Volterra integral equations

This paper considers the problem of pricing American options when the dynamics of the underlying are driven by both stochastic volatility following a square root process as used by , and by a Poisson jump process as introduced by Merton (1976). Probability arguments are invoked to find a representation of the solution in terms of expectations over the joint distribution of the underlying process.

In the present paper, we solve numerically Volterra integral equations of second kind with regular
and singular kernels by given a numerical algorithm to solve the equation. Numerical example are
considered to verify the effectiveness of the proposed derivations and numerical solution is compared with the
existing method available in the literature.

Volterra series provides a strong platform for non-linear analysis and higher order frequency response functions. However, limited convergence is an inherent di$culty associated with the series and needs to be addressed rigorously, prior to its application to a physical system. The power series representation of the response of non-linear systems, subjected to harmonic excitation is investigated in this study. The problem of convergence is addressed in terms of the convergence of individual frequency harmonics of the non-linear response. Though the procedure is applicable to general polynomial form non-linearity, it is illustrated for a Du$ng oscillator subjected to harmonic excitation. A general and structured series expression is obtained for amplitudes of all the response harmonics and convergence is investigated in terms of a non-dimensional non-linear parameter. Critical values of this parameter, representing the upper limit of excitation level for the convergence, are de"ned for a wide range of excitation frequencies. Zones of convergence and divergence of the response series are presented graphically, for a range of the non-dimensional non-linear parameter and the number of terms included in the approximation of a response harmonic. An algorithm based on ratio test is presented to compute the critical value of the non-dimensional non-linear parameter. Results obtained from the suggested algorithm are found to be in close agreement with the exact values. The method gives better results compared to previous methods and has wider application in terms of excitation frequency. The procedure is also investigated for a two-degree-of-freedom system.

This paper considers the problem of pricing American options when the dynamics of the underlying are driven by both stochastic volatility following a square root process as used by Heston (1993), and by a Poisson jump process as introduced by Merton (1976). Probability arguments are invoked to find a representation of the solution in terms of expectations over the joint distribution of the underlying process. A combination of Fourier transform in the log stock price and Laplace transform in the volatility is then applied to find the transition probability density function of the underlying process. It turns out that the price is given by an integral dependent upon the early exercise surface, for which a corresponding integral equation is obtained. The solution generalises in an intuitive way the structure of the solution to the corresponding European option pricing problem in the case of a call option and constant interest rates obtained by Scott (1997).

A computer-based numerical procedure is presented which has been derived for the static analysis of structures built with linearly v&o-elastic materials that exhibit ageing, i.e. concrete. The method, based on Gauss quadrature formulas and Lagrange interpolation formulas, improves a general method, based on the trapezoidal rule, generally used for the solution of the Volterra integrals governing the problem in time. This procedure leads to an easy, recurrent algebraic formula that applies to any problem concerning delayed behaviour of reinforced concrete and prestressed concrete sections and structures under sustained loads. Some numerical tests show the improvement in precision gathered.

Estimates for step-by-step interpolation projections are established. Depending on the spectrum of the transfer matrix these estimates allow to obtain the pointwise convergence of the projectors to the identity operator or, in some limit cases, to prove stable convergence of the corresponding approximate operators of integral equations. This, via general convergence theorems for operator equations, permits to get the convergence of collocation method for Volterra integral equations of the second kind in spaces of continuous or certain times continuously differentiable functions. Applications in the case of the most practical types of splines are analyzed.

We prove the existence and uniqueness, local in time, of the solution of a one-phase Stefan problem for a non-classical heat equation for a semi-infinite material with a heat flux boundary condition at the fixed face x = 0. Here the heat source depends on the temperature at the fixed face x = 0. We use the Friedman-Rubinstein integral representation method and the Banach contraction theorem in order to solve an equivalent system of two Volterra integral equations.

The use of common reinforced concrete shear walls in high rise buildings is sometimes limited because of the large amount of reinforcement localized at the end of the element. A good alternative in avoiding this disadvantage is to use composite steel concrete structural shear walls with steel encased profiles. This solution used for high rise buildings, offers to designers lateral stiffness, shear capacity and high bending resisting moment of structural walls. The encasement of the steel shapes in concrete is applied also for the following purposes: flexural stiffening and strengthening of compression elements; fire protection; potentially easier repairs after moderate damage; economy with respect both to material and construction. Until now in the national and international literature poor information about nonlinear behaviour of composite steel concrete structural shear walls with steel encased profiles is available. A theoretical and experimental program related to the behaviour of steel concrete structural shear walls with steel encased profiles is developed at "Politehnica" University of Timişoara. The program refers to six different elements, which differ by the shape of the steel encased profile and also by the arrangement of steel shapes on the cross section of the element. In order to calibrate the elements for experimental study some numerical analysis were made. The paper presents the results of numerical analysis with details of stress distribution, crack distribution, structural stiffness at various loads, and load bearing capacity of the elements.

Some properties of non-locally bounded solutions for Abel integral equations are given. The case in which there exists two non-trivial solutions for such equations is also studied. Besides, some known results about existence, uniqueness and attractiveness of solutions for some Volterra equations are improved.

Extended one-step schemes of exponential type are introduced for solving singularly perturbed Volterra integro-differential problems. These schemes are of order (m + 1), m = 0, 1, 2, . . ., when the perturbation parameter, e, is fixed. These schemes have the property that if e is of order h they reduced to first order of accuracy and optimal when e ! 0. Stability analysis of these schemes are presented. Numerical results and comparisons with other schemes are presented.

A new approach to the construction of finite-difference methods is presented. It is shown how the multi-point differentiators can generate regularizing algorithms with stepsize h being a regularization parameter. The explicitly computable estimation constants are given. Also an iteratively regularized scheme for solving the numerical differentiation problem in the form of Volterra integral equation is developed.

We study S-asymptotically ω-periodic mild solutions of the semilinear Volterra equation u (t) = (a * Au)(t) + f (t, u(t)), considered in a Banach space X, where A is the generator of an (exponentially) stable resolvent family. In particular, we extend recent results for semilinear fractional integro-differential equations considered in [4] and for semilinear Cauchy problems of first order given in . Applications to integral equations arising in vicoelasticity theory are shown.

In this letter, a Painlevé integrable coupled KdV equation is proved to be also Lax integrable by a prolongation technique. The Miura transformation and the corresponding coupled modified KdV equation associated with this equation are derived.

The paper deals with optimal control in a linear integral age-dependent model of population dynamics. A problem for maximizing the harvesting return on a finite time horizon is formulated and analyzed. The optimal controls are the harvesting age and the rate of population removal by harvesting. The gradient and necessary condition for an extremum are derived. A qualitative analysis of

The paper investigates a nonlinear optimal control problem for the Volterra integral equations with an unknown function in the lower limit of integration. The applied interpretation of the problem consists of the optimal renovation of vintage capital under the embodied technological change. An explicit solution is constructed and its properties are analyzed in the case of a nonlinear revenue dependence on product output.

Some results about existence, uniqueness, and attractive behaviour of solutions for nonlinear Volterra integral equations with non-convolution kernels are presented in this paper. These results are based on similar ones about nonlinear Volterra integral equations with convolution kernels and some comparison techniques. Therefore, this paper is devoted to find a wide class of nonconvolution Volterra integral equations where their solutions behave like those of Volterra equations with convolution kernels.

For certain cyclic cubic fields k, we verified that Iwasawa invariants λ 3 (k) vanished by calculating units of abelian number field of degree 27. Our method is based on the explicit representation of a system of cyclotomic units of those fields.

In this work the numerical solution of a Volterra integral equation with a certain weakly singular kernel, depending on a real parameter , is considered. Although for certain values of this equation possesses an inÿnite set of solutions, we have been able to prove that Euler's method converges to a particular solution. It is also shown that the error allows an asymptotic expansion in fractional powers of the stepsize, so that general extrapolation algorithms, like the E-algorithm, can be applied to improve the numerical results. This is illustrated by means of some examples.

Volterra series provides a strong platform for non-linear analysis and higher order frequency response functions. However, limited convergence is an inherent di$culty associated with the series and needs to be addressed rigorously, prior to its application to a physical system. The power series representation of the response of non-linear systems, subjected to harmonic excitation is investigated in this study. The problem of convergence is addressed in terms of the convergence of individual frequency harmonics of the non-linear response. Though the procedure is applicable to general polynomial form non-linearity, it is illustrated for a Du$ng oscillator subjected to harmonic excitation. A general and structured series expression is obtained for amplitudes of all the response harmonics and convergence is investigated in terms of a non-dimensional non-linear parameter. Critical values of this parameter, representing the upper limit of excitation level for the convergence, are de"ned for a wide range of excitation frequencies. Zones of convergence and divergence of the response series are presented graphically, for a range of the non-dimensional non-linear parameter and the number of terms included in the approximation of a response harmonic. An algorithm based on ratio test is presented to compute the critical value of the non-dimensional non-linear parameter. Results obtained from the suggested algorithm are found to be in close agreement with the exact values. The method gives better results compared to previous methods and has wider application in terms of excitation frequency. The procedure is also investigated for a two-degree-of-freedom system.

The purpose of this paper is to present some fixed point results for self-generalized contractions in ordered metric spaces. Our results generalize and extend some recent results of A.C.M. Ran, M.C. Reurings [A.C.M. Ran, M.C. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435-1443], J.J. Nieto, R. Rodríguez-López [J.J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223-239; J.J. Nieto, R. Rodríguez-López, Existence and uniqueness of fixed points in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (2007) 2205-2212], J.J. Nieto, R.L. Pouso, R. Rodríguez-López [J.J. Nieto, R.L. Pouso, R. Rodríguez-López, Fixed point theorem theorems in ordered abstract sets, Proc. Amer. Math. Soc. 135 (2007) 2505-2517], A. Petruşel, I.A. Rus [A. Petruşel, I.A. Rus, Fixed point theorems in ordered L-spaces, Proc. Amer. Math. Soc. 134 (2006) 411-418] and Generalized contractions in partially ordered metric spaces, Appl. Anal., in press]. As applications, existence and uniqueness results for Fredholm and Volterra type integral equations are given.

We consider Runge-Kutta methods for second-kind Volterra Integral Equations with weakly singular kernel. Order conditions, whose number and structure depend on the singularity of the equation, are derived in a recursive manner using an approach originally devised by P. Albrecht for Ordinary Differential Equations. Order conditions are hence generated, in an automatic way, by means of a symbolic algorithm and some numerical experiments are presented.

Some results about existence, uniqueness, and attractive behaviour of solutions for nonlinear Volterra integral equations with non-convolution kernels are presented in this paper. These results are based on similar ones about nonlinear Volterra integral equations with convolution kernels and some comparison techniques. Therefore, this paper is devoted to find a wide class of nonconvolution Volterra integral equations where their solutions behave like those of Volterra equations with convolution kernels.

In this letter, a Painlevé integrable coupled KdV equation is proved to be also Lax integrable by a prolongation technique. The Miura transformation and the corresponding coupled modified KdV equation associated with this equation are derived.

One of the methods for solving definite integrals is modified trapezoid method, which is obtained by using Hermit interpolation [J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Second Edition, Springer-Verlag, 1993]. In this article we intend to make a quadrature method for solving the linear integral equations such as repeated trapezoid and repeated Simpson quadrature by using repeated modified trapezoid formula and by doing so, we solve the linear integral equations more accurately. For further information on quadrature methods see [

An optimization problem in an economic vintage capital model with nonlinear utility is investigated. It is described by non-linear Volterra integral equations with an unknown in the limits of integration. The concavity of the problem is proven, the condition for an extremum is established, and their relevance to applications is demonstrated.

In this work the numerical solution of a Volterra integral equation with a certain weakly singular kernel, depending on a real parameter , is considered. Although for certain values of this equation possesses an inÿnite set of solutions, we have been able to prove that Euler's method converges to a particular solution. It is also shown that the error allows an asymptotic expansion in fractional powers of the stepsize, so that general extrapolation algorithms, like the E-algorithm, can be applied to improve the numerical results. This is illustrated by means of some examples.

In this paper we prove that positive solutions of some nonlinear Volterra integral equations must be locally bounded and global attractors of positive functions. These results complete previous results about the existence and uniqueness of solutions and their attractive behavior.

It is known that Alekseev's variation of parameters formula for ordinary differential equations can be generalized to other types of causal equations (including delay differential equations and Volterra integral equations), and corresponding discrete forms. Such variation of parameters formulae can be employed, together with appropriate inequalities, in discussing the behaviour of solutions of continuous and discretized problems, the significance of parameters in mathematical models, sensitivity and stability issues, bifurcation, and (in classical numerical analysis) error control, convergence and super-convergence of densely defined approximations and error analysis in general. However, attempts to extend Alekseev's formula to nonlinear Volterra integral equations are not straightforward, and difficulties can recur in attempts to analyze the sensitivity of functionally-dependent or structurally-dependent solutions. In analyzing sensitivity we discuss behaviour for infinitesimally small perturbations. In discussions of stability we need to establish the existence of bounds on changes to solutions (or their decay in the limit as t → ∞) that ensue from perturbations in the problem. Yet the two topics are related, not least through variation of parameters formulae, and (motivated by some of our recent results) we discuss this and related issues within a general framework.

It is known that Alekseev's variation of parameters formula for ordinary differential equations can be generalized to other types of causal equations (including delay differential equations and Volterra integral equations), and corresponding discrete forms. Such variation of parameters formulae can be employed, together with appropriate inequalities, in discussing the behaviour of solutions of continuous and discretized problems, the significance of parameters in mathematical models, sensitivity and stability issues, bifurcation, and (in classical numerical analysis) error control, convergence and super-convergence of densely defined approximations and error analysis in general. However, attempts to extend Alekseev's formula to nonlinear Volterra integral equations are not straightforward, and difficulties can recur in attempts to analyze the sensitivity of functionally-dependent or structurally-dependent solutions. In analyzing sensitivity we discuss behaviour for infinitesimally small perturbations. In discussions of stability we need to establish the existence of bounds on changes to solutions (or their decay in the limit as t → ∞) that ensue from perturbations in the problem. Yet the two topics are related, not least through variation of parameters formulae, and (motivated by some of our recent results) we discuss this and related issues within a general framework.

We first introduce the notion of positive linear Volterra integral equations. Then, we offer a criterion for positive equations in terms of the resolvent. In particular, equations with nonnegative kernels are positive. Next, we obtain a variant of the Paley-Wiener theorem for equations of this class and its extension to perturbed equations. Furthermore, we get a Perron-Frobenius type theorem for linear Volterra integral equations with nonnegative kernels. Finally, we give a criterion for positivity of the initial function semigroup of linear Volterra integral equations and provide a necessary and sufficient condition for exponential stability of the semigroups.

We first introduce the notion of positive linear Volterra integral equations. Then, we offer a criterion for positive equations in terms of the resolvent. In particular, equations with nonnegative kernels are positive. Next, we obtain a variant of the Paley-Wiener theorem for equations of this class and its extension to perturbed equations. Furthermore, we get a Perron-Frobenius type theorem for linear Volterra integral equations with nonnegative kernels. Finally, we give a criterion for positivity of the initial function semigroup of linear Volterra integral equations and provide a necessary and sufficient condition for exponential stability of the semigroups.

On the basis of the Volterra integral equation approach a universal method for the calculation of sorption uptake curves of fluid multi-component mixtures in porous solids, at both constant and variable concentration of the mixture component, at the outer surface of the porous medium has been proposed. The method, being independent of the system geometry, is valid for the isothermic case with constant elements of matrix of diffusion coefficients. For the variable concentration case, the solution of diffusion equation at constant surface concentration is needed.

In this work the numerical solution of a Volterra integral equation with a certain weakly singular kernel, depending on a real parameter , is considered. Although for certain values of this equation possesses an inÿnite set of solutions, we have been able to prove that Euler's method converges to a particular solution. It is also shown that the error allows an asymptotic expansion in fractional powers of the stepsize, so that general extrapolation algorithms, like the E-algorithm, can be applied to improve the numerical results. This is illustrated by means of some examples.

This article presents two theoretical approaches that simulate the visco-elastic behaviour of elastomer specimens. The first approach, based on an equivalent rheological model, provides a dynamic modulus extracted from a Volterra development of the visco-elastic constitutive law using either relaxation or creep kernels. The second approach establishes a restoring force model based on a first-order differential equation that relates the restoring force to the deflection, the forcing frequency and deflection amplitude dependence being taken into account by the envelope curves of the force-deflection loop.

In this paper, for the “critical case” with two delays, we establish two relations between any two solutions y(t) and y∗(t) for the Volterra integral equation of non-convolution type y(t)=f(t)+∫t−τt−δk(t,s)g(y(s))ds and a solution z(t) of the first order differential equation ż(t)=β(t)[z(t−δ)−z(t−τ)], and offer a sufficient condition that limt→+∞(y(t)−y∗(t))=0.

In this paper stochastic Volterra equations admitting exponentially bounded resolvents are studied. After obtaining convergence of resolvents, some properties for stochastic convolutions are studied. Our main result provide sufficient conditions for strong solutions to stochastic Volterra equations.

The Romberg extrapolation is applied on quadrature method solution of linear Volterra integral equations (VIE) of the second kind. An algorithm for new solution calculated by Romberg method is given. The calculated solutions show more efficiency and accuracy with less computation than solution with quadrature method over more points of integration.

The use of common reinforced concrete shear walls in high rise buildings is sometimes limited because of the large amount of reinforcement localized at the end of the element. A good alternative in avoiding this disadvantage is to use composite steel concrete structural shear walls with steel encased profiles. This solution used for high rise buildings, offers to designers lateral stiffness, shear capacity and high bending resisting moment of structural walls. The encasement of the steel shapes in concrete is applied also for the following purposes: flexural stiffening and strengthening of compression elements; fire protection; potentially easier repairs after moderate damage; economy with respect both to material and construction. Until now in the national and international literature poor information about nonlinear behaviour of composite steel concrete structural shear walls with steel encased profiles is available. A theoretical and experimental program related to the behaviour of steel concrete structural shear walls with steel encased profiles is developed at "Politehnica" University of Timişoara. The program refers to six different elements, which differ by the shape of the steel encased profile and also by the arrangement of steel shapes on the cross section of the element. In order to calibrate the elements for experimental study some numerical analysis were made. The paper presents the results of numerical analysis with details of stress distribution, crack distribution, structural stiffness at various loads, and load bearing capacity of the elements.

This article presents two theoretical approaches that simulate the visco-elastic behaviour of elastomer specimens. The first approach, based on an equivalent rheological model, provides a dynamic modulus extracted from a Volterra development of the visco-elastic constitutive law using either relaxation or creep kernels. The second approach establishes a restoring force model based on a first-order differential equation that relates the restoring force to the deflection, the forcing frequency and deflection amplitude dependence being taken into account by the envelope curves of the force-deflection loop.

In this paper we deal with almost periodic functions with values in a Fréchet space. We apply obtained results to prove the existence of solutions of the initial value problem as well as the Volterra integral equation in this class of functions. We also introduce and investigate asymptotically almost periodic functions with values in a Fréchet space.