american fractional lookback options valuation and premium decomposition this paper deals with valuation and premium decomposition of american fractional lookback options written on dividend paying assets for which exact formulas are unknown except for the perpetual case via a pde approach we derive laplace transforms of the values of lookback call and put options which can be decomposed into the associated european values plus the early exercise premiums using abelian theorems of laplace transforms we characterize asymptotic behaviors of the early exercise boundaries at a time to close to expiration and at infinite time to expiration based on the gaver stehfest inversion method combined with the newton method we develop a fast and accurate algorithm for computing both the option value and the early exercise boundary numerical analysis reveals some notable features of the american fractional lookback options

bounds for the zeros of polynomials from eigenvalues and singular values of some companion matrices abstract let a monic polynomial pn x xn a1xn 1 an aj plane1d 49e j 1 2 n be given bounds for the eigenvalues and the computation of the singular values of some special companion matrices give bounds for the zeros of the polynomial pn which improve classical and recently found ones also some new bounds for the zeros are given

the nilpotent centralizer method for spectrally arbitrary patterns this paper introduces a new criterion for an n n sign pattern respectively zero nonzero pattern a to be spectrally arbitrary that is to have the property that for each monic real polynomial r x of degree n there exists a matrix with sign pattern respectively zero nonzero pattern a that has r x as its characteristic polynomial to date analytic properties of a certain polynomial map associated with a have been used to prove that a is spectrally arbitrary in this paper we derive a method that uses the algebraic structure of a certain nilpotent matrix to show that a sign pattern respectively zero nonzero pattern is spectrally arbitrary

valuation of stock loans with regime switching this paper is concerned with stock loan valuation in which the underlying stock price is dictated by geometric brownian motion with regime switching the stock loan pricing is quite different from that for standard american options because the associated variational inequalities may have infinitely many solutions in addition the optimal stopping time equals infinity with positive probability variational inequalities are used to establish values of stock loans and reasonable values of critical parameters such as loan sizes loan rates and service fees in terms of certain algebraic equations numerical examples are included to illustrate the results

exponential splitting for nonautonomous linear discrete time systems in banach spaces in this paper we consider some concepts of exponential splitting for nonautonomous linear discrete time systems these concepts are generalizations of some well known concepts of uniform and nonuniform exponential dichotomies connections between these concepts are presented and some illustrating examples prove that these are distinct

potentially nilpotent patterns and the nilpotent jacobian method a nonzero pattern is a matrix with entries in 0 a pattern is potentially nilpotent if there is some nilpotent real matrix with nonzero entries in precisely the entries indicated by the pattern we develop ways to construct some potentially nilpotent patterns including some balanced tree patterns we explore the index of some of the nilpotent matrices constructed and observe that some of the balanced trees are spectrally arbitrary using the nilpotent jacobian method inspired by an argument of pereira nilpotent matrices and spectrally arbitrary sign patterns electron j linear algebra 16 2007 235 we uncover a feature of the nilpotent jacobian method in particular we show that if n is the nilpotent matrix employed in this method to show that a pattern is a spectrally arbitrary pattern then n must have full index

an ergodic theorem in the qualitative behavior of non linear semiflows abstract take to be a continuous semiflow on the locally compact metric space and let a be a family of possibly unbounded densely defined closed operators on the banach space x we show that the solutions of the variational equation u a t u t r have a uniform exponential decay if and only if there exist p 1 and a sequence of positive real numbers n n n with n satisfying lim n n n 0 n x p d 0 uniformly w r t for all x x where t t is a strongly continuous cocycle solving in the classical sense the above variational equation thus we generalize some known results obtained by datko and pazy

far field boundary conditions for black scholes equations in this paper we consider the partial differential equations approach for valuing european style options in order to solve the equations numerically by finite difference or finite element methods it is necessary to introduce an artificial boundary in order to make the computational domain bounded we derive pointwise bounds for the error caused by various boundary conditions imposed on the artificial boundary the error estimates can be used to determine a priori a suitable location for the artificial boundary in terms of a given error tolerance

lyapunov functions for strong exponential contractions abstract our main aim is to give a complete characterization of an exponential contraction in terms of lyapunov sequences for discrete time or lyapunov functions for continuous time in particular we obtain inverse theorems giving explicitly a quadratic lyapunov function for each exponential contraction we consider the general cases of 1 nonautonomous dynamics either obtained from a product of linear operators or from a nonautonomous differential equation respectively for discrete and continuous time 2 nonuniform exponential contractions in which the uniform stability is replaced by a nonuniform stability 3 strong exponential behavior in the sense that we have simultaneously lower and upper contraction bounds we emphasize that the two last properties are ubiquitous in the context of ergodic theory and in particular for almost all trajectories of any measure preserving flow which justifies completely the study of this general type of exponential behavior as a nontrivial application of our work we establish the robustness of any strong nonuniform exponential contraction that is the persistence of the asymptotic stability of a strong nonuniform exponential contraction under sufficiently small linear perturbations

valuing finite lived russian options abstract this paper deals with the valuation of the russian option with finite time horizon in the framework of the black scholes merton model on the basis of the pde approach to a parabolic free boundary problem we derive laplace transforms of the option value the early exercise boundary and some hedging parameters using abelian theorems of laplace transforms we characterize the early exercise boundary at a time to close to expiration as well as the well known perpetual case in a unified way furthermore we obtain a symmetric relation in the perpetual early exercise boundary combining the gaver stehfest inversion method and the newton method we develop a fast algorithm for computing both the option value and the early exercise boundary in the finite time horizon

construction of matrices with a given graph and prescribed interlaced spectral data abstract a result of duarte linear algebra appl 113 1989 173 182 asserts that for real 1 n 1 n 1 with 1 1 2 2 n 1 n and each tree t on n vertices there exists an n n real symmetric matrix a whose graph is t such that a has eigenvalues 1 2 n and the principal submatrix obtained from a by deleting its last row and column has eigenvalues 1 n 1 this result is extended to connected graphs through the use of the implicit function theorem

convexity of the optimal stopping boundary for the american put option we show that the optimal stopping boundary for the american put option is convex in the standard black scholes model the methods are adapted from ice melting problems and rely upon studying the behavior of level curves of solutions to certain parabolic differential equations

on convergence of laplace inversion for the american put option under the cev model in this paper we study the convergence of the inverse laplace transform for valuing american put options when the dynamics of the risky asset is governed by the constant elasticity of variance cev model the cev model is one popular alternative of the black scholes model to describe well the real financial market we calculate various coefficients explicitly and prove that the inverse laplace transform converges absolutely using the properties of whittaker functions

spectrally and inertially arbitrary sign patterns we introduce some n by n sign patterns which allow for arbitrary spectrum and hence also arbitrary inertia consequently we demonstrate that some known inertially arbitrary patterns are in fact spectrally arbitrary we demonstrate that all inertially arbitrary patterns of order 3 are spectrally arbitrary and classify all spectrally arbitrary patterns of order 3 we illustrate that in general the class of spectrally arbitrary patterns is distinct from the inertially arbitrary patterns and present some observations about inertially arbitrary patterns

asymptotic expansion of solutions to the black scholes equation arising from american option pricing near the expiry our aim in this paper is to approximate the price of an american call option written on a dividend paying stock close to expiry using an asymptotic analytic approach we use the heat equation equivalent of the black scholes partial differential equation defined on an unbounded spatial domain and decompose it into inner and outer problems we extend the idea presented in h han and x wu a fast numerical method for the black scholes equation of american options siam journal on numerical analysis 41 6 2003 2081 2095 in which a weakly singular memory type transparent boundary condition tbc is obtained for the special case that the initial condition is equal to zero we first derive this tbc in the general case and then focus on the outer problem in conjunction with an equivalent non singular version of the tbc dubbed etbc which is more tractable for analytical purposes we then obtain the general solution of the outer problem in series form based on the repeated integrals of the complementary error function which also satisfies the introduced etbc as the next step using the machinery of poincare asymptotic expansion and taking time to expiry as the expansion parameter we find the general term of this series in closed form when the risk free interest rate r is less than the dividend yield we also obtain the first five terms in the opposite case r in a systematic manner we also prove the convergence properties of the obtained series rigorously under some general conditions our numerical experiments based on the obtained asymptotic series demonstrate the applicability and effectiveness of the results in valuation of a wide range of american option problems

lyapunov type characterization of hyperbolic behavior abstract we give a complete characterization of the uniform hyperbolicity and nonuniform hyperbolicity of a cocycle with values in the space of bounded linear operators acting on a hilbert space in terms of the existence of appropriate quadratic forms our work unifies and extends many results in the literature by considering the general case of not necessarily invertible cocycles acting on a hilbert space over an arbitrary invertible dynamics as a nontrivial application of we study the persistence of hyperbolicity under small linear perturbations

sets of refined inertias of zero nonzero patterns abstract a set h n of refined inertias for zero nonzero patterns is introduced that is analogous to the set h n previously considered for sign patterns for n 3 and 4 a complete characterization of irreducible zero nonzero patterns that allow or require h n is given and each zero nonzero pattern that allows h n has a signing that allows h n in contrast for n 5 a family of irreducible zero nonzero patterns is given that allows h n but for which no one signing allows h n

spectrally arbitrary zero nonzero patterns and field extensions abstract an n n matrix pattern is said to be spectrally arbitrary over a field f provided for every monic polynomial p t of degree n with coefficients from f there exists a matrix with entries from f in the given pattern that has characteristic polynomial p t let e f k be an extension of fields it is natural to ask whether a pattern that is spectrally arbitrary over f must also be spectrally arbitrary over e or k in this article it is shown that if f is dense in k and k is a complete metric space then any spectrally arbitrary or relaxed spectrally arbitrary pattern over f is relaxed spectrally arbitrary over k it is also established that if e is an algebraically closed subfield of a field f then any spectrally arbitrary pattern over f is spectrally arbitrary over e the 2n conjecture and the superpattern conjecture are explored over fields other than the real numbers in particular examples are provided to show that the superpattern conjecture is false over the field with 3 elements

a multiquadric quasi interpolations method for cev option pricing model abstract the pricing of option contracts when the underlying process follows the constant elasticity of variance cev model is considered for cev european options the closed form solutions involve the non central chi square distribution whose computations by the current literatures are rather unstable and extremely expensive based on multiquadric quasi interpolation methods this study suggests a stable and fast numerical algorithm for cev option pricing model the method is confirmed to be a multinomial tree in which the underlying variable moves from its initial value to an infinity of possible values of the next time step the probabilities in the associated tree are ensured to be positive which is a sufficient condition for stability and convergence the method is flexible since it is simple to implement with the nonuniform knots moreover the method is easy to value the greek letters which are important parameters in financial engineering as the multiquadric function is infinitely continuously differentiable besides the method does not require solving a resultant full matrix the ill conditioning problem arising when using the radial basis functions as a global interpolant can be avoided numerical experiments imply that the method is highly effective to calculate the stock options and its greeks under the cev model

cev asymptotics of american options abstract the constant elasticity of variance cev model is a practical approach to option pricing by fitting to the implied volatility smile its application to american style derivatives however poses analytical and numerical challenges by taking the laplace carson transform lct to the free boundary value problem characterizing the option value function and the early exercise boundary the analytical result involves confluent hyper geometric functions thus the numerical computation could be unstable and inefficient for certain set of parameter values we solve this problem by an asymptotic approach to the american option pricing problem under the cev model we demonstrate the use of the proposed approach using perpetual and finite time american puts

a proof of the tn conjecture centralizers jacobians and spectrally arbitrary sign patterns we prove the so called tn conjecture for every real monic polynomial p x of degree n 2 there exists an n by n matrix with sign pattern tn 0 0 0 0 0 0 0 0 whose characteristic polynomial is p x the proof converts the problem of determining the nonsingularity of a certain jacobi matrix to the problem of proving the non existence of a nonzero matrix b that commutes with a nilpotent matrix with sign pattern tn and has zeros in positions 1 1 and j 1 j for j 2 n 1

asymptotic option pricing under the cev diffusion abstract in finance many option pricing models generalizing the black scholes model do not have closed form analytic solutions so that it is hard to compute the solutions or at least it requires much time to compute the solutions therefore asymptotic representation of options prices of various type has important practical implications in finance this paper presents asymptotic expansions of option prices in the constant elasticity of variance model as the parameter appearing in the exponent of the diffusion coefficient tends to 2 which corresponds to the well known black scholes model we use perturbation theory for partial differential equations to obtain the relevant results for european vanilla barrier and lookback options we make our application of perturbation theory mathematically rigorous by supplying error bounds

spectrally arbitrary patterns over rings with unity abstract a zero nonzero pattern a is a matrix with entries from the set 0 a square pattern a is spectrally arbitrary over r a commutative ring with unity if for each n th degree monic polynomial f x r x there exists a matrix a over r with pattern a such that the characteristic polynomial p a x f x a pattern a is relaxed spectrally arbitrary over r if for each n th degree monic polynomial f x r x there exists a matrix a over r with either pattern a or a subpattern of a such that the characteristic polynomial p a x f x we evaluate how the structure of rings affects how we determine if a pattern is spectrally arbitrary we consider whether a pattern a that is spectrally arbitrary over r is spectrally arbitrary or relaxed spectrally arbitrary over another commutative ring s with unity in particular we establish that a pattern that is spectrally arbitrary over z is relaxed spectrally arbitrary over z m for all m z and spectrally arbitrary over q similarly a pattern that is spectrally arbitrary over the p adic integers is spectrally arbitrary over the p adic numbers

uni mode uniqueness conditions for candecomp parafac decomposition of n way arrays with linearly dependent loadings abstract guo miron brie and stegeman x guo s miron d brie a stegeman uni mode and partial uniqueness conditions for candecomp parafac of three way arrays with linearly dependent loadings siam j matrix anal appl 33 2012 111 129 give three sufficient conditions for the three way candecomp parafac cp model which ensure uniqueness in one of the three modes uni mode uniqueness in this paper we generalize these uniqueness conditions to n 3 based on these conditions a partial uniqueness condition is given which allows collinear loadings in only one mode

singular perturbations in option pricing after the celebrated black scholes formula for pricing call options under constant volatility the need for more general nonconstant volatility models in financial mathematics motivated numerous works during the 1980s and 1990s in particular a lot of attention has been paid to stochastic volatility models in which the volatility is randomly fluctuating driven by an additional brownian motion we have shown in derivatives in financial markets with stochastic volatility cambridge university press cambridge uk 2000 internat j theoret appl finance 13 2000 pp 101 142 that in the presence of a separation of time scales between the main observed process and the volatility driving process asymptotic methods are very efficient in capturing the effects of random volatility in simple robust corrections to constant volatility formulas from the point of view of pdes this method corresponds to a singular perturbation analysis the aim of this paper is to deal with the nonsmoothness of the payoff

stochastic volatility asymptotics of stock loans valuation and optimal stopping abstract a stock loan or equity security lending service is a loan which uses stocks as collateral the borrower has the right to repay the principal with interest and regain the stock or make no repayment and surrender the stock therefore the valuation of stock loan is an optimal stopping problem related to a perpetual american option with a negative effective interest rate the negative effective interest rate makes standard techniques for perpetual american option pricing failure using a fast mean reverting stochastic volatility model we applied a perturbation technique to the free boundary value problem for the stock loan price an analytical pricing formula and optimal exercise boundary are derived by means of asymptotic expansion

valuing continuous installment options abstract installment options are path dependent contingent claims in which the premium is paid discretely or continuously in installments instead of paying a lump sum at the time of purchase this paper deals with valuing european continuous installment options written on dividend paying assets in the standard black scholes merton framework the valuation of installment options can be formulated as a free boundary problem due to the flexibility of continuing or stopping to pay installments on the basis of a pde for the initial premium we derive an integral representation for the initial premium being expressed as a difference of the corresponding european vanilla value and the expected present value of installment payments along the optimal stopping boundary applying the laplace transform approach to this pde we obtain explicit laplace transforms of the initial premium as well as its greeks which include the transformed stopping boundary in a closed form abelian theorems of laplace transforms enable us to characterize asymptotic behaviors of the stopping boundary close and at infinite time to expiry we show that numerical inversion of these laplace transforms works well for computing both the option value and the optimal stopping boundary

an extension of a theorem of r datko to the case of non uniform exponential stability of linear skew product semiflows abstract in 1970 r datko proved that the trajectories of a c 0 semigroup t t t 0 on a hilbert space x exhibit an exponential decay if and only if they stay in l 2 r x datko s theorem was crucial in extending the lyapunov operator equation to the case of autonomous systems x a x with unbounded a extensions have been done to the case of uniform exponential stability of evolution families and more recently of lsps linear skew product semiflow the aim of the present paper is to extend datko s result to the general case of non uniform exponential stability of the lsps that does not necessarily possess a uniform exponential growth

a fast numerical method for the black scholes equation of american options this paper introduces a fast numerical method for computing american option pricing problems governed by the black scholes equation the treatment of the free boundary is based on some properties of the solution of the black scholes equation an artificial boundary condition is also used at the other end of the domain the finite difference method is used to solve the resulting problem computational results are given for some american call option problems the results show that the new treatment is very efficient and gives better accuracy than the normal finite difference method

potentially nilpotent and spectrally arbitrary even cycle sign patterns abstract an n n sign pattern s n is potentially nilpotent if there is a real matrix having sign pattern s n and characteristic polynomial x n a new family of sign patterns c n with a cycle of every even length is introduced and shown to be potentially nilpotent by explicitly determining the entries of a nilpotent matrix with sign pattern c n these nilpotent matrices are used together with a jacobian argument to show that c n is spectrally arbitrary i e there is a real matrix having sign pattern c n and characteristic polynomial x n i 1 n 1 i i x n i for any real i some results and a conjecture on minimality of these spectrally arbitrary sign patterns are given

on stability of linear time varying infinite dimensional discrete time systems the asymptotic behaviour of linear time varying infinite dimensional discrete lime systems is considered the introduced notions are weak power equistability power equistability uniform power equistability l p equistability uniform l p equistability and l p x equistability it is shown that they are identical a generalization of the concept of spectral radius of a single operator is also proposed it is proven that any time varying system is uniformly power equistable if and only if the generalized spectral radius of the sequence of the operators which define the system considered is less than one

spectrally arbitrary patterns reducibility and the 2n conjecture for n 5 a sign pattern z a matrix whose entries are elements of 0 is spectrally arbitrary if for any self conjugate spectrum there is a real matrix with sign pattern z having the given spectrum spectrally arbitrary sign patterns were introduced in 5 where it was incorrectly stated that if a sign pattern z is reducible and each of its irreducible components is a spectrally arbitrary sign pattern then z is a spectrally arbitrary sign pattern and it was conjectured that the converse is true as well we present counterexamples to both of these statements in 2 it was conjectured that any n n spectrally arbitrary sign pattern must have at least 2n nonzero entries we establish that this conjecture is true for 5 5 sign patterns we also establish analogous results for nonzero patterns

lyapunov sequences for exponential trichotomies abstract for a linear cocycle with discrete time we give a complete characterization of nonuniform exponential trichotomies in terms of strict lyapunov sequences we also obtain inverse theorems by constructing explicitly strict lyapunov sequences for each nonuniform exponential trichotomy these are constructed in terms of lyapunov norms with respect to which the nonuniform behavior of the trichotomies becomes uniform we also obtain a corresponding version of the results for cocyles over measure preserving transformations

bordering for spectrally arbitrary sign patterns abstract we develop a matrix bordering technique that can be applied to an irreducible spectrally arbitrary sign pattern to construct a higher order spectrally arbitrary sign pattern this technique generalizes a recently developed triangle extension method we describe recursive constructions of spectrally arbitrary patterns using our bordering technique and show that a slight variation of this technique can be used to construct inertially arbitrary sign patterns

exponential dichotomy roughness on banach spaces in the present paper we extend existing results on exponential dichotomy roughness for linear ode systems to infinite dimensional banach space we give new conditions for the existence of exponential dichotomy roughness in infinite dimensional space and in the finite interval case we also improve previous results by indicating the exact values of the dichotomic constants of the perturbed equation

integrally normalizable matrices and zero nonzero patterns abstract the problem of determining whether or not an n n integer matrix is diagonally similar to an integer matrix with n 1 off diagonal entries equal to 1 is studied such a matrix is called integrally normalizable and a zero nonzero pattern is integrally normalizable if each matrix with this zero nonzero pattern is integrally normalizable with respect to the same set of n 1 off diagonal entries matrices that are integrally normalizable with respect to a fixed spanning tree and integrally normalizable zero nonzero patterns are characterized the maximum number of nonzero entries in an n n integrally normalizable zero nonzero pattern is shown to be n 2 3 n 2 2 extensions to matrices over other integral domains are also presented

spectrally arbitrary star sign patterns abstract an n n sign pattern s n is spectrally arbitrary if for any given real monic polynomial g x of degree n there is a real matrix having sign pattern s n and characteristic polynomial g x all n n star sign patterns that are spectrally arbitrary and all minimal such patterns are characterized this subsequently leads to an explicit characterization of all n n star sign patterns that are potentially nilpotent it is shown that any super pattern of a spectrally arbitrary star sign pattern is also spectrally arbitrary

corrigendum to companion matrix patterns linear algebra appl 463 2014 255 272 abstract we correct an error in the statements of corollary 2 5 and corollary 4 3 of 1

an artificial boundary method for american option pricing under the cev model the black scholes asset price dynamics is well known to be inadequate for capturing the volatility smile in the financial market therefore the constant elasticity of variance cev model has become a popular alternative to valuing options fitting to the smile american option pricing however is computationally intensive as there are no analytical formulas available this paper proposes an artificial boundary method for partial differential equations pdes to compute american option prices and greeks under the cev model the idea is to reduce the infinite computational domain to a finite one by introducing an artificial boundary on which an exact boundary condition is derived we then develop a crank nicolson scheme to solve the pde with the artificial boundary condition implemented with a numerical laplace inversion with a finite computational domain the optimal exercise boundary can be determined efficiently our numerical examples show that the proposed scheme is accurate robust with respect to

strong nonuniform behaviour a datko type characterization abstract we obtain a datko type characterization of several classes of a strong nonuniform exponential behaviour we consider separately the case of exponential contractions expansions and dichotomies in addition we consider both discrete and continuous time as a nontrivial application of our work we study the robustness property of strong nonuniform exponential contractions expansions and dichotomies

an artificial boundary method for the hull white model of american interest rate derivatives abstract the hull white hw model is a widely used one factor interest rate model because of its analytical tractability on liquidly traded derivatives super calibration ability to the initial term structure and elegant tree building procedure as an explicit finite difference scheme lattice method is subject to some stability criteria which may deteriorate the computational efficiency for early exercisable derivatives this paper proposes an artificial boundary method based on the partial differential equations pdes to price interest rate derivatives with early exercise american feature under the hw model we construct conversion factors to extract the market information from the zero coupon curve and then reduce the infinite computational domain into a finite one by using an artificial boundary on which an exact boundary condition is derived we then develop an implicit scheme with unconditional stability to solve the pde in the reduced bounded domain with a finite computational domain the optimal exercise strategy can be determined efficiently our numerical examples show that the proposed scheme is accurate robust to the truncation size and more efficient than the popular lattice method for accurate derivative prices in addition the singularity separating technique is incorporated into the artificial boundary method to enhance accuracy and flexibility of the numerical scheme

optimal dividends and bankruptcy procedures analysis of the ornstein uhlenbeck process abstract this paper investigates the impact of bankruptcy procedures on optimal dividend barrier policies we specifically focus on chapter 11 of the us bankruptcy code which allows a firm in default to continue its business for a certain period of time our model is based on the surplus of a firm that earns investment income at a constant rate of credit interest when it is in a creditworthy condition the firm pays a debit interest rate that depends on the deficit level when it is in financial distress thus the surplus follows an ornstein uhlenbeck ou process with a negative surplus dependent mean reverting rate default and liquidation are modeled as distinguishable events by using an excursion time or occupation time framework this paper demonstrates how the optimal dividend barrier can be obtained by deriving a closed form solution for the dividend value function it also characterizes the distributional property and expectation of bankruptcy time subject to the bankruptcy procedure our numerical examples show that under an optimal dividend barrier strategy the bankruptcy procedure may not prolong the expected bankruptcy time in some situations

spectrally arbitrary pattern extensions abstract a matrix pattern is often either a sign pattern with entries in 0 or more simply a nonzero pattern with entries in 0 a matrix pattern a is spectrally arbitrary if for any choice of a real matrix spectrum there is a real matrix having the pattern a and the chosen spectrum we describe a graphical technique a triangle extension for constructing spectrally arbitrary patterns out of some known lower order spectrally arbitrary patterns these methods provide a new way of viewing some known spectrally arbitrary patterns as well as providing many new families of spectrally arbitrary patterns we also demonstrate how the technique can be applied to certain inertially arbitrary patterns to obtain larger inertially arbitrary patterns we then provide an additional extension method for zero nonzero patterns patterns with entries in 0

on the spectra of close to schwarz matrices several inverse eigenvalue problems for tridiagonal matrices are discussed in particular it is shown that for every set s of n complex numbers which is symmetric with respect to the real axis there exists a symmetric in modulus tridiagonal n n matrix a which has a nonnegative super diagonal a nonpositive subdiagonal all diagonal elements zero except for a11 which is nonnegative and ann which is nonpositive and with spectrum s 2002 elsevier science inc all rights reserved

stability of dichotomies in difference equations with infinite delay abstract for delay difference equations with infinite delay we consider the notion of nonuniform exponential dichotomy this includes the notion of uniform exponential dichotomy as a very special case our main aim is to establish a stable manifold theorem under sufficiently small nonlinear perturbations we also establish the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations finally we characterize the nonuniform exponential dichotomies in terms of strict lyapunov sequences in particular we construct explicitly a strict lyapunov sequence for each exponential dichotomy

averaging theorems for the large time behavior of the solutions of nonautonomous systems abstract we obtain some averaging theorems for the large time behavior of an evolution family u t s t s 0 acting on a banach space it is known that if a trajectory u t 0 x 0 is asymptotically stable then its p mean tends to zero we will show here that if the uniformly weighted p means of all the trajectories starting on the unit sphere are bounded then u t s t s 0 is uniformly exponentially stable while the converse statement is a simple verification discrete time versions of this result are given also variants for the uniform exponential blow up are obtained thus we generalize some known results obtained by r datko a pazy and v pata

spectrally arbitrary patterns abstract an n n sign pattern matrix s is an inertially arbitrary pattern iap if each nonnegative triple n 1 n 2 n 3 with n 1 n 2 n 3 n is the inertia of a matrix with sign pattern s analogously s is a spectrally arbitrary pattern sap if for any given real monic polynomial r x of order n there is a matrix with sign pattern s and characteristic polynomial r x focusing on tree sign patterns consider the n n tridiagonal sign pattern t n with each superdiagonal entry positive each subdiagonal entry negative the 1 1 entry negative the n n entry positive and every other entry zero it is conjectured that t n is an iap by constructing matrices a n with pattern t n it is proved that t n allows any inertia with n 3 0 1 2 n 1 n for all n 2 this leads to a proof of the conjecture for n 5 the truth of the conjecture is extended to n 7 by showing the stronger result that t n is a sap the proof of this latter statement involves finding a matrix a n with pattern t n that is nilpotent further questions about patterns that are saps and iaps are considered

companion matrix patterns abstract companion matrices especially the frobenius companion matrices are used in algorithms for finding roots of polynomials and are also used to find bounds on eigenvalues of matrices in 2003 fiedler introduced a larger class of companion matrices that includes the frobenius companion matrices one property of the combinatorial pattern of these companion matrices is that up to diagonal similarity they uniquely realize every possible spectrum of a real matrix we characterize matrix patterns that have this property and consequently introduce more companion matrix patterns we observe that each fiedler companion matrix is permutationally similar to a unit hessenberg matrix we provide digraph characterizations of the classes of patterns described and in particular all sparse companion matrices noting that there are companion matrices that are not sparse

admissibility criteria for nonuniform dichotomic behavior of nonautonomous systems on the whole line abstract we give new criteria for nonuniform dichotomy of nonautonomous systems on the whole line in terms of admissibility relative to an integral equation in our approach the input space i r x is an intersection of spaces that can be successively minimized and the output space c r x can be one of some well known spaces of continuous functions using computational arguments we show that the admissibility of c r x i r x leads to a nonuniform exponential dichotomy we expose a complete analysis of the connections between admissibility and nonuniform dichotomy on the whole line and we also discuss several interesting consequences moreover we obtain the explicit expression of the growth rates for dichotomy in terms of the initial exponential growth and the norm of the input output operator finally we present a direct application of the main result in the case of evolution families which admit uniform exponential growth

stock loan valuation under a stochastic interest rate model stock loans are loans collateralized by stocks they are modern financial products designed for investors with large equity positions mathematically stock loans can be regarded as american call options with a time dependent strike price this study is the first in the literature that considers the valuation of stock loans in a stochastic interest rate framework based on portfolio analysis a partial differential equation pde governing the value of stock loans is derived a set of appropriate boundary conditions particularly in the interest rate direction are also proposed to close the pricing system a sound justification is provided for the proposed boundary conditions mathematically as well as financially to solve the proposed nonlinear pde system a predictor corrector finite difference method is adopted moreover an alternating direction implicit adi method is used to improve the computational efficiency numerical results suggest that the current method is reliable and the stochastic interest rate leads to a higher optimal exercise price of the stock loan in comparison with that calculated from the black scholes model

a high order finite difference method for option valuation in this paper we propose the use of an efficient high order finite difference algorithm to price options under several pricing models including the blackscholes model the mertons jumpdiffusion model the hestons stochastic volatility model and the nonlinear transaction costs or illiquidity models we apply a local mesh refinement strategy at the points of singularity usually found in the payoff of most financial derivatives to improve the accuracy and restore the rate of convergence of a non uniform high order five point stencil finite difference scheme for linear models the time stepping is dealt with by using an exponential time integration scheme with carathodoryfejr approximations to efficiently evaluate the product of a matrix exponential with a vector of option prices nonlinear blackscholes equations are solved using an efficient iterative scheme coupled with a richardson extrapolation our numerical experiments clearly demonstrate the high order accuracy of the proposed finite difference method making the latter a method of choice for solving both linear and nonlinear partial differential equations in financial engineering problems

robustness of nonuniform exponential dichotomies in banach spaces we give conditions for the robustness of nonuniform exponential dichotomies in banach spaces in the sense that the existence of an exponential dichotomy for a given linear equation x a t x persists under a sufficiently small linear perturbation we also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy and of the angles between the stable and unstable subspaces our proofs exhibit implicitly the exponential dichotomies of the perturbed equations in terms of fixed points of appropriate contractions we emphasize that we do not need the notion of admissibility of bounded nonlinear perturbations we also obtain related robustness results in the case of nonuniform exponential contractions in addition we establish an appropriate version of robustness for nonautonomous dynamical systems with discrete time

a version of a theorem of r datko for stability in average in this note we obtain a version of the well known theorem of r datko for the notion of the exponential stability in average we consider both cocycles over flows as well as cocycles over maps

mathematical analysis and numerical methods for a pde model of a stock loan pricing problem in this paper the mathematical analysis of a model for pricing stock loan contracts when the accumulative dividend yield associated to the stock is returned by the lender to the borrower on redemption is carried out more precisely the model is formulated in terms of an obstacle problem associated to a kolmogorov equation and the existence and uniqueness in the set of solutions with polynomial growth are obtained also some regularity properties of the solution are analyzed next for the numerical solution of the problem the combination of crank nicolson lagrange galerkin with the augmented lagrangian active set method is described finally some numerical examples illustrate the theoretical properties of the optimal redeeming boundary previously stated in the literature

singular perturbations for boundary value problems arising from exotic options we study the pricing of three exotic derivative securities barrier lookback and passport options which can be characterized by boundary value pde problems in the context of popular markovian sv models of stock prices by extending the fast mean reverting asymptotic analysis in j p fouque g papanicolaou and k r sircar derivatives in financial markets with stochastic volatility cambridge university press london 2000 the usual greek correction to the black scholes prices of these contracts is further corrected by a boundary integral term that is rapidly computed numerically in the case of the passport option the asymptotic method is effective in accounting for sv effects in a simple and robust fashion even in the presence of a highly nonlinear embedded stochastic control problem

low rank perturbations and the spectrum of a tridiagonal sign pattern abstract the n by n tridiagonal sign pattern t n has every superdiagonal entry positive every subdiagonal entry negative the 1 1 entry negative the n n entry positive and every other diagonal entry zero inertia and spectral results for matrices a n having the sign pattern t n are proved using new techniques on low rank perturbations it is also shown by using maple that for 8 n 16 t n allows any spectrum these results extend those previously in the literature and strengthen the conjecture that t n allows any spectrum for all values of n in addition bounds on the algebraic multiplicity of an eigenvalue of a low rank perturbation of a general matrix are obtained

quadratic lyapunov functions and nonuniform exponential dichotomies abstract for nonautonomous linear equations x a t x we show how to characterize completely nonuniform exponential dichotomies using quadratic lyapunov functions the characterization can be expressed in terms of inequalities between matrices in particular we obtain converse theorems by constructing explicitly quadratic lyapunov functions for each nonuniform exponential dichotomy we note that the nonuniform exponential dichotomies include as a very special case uniform exponential dichotomies in particular we recover in a very simple manner a complete characterization of uniform exponential dichotomies in terms of quadratic lyapunov functions we emphasize that our approach is new even in the uniform case furthermore we show that the instability of a nonuniform exponential dichotomy persists under sufficiently small perturbations the proof uses quadratic lyapunov functions and in particular avoids the use of invariant unstable manifolds which to the best of our knowledge are not known to exist in this general setting

three remarks on spectra of zero nonzero patterns abstract using standard techniques from combinatorics model theory and algebraic geometry we prove generalized versions of several basic results in the theory of spectrally arbitrary matrix patterns also we point out a counterexample to a conjecture proposed recently by mcdonald and melvin

lyapunov functions for strong exponential dichotomies we give a complete characterization of a general type of exponential dichotomies in terms of lyapunov functions both for discrete and continuous time this includes constructing explicitly quadratic lyapunov functions for each exponential dichotomy we consider the general cases of nonautonomous dynamics nonuniform exponential dichotomies and motivated by ergodic theory strong exponential dichotomies in the sense that there exist simultaneously lower and upper contraction and expansion bounds as a nontrivial application we establish the persistence of the asymptotic stability of a strong nonuniform exponential dichotomy under sufficiently small linear perturbations

a version of a theorem of r datko for nonuniform exponential contractions abstract we point out versions of a well known theorem of r datko for nonuniform exponential contractions more precisely we obtain theorems that characterize the nonuniform exponential stability and the nonuniform exponential dichotomy for nonuniform exponential contractions thus we generalize the results in datko 1972 8 and preda and megan 1985 14

lyapunov sequences for exponential dichotomies abstract for a linear nonautonomous dynamics with discrete time we study the relation between nonuniform exponential dichotomies and strict lyapunov sequences given such a sequence we obtain the stable and unstable subspaces from the intersection of the images and preimages of the cones defined by each element of the sequence the main difficulty is to extract some information about the angles between the stable and unstable subspaces or some appropriate notion in the case of banach spaces from the lyapunov sequence in particular for a large class of nonuniform exponential dichotomies we give a complete characterization in terms of strict quadratic lyapunov sequences that is strict lyapunov sequences defined by quadratic forms we also construct explicitly families of strict lyapunov sequences for each nonuniform exponential dichotomy in terms of lyapunov norms

matching asymptotics in path dependent option pricing abstract the valuation of path dependent options in finance creates many interesting mathematical challenges among them are a large delta and gamma near the expiry leading to a big error in pricing those exotic options as well as european vanilla options also the higher order corrections of the asymptotic prices of the derivatives in some stochastic volatility models are difficult to be evaluated in this paper we use the method of matched asymptotic expansions to obtain more practical values of lookback and barrier option prices near the expiry our results verify that matching asymptotics is a useful tool for pde methods in path dependent option pricing

allow problems concerning spectral properties of sign pattern matrices a survey abstract an n n sign pattern matrix has entries in 0 this paper surveys the following problems concerning spectral properties of sign pattern matrices sign patterns that allow all possible spectra spectrally arbitrary sign patterns sign patterns that allow all inertias inertially arbitrary sign patterns sign patterns that allow nilpotency potentially nilpotent sign patterns and sign patterns that allow stability potentially stable sign patterns relationships between these four classes of sign patterns are given and several open problems are identified

efficient and high accuracy pricing of barrier options under the cev diffusion binomial and trinomial lattices are popular techniques for pricing financial options these methods work well for european and american options but for barrier options the need to place a tree node very close to a barrier brings difficulties in their implementation and a large number of time steps are usually required when the barrier is close to the current asset price a finite difference implementation is simpler and we propose a fourth order numerical scheme for continuously and discretely monitored barriers we demonstrate the superior performance of our technique over existing procedures for the black scholes model and we then price barriers under constant elasticity of variance cev diffusion continuously monitored barriers under cev admit an analytical solution but evaluation via this formula is not straightforward furthermore discretely monitored barriers have to be priced numerically our main contribution is therefore a highly accurate and efficient numerical scheme for barrier options under cev and we provide several numerical examples to illustrate the merit of the new technique

quadratic convergence for valuing american options using a penalty method the convergence of a penalty method for solving the discrete regularized american option valuation problem is studied sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution these conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem the efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the american constraint explicitly convergence rates are studied as the timestep and mesh size tend to zero it is observed that an implicit treatment of the american constraint does not converge quadratically as the timestep is reduced if constant timesteps are used a timestep selector is suggested which restores quadratic convergence