by applying our quantum model

a quantum model for the stock market abstract beginning with several basic hypotheses of quantum mechanics we give a new quantum model in econophysics in this model we define wave functions and operators of the stock market to establish the schrodinger equation for stock price based on this theoretical framework an example of a driven infinite quantum well is considered in which we use a cosine distribution to simulate the state of stock price in equilibrium after adding an external field into the hamiltonian to analytically calculate the wave function the distribution and the average value of the rate of return are shown

a stochastic sirs epidemic model with infectious force under intervention strategies abstract in this paper we extend a classical sirs epidemic model with the infectious forces under intervention strategies from a deterministic framework to a stochastic differential equation sde one through introducing random fluctuations the value of our study lies in two aspects mathematically by using the markov semigroups theory we prove that the reproduction number r 0 s can be used to govern the stochastic dynamics of sde model if r 0 s 1 under mild extra conditions the sde system has a disease free absorbing set which means the extinction of disease with probability one if r 0 s 1 under mild extra conditions it has an endemic stationary distribution which leads to the stochastical persistence of the disease epidemiologically we find that random fluctuations can suppress disease outbreak which can provide us some useful control strategies to regulate disease dynamics

threshold behavior in a stochastic sir epidemic model with logistic birth abstract in this paper we consider a stochastic sir epidemic model with logistic birth by using the stochastic lyapunov function method we show that the stochastic basic reproduction number r 0 s can be used to determine the threshold dynamics of the stochastic system if r 0 s 1 we establish sufficient conditions for the existence of a stationary distribution of the positive solutions to the model while if r 0 s 1 under some extra conditions we obtain sufficient conditions for extinction of the disease finally some examples and numerical simulations are provided to illustrate the theoretical results

simulation of nonlinear interest rates in quantum finance libor market model abstract the simulation of the libor market model lmm is extensively studied in the framework of quantum finance the imperfectly correlated libor rates are simulated based on a gaussian quantum field and a recursion equation of nontrivial stochastic drift the libor options are studied using both the simulation method and the analytical formula the caplet price of simulation is compared with black s caplet formula which can be exactly derived from the lmm the invariance of caplet price for different forward bond numeraire is verified by using the simulation the simulation results for coupon bond options and swaptions are compared with the approximate price which are limited for the reason that the approximate price is derived using the small volatility expansion the simulation method is shown to have great potential in the application of pricing interest rate instruments

a stochastic sirs epidemic model with nonlinear incidence rate a stochastic sirs model with ratio dependent incidence rate is developed the global dynamics of the deterministic model is shown the stochastic dynamics of the sde model is given the existence of a unique stationary distribution of the sde model is displayed in this paper we investigate the global dynamics of a general sirs epidemic model with a ratio dependent incidence rate and its corresponding stochastic differential equation version for the deterministic model we show that the basic reproduction number r0 determines whether there is an endemic outbreak or not if r0 1 the disease persists for the stochastic model we show that its related reproduction number r0s can determine whether there is a unique disease free stationary distribution or a unique endemic stationary distribution in addition we provide analytic results regarding the stochastic boundedness and permanence extinction one of the most interesting findings is that random fluctuations introduced in our stochastic model can suppress disease outbreak which can provide us some useful control strategies to regulate disease dynamics

a quantum statistical approach to simplified stock markets we use standard perturbation techniques originally formulated in quantum statistical mechanics in the analysis of a toy model of a stock market which is given in terms of bosonic operators in particular we discuss the probability of transition from a given value of the portfolio of a certain trader to a different one this computation can also be carried out using some kind of feynman graphs adapted to the present context

quantum brownian motion model for the stock market abstract it is believed by the majority today that the efficient market hypothesis is imperfect because of market irrationality using the physical concepts and mathematical structures of quantum mechanics we construct an econophysical framework for the stock market based on which we analogously map massive numbers of single stocks into a reservoir consisting of many quantum harmonic oscillators and their stock index into a typical quantum open system a quantum brownian particle in particular the irrationality of stock transactions is quantitatively considered as the planck constant within heisenberg s uncertainty relationship of quantum mechanics in an analogous manner we analyze real stock data of shanghai stock exchange of china and investigate fat tail phenomena and non markovian behaviors of the stock index with the assistance of the quantum brownian motion model thereby interpreting and studying the limitations of the classical brownian motion model for the efficient market hypothesis from a new perspective of quantum open system dynamics

pseudo hermitian formulation of the quantum black scholes hamiltonian abstract we show that the non hermitian black scholes hamiltonian and its various generalizations are pseudo hermitian the metric operator is explicitly constructed for this class of hamiltonians it is also shown that the effective black scholes hamiltonian and its partner form a pseudo supersymmetric system

valuation of caps and swaptions under a stochastic string model abstract we develop a gaussian stochastic string model that provides exact closed form expressions for the prices and hedging portfolios of caps and swaptions under certain conditions our pricing expressions reduce to black 1976 formulas we also propose a stochastic string libor market model that generalizes the models of brace et al 1997 and longstaff et al 2001a we provide a possible explanation for some problems involved in the relative valuation of these derivatives we also attain the observational equivalence of kerkhof and pelsser 2002 and prove that in our setup the models proposed by brace et al 1997 and longstaff et al 2001a are more parsimonious than stated in the original papers