biorthogonal systems and the infinite companion matrix abstract a nonsymmetric analogue of a gram matrix is used to represent the infinite companion matrix introduced by the author in 4 as a projection operator in h2 the method consists in constructing for a given set 0 fn 1 of linearly independent vectors another set g0 gn 1 which lies in a given space and satisfies fi gk ik the infinite companion matrix may then be interpreted as an interpolation operator connections with lyapunov type equations are also explained

an algebraic method to solve the minimal partial realization problem for scalar sequences abstract we give an algebraic method to construct all minimal partial realizations of a finite sequence of scalar numbers we use the minimal number of parameters to characterize all these minimal partial realizations three examples are worked out to illustrate the method

generalized inversion of finite rank hankel and toeplitz operators with rational matrix symbols abstract the goal of the paper is a generalized inversion of finite rank hankel operators and hankel or toeplitz operators with block matrices having finitely many rows to attain it a left coprime fractional factorization of a strictly proper rational matrix function and the bezout equation are used generalized inverses of these operators and generating functions for the inverses are explicitly constructed in terms of the fractional factorization

sine gordon theory in a semi strip abstract initial boundary value problems for sine gordon and complex sine gordon equations in a semi strip are treated the evolution of the weyl function and a uniqueness result are obtained for the complex sine gordon equation the evolution of the weyl function as well as an existence result and a procedure to recover solution are given for the sine gordon equation it is shown that for a wide class of examples the solutions of the sine gordon equation are unbounded in the quarter plane

nonlinear systems related to an arbitrary space time dependence of the spectral transform a general algebraic analytic scheme for the spectral transform of solutions of nonlinear evolution equations is proposed this allows one to give the general nonlinear evolution corresponding to an arbitrary time and space dependence of the spectral transform in general nonlinear and with nonanalytic dispersion relations the main theorem is that the compatibility conditions always give a true nonlinear evolution because it can always be written as an identity between polynomials in the spectral variable k this general result is then used to obtain first a method to generate a new class of solutions to the nonlinear schrodinger equation and second to construct the spectral transform theory for solving initial boundary value problems for resonant wave coupling processes like self induced transparency in two level media or stimulated brillouin scattering of plasma waves or else stimulated raman scattering in nonlinear optics etc

the group inverse of the transformation j x ax xb abstract the group inverse j of the sylvester transformation j x ax xb is provided that it exists represented in polynomial form j y ij c ij a i yb j in terms of the minimal polynomials of a and b in one of the representations the coefficient matrix c ij is expressed with the help of a new class of derived hankel matrices

inversion des matrices de hankel abstract hankel determinants can be viewed as special schur symmetric functions this provides without computations of determinants or matrices most of the algebra of hankel determinants and explains their links with other classical fields

a new finite difference method for pricing and hedging fixed income derivatives we propose a second order accurate numerical finite difference method to replace the classical schemes used to solve pdes in financial engineering we name it modified fully implicit method the motivation for doing so stems from the accuracy loss while trying to stabilize the solution via the up wind scheme in the convective term as well as the fact that spurious oscillations solutions occur when volatilities are low this is actually the range that is commonly observed in interest rate markets unlike the classical schemes our method covers the whole spectrum of volatilities in the interest rate dynamics we obtain analytical and numerical results for pricing and hedging a zero coupon bond and an asian interest rate option in the case of the asian option we compare the realistic discrete compounding interest rate scheme associated with the modified fully implicit method with the continuous compounding scheme often exploited in the literature obtaining relative discrepancies between prices exceeding 50 this indicates that the former scheme is more appropriate then the latter to price more complicate derivatives than straight bonds

lyapunov equations for companion matrices abstract matrix equations of the form c a x xc t b y where ca and cb are companion matrices are considered in a first step 1 is reduced to an equation for which the solution is hankel and the right side has rank 2 at most in a second step a solvability condition and a general solution are given in terms of the polynomials a b and the solution of some polynomial congruence an application to resultant matrix equations is included

hankel matrices generated by the markov parameters of rational functions abstract finite hankel matrices s i j are considered for which s i are the markov parameters of a given rational function p a the problems of inversion and kernel description are reduced to the problem of solving some polynomial congruences in terms of p and a an application to hankel matrices of the form y t c i j a x where c a is a companion matrix is included

lyapunov bezout and hankel abstract given a polynomial f of degree n we denote by c its companion matrix and by s the truncated shift operator of order n we consider lyapunov type equations of the form x sxc w and x cxs w we derive some properties of these equations which make it possible to characterize bezoutian matrices as solutions of the first equation with suitable right hand sides w similarly for hankel and the second equation and to write down explicit expressions for these solutions this yields explicit factorization formulae for polynomials in c for the schur cohn matrix and for matrices satisfying certain intertwining relations as well as for bezoutian matrices

solution of the boundary value problem for the integrable discrete srs system on the semi line the complete solution of the initial boundary value problem for the integrable discrete version of the stimulated raman scattering system on the semi line is constructed by means of the inverse spectral transform the spectral data obey a riccati time evolution equation which allows for soliton generation out of a medium initially at rest it is also proved that the construction of the solution at any finite distance actually results in solving an algebraic system

note on products of bezoutians and hankel matrices abstract a quantitative theorem due to fiedler and ptak on products of bezoutians and their inverses is extended to a more general case using as the main tool an alternative to fiedler s assertion on the quasidirect decomposition of hankel matrices

a series form solution for pricing variance and volatility swaps with stochastic volatility and stochastic interest rate abstract in this paper we present analytical pricing formulae for variance and volatility swaps when both of the volatility and interest rate are assumed to be stochastic and follow a cir cox ingersoll ross process forming a heston cir hybrid model the solutions are written in a series form with a theoretical proof of their convergence ensuring the accuracy of the determined swap prices the application of the formulae in practice is also demonstrated through the designed numerical experiments

hankel matrices and the infinite companion abstract a formula of barnett type relating the bezoutian b f g to the hankel matrix h g f is extended to rectangular bezoutians the proof is based on an interesting relation between the family of all hankel matrices corresponding to the markov parameters of g f and the infinite companion matrix corresponding to f

a numerical method for pricing discrete double barrier option by legendre multiwavelet in this article a fast numerical algorithm for pricing discrete double barrier option is presented according to blackscholes model the price of option in each monitoring date can be evaluated by a recursive formula upon the heat equation solution these recursive solutions are approximated by using legendre multiwavelets as orthonormal basis functions and expressed in operational matrix form the most important feature of this method is that its cpu time is nearly invariant when monitoring dates increase besides the rate of convergence of presented algorithm was obtained the numerical results verify the validity and efficiency of the numerical method

elbow scattering and inverse scattering applications to lkdv and kdv it is well known that many integrable partial differential equations are consistency conditions of a lax pair of linear differential equations whose variables x t can be considered as coordinates in a plane the standard inverse scattering method and the standard direct linearization methods are related to the scattering problem defined on the x axis in a previous paper the scattering problem defined on the t axis was considered here the scattering on an unbounded line of the plane an elbow is defined and it is shown how it can yield new reconstruction methods for the solutions of the consistency condition the results are neither as complete or as much interesting as the standard ist but their derivation shows that defining the wanted class of solutions is crucial and it helps understanding why the standard ist is so simple

inverse problems for hankel and toeplitz matrices abstract the problem is considered how to construct a toeplitz or hankel matrix a from one or two equations of the form au g the general solution is described explicitly special cases are inverse spectral problems for hankel and toeplitz matrices and problems from the theory of orthogonal polynomials

resultant matrices and inversion of bezoutians abstract the goal of this paper is to construct inverses of toeplitz and hankel bezoutians we obtain quite explicit formulas which are also specified to cases where the bezoutians possess additional symmetries the construction relies on a result about the nullspace of generalized resultant matrices which can be described by means of the solutions of corresponding bezout equations

efficient and fast numerical method for pricing discrete double barrier option by projection method in this paper we introduce a new and considerably fast numerical method based on projection method in pricing discrete double barrier option according to the blackscholes framework the price of option in each monitoring dates is the solution of well known partial differential equation that can be expressed recursively upon the heat equation solution these recursive solutions are approximated by projection method and expressed in operational matrix form the most important advantage of this method is that its computational time is nearly fixed against monitoring dates increase afterward in implementing projection method we use legendre polynomials as an orthogonal basis finally the numerical results show the validity and efficiency of presented method in comparison with some others

numerical valuation of discrete double barrier options in the present paper we explore the problem for pricing discrete barrier options utilizing the black scholes model for the random movement of the asset price we postulate the problem as a path integral calculation by choosing approach that is similar to the quadrature method thus the problem is reduced to the estimation of a multi dimensional integral whose dimension corresponds to the number of the monitoring dates we propose a fast and accurate numerical algorithm for its valuation our results for pricing discretely monitored one and double barrier options are in agreement with those obtained by other numerical and analytical methods in finance and literature a desired level of accuracy is very fast achieved for values of the underlying asset close to the strike price or the barriers the method has a simple computer implementation and it permits observing the entire life of the option

the zakharov shabat spectral problem on the semi line hilbert formulation and applications the inverse spectral transform for the zakharov shabat equation on the semi line x 0 is reconsidered as a hilbert problem the boundary data induce an essential singularity as k to one of the basic solutions then solving the inverse problem means solving a hilbert problem with particular prescribed behaviour it is demonstrated that the direct and inverse problems are solved in a consistent way as soon as the spectral transform vanishes as 1 k at infinity in the whole upper half plane where it may possess single poles and is continuous and bounded on the real k axis the method is applied to stimulated raman scattering and sine gordon light cone for which it is demonstrated that time evolution conserves the properties of the spectral transform

numerical method for discrete double barrier option pricing with time dependent parameters in this paper the main purpose is pricing of discrete double barrier option under black scholes framework with time dependent parameters by some conventional transforms in each monitoring interval the problem is reduced to well known black scholes partial differential equations with convenient constant coefficient such that the solution can be expressed recursively upon the heat equation solution finally a numerical method is proposed to compute the obtained recursive formula efficiently also the greeks of contract are calculated