ANTICIPATED BACKWARD STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH JUMPS AND APPLICATIONS TO DYNAMIC RISK MEASURES

In this paper, we focus on anticipated backward stochastic Volterra integral equations(ABSVIEs) with jumps. We solve the problem of the well-posedness of so-called M-solutions to this class of equation, and analytically derive a comparison theorem for them and for the continuous equilibrium consumption process. These continuous equilibrium consumption processes can be described by the solutions to this class of ABSVIE with jumps.Motivated by this, a class of dynamic risk measures induced by ABSVIEs with jumps are discussed.

GLOBAL SOLUTIONS OF SYSTEMS OF NONLINEAR IMPULSIVE VOLTERRA INTEGRAL EQUATIONS IN BANACH SPACES

The existence of solutions for systems of nonlinear impulsive Volterra integral equations on the infinite interval R+ with an infinite number of moments of impulse effect in Banach spaces is studied. Some existence theorems of extremal solutions are obtained, which extend the related results for this class of equations on a finite interval with a finite. number of moments of impulse effect. The results are demonstrated by means of an example of an infinite systems for impulsive integral equations.

Solution and Analysis of the Fuzzy Volterra Integral Equations viaHomotopy Analysis Method

Homotopy Analysis Method(HAM)is semi-analytic method to solve the linear and nonlinear mathematical models which can be used to obtain the approximate solution.The HAM includes an auxiliary parameter,which is an efficient way to examine and analyze the accuracy of linear and nonlinear problems.The main aim of this work is to explore the approximate solutions of fuzzy Volterra integral equations(both linear and nonlinear)with a separable kernel via HAM.This method provides a reliable way to ensure the convergence of the approximation series.A new general form of HAM is presented and analyzed in the fuzzy domain.A qualitative convergence analysis based on the graphical method of a fuzzy HAM is discussed.The solutions sought by the proposed method show that the HAM is easy to implement and computationally quite attractive.Some solutions of fuzzy second kind Volterra integral equations are solved as numerical examples to show the potential of the method.The results also show that HAM provides an easy way to control and modify the convergence area in order to obtain accurate solutions.

EXTRAPOLATION FOR COLLOCATION METHOD OF THE FIRST KIND VOLTERRA INTEGRAL EQUATIONS

1. Introduction It is known that the following Cauchy problem for a parabolic partial differential equation (where the values at the right boundary, u.(1, t)=v(t) are unknown and sought for) is ill-posed: the solution (v) does not depend continuously on the data (g). In order to treat the ill-posedness and develop the numerical method, one reformulates the problem as a Volterra integral equation of the first kind wish a convolution type kernel (see Sneddon [1], Carslaw and Jaeger [2])

On Solving a System of Volterra Integral Equations with Relaxed Monte Carlo Method

A random simulation method was used for treatment of systems of Volterra integral equations of the second kind. Firstly, a linear algebra system was obtained by discretization using quadrature formula. Secondly, this algebra system was solved by using relaxed Monte Carlo method with importance sampling and numerical approximation solutions of the integral equations system were achieved. It is theoretically proved that the validity of relaxed Monte Carlo method is based on importance sampling to solve the integral equations system. Finally, some numerical examples from literatures are given to show the efficiency of the method.

Backward stochastic Volterra integral equations——a brief survey

In this paper, we present a brief survey on the updated theory of backward stochas-tic Volterra integral equations （BSVIEs, for short）. BSVIEs are a natural generalization of backward stochastic diff erential equations （BSDEs, for short）. Some interesting motivations of studying BSVIEs are recalled. With proper solution concepts, it is possible to establish the corresponding well-posedness for BSVIEs. We also survey various comparison theorems for solutions to BSVIEs.

SINGULAR CONTROL OF STOCHASTIC VOLTERRA INTEGRAL EQUATIONS

This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations,where the solution X^(u,ξ)(t)=X(t)is given X(t)=φ(t)+∫_(0)^(t) b(t,s,X(s),u(s))ds+∫_(0)^(t)σ(t,s,X(s),u(s))dB(s)+∫_(0)^(t)h(t,s)dξ(s).by Here d B(s)denotes the Brownian motion It?type differential,ξdenotes the singular control(singular in time t with respect to Lebesgue measure)and u denotes the regular control(absolutely continuous with respect to Lebesgue measure).Such systems may for example be used to model harvesting of populations with memory,where X(t)represents the population density at time t,and the singular control processξrepresents the harvesting effort rate.The total income from the harvesting is represented by J(u, ξ) = E[∫_(0)^(t) f_(0)(t,X(t), u(t))dt + ∫_(0)^(t)f_(1)(t,X(t))dξ(t) + g(X(T))] for the given functions f0,f1 and g,where T>0 is a constant denoting the terminal time of the harvesting.Note that it is important to allow the controls to be singular,because in some cases the optimal controls are of this type.Using Hida-Malliavin calculus,we prove sufficient conditions and necessary conditions of optimality of controls.As a consequence,we obtain a new type of backward stochastic Volterra integral equations with singular drift.Finally,to illustrate our results,we apply them to discuss optimal harvesting problems with possibly density dependent prices.

EXISTENCE OF SOLUTIONS FOR MIXED MONOTONE IMPULSIVE VOLTERRA INTEGRAL EQUATIONS IN BANACH SPACES

This paper studies the existence of solutions for mixed monotone impulsive Volterra integral equations on the infinite interval R+ with an infinite number of moments of impulse effect in Banach spaces. By using the mixed monotone iterative technique and Monch fixed point theorem, Some existence theorems of solutions and coupled minimal and maximal quasisolutions are obtained. Finally, an example is worked out.

ON SPECTRAL METHODS FOR VOLTERRA INTEGRAL EQUATIONS AND THE CONVERGENCE ANALYSIS

The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical results confirm the theoretical prediction of the exponential rate of convergence. The result in this work seems to be the first successful spectral approach （with theoretical justification） for the Volterra type equations.

A NOTE ON JACOBI SPECTRAL-COLLOCATION METHODS FOR WEAKLY SINGULAR VOLTERRA INTEGRAL EQUATIONS WITH SMOOTH SOLUTIONS

This work is concerned with spectrM Jacobi-collocation methods for Volterra integral equations of the second kind with a weakly singular of the form （t - s）-a When the underlying solutions are sufficiently smooth, the convergence analysis was carried out in [Chen ＆ Tang, J. Comput. Appl. Math., 233 （2009）, pp. 938-950]; due to technical reasons 1 In this work, we will improve the results to the the results are restricted to 0 〈 μ 〈 1/2. general case 0 〈 μ 〈 1 and demonstrate that the numericl errors decay exponentially in the infinity and weighted norms when the smooth solution is involved.

Convergence Analysis for the Chebyshev Collocation Methods to Volterra Integral Equations with a Weakly Singular Kernel

In this paper,a Chebyshev-collocation spectral method is developed for Volterra integral equations(VIEs)of second kind with weakly singular kernel.We first change the equation into an equivalent VIE so that the solution of the new equation possesses better regularity.The integral term in the resulting VIE is approximated by Gauss quadrature formulas using the Chebyshev collocation points.The convergence analysis of this method is based on the Lebesgue constant for the Lagrange interpolation polynomials,approximation theory for orthogonal polynomials,and the operator theory.The spectral rate of convergence for the proposed method is established in the L^(∞)-norm and weighted L^(2)-norm.Numerical results are presented to demonstrate the effectiveness of the proposed method.

L^p Solutions of Backward Stochastic Volterra Integral Equations

This paper is devoted to the unique solvability of backward stochastic Volterra integral equations （BSVIEs, for short）, in terms of both M-solution and the adapted solutions. We prove the existence and uniqueness of M-solutions of BSVIEs in Lp （1 〈 p 〈 2）, which extends the existing results on M-solutions. The unique solvability of adapted solutions of BSVIEs in Lp （p 〉 1） is also considered, which also generalizes the results in the existing literature.

On Discontinuous and Continuous Approximations to Second-Kind Volterra Integral Equations

Collocation and Galerkin methods in the discontinuous and globally continuous piecewise polynomial spaces,in short,denoted as DC,CC,DG and CG methods respectively,are employed to solve second-kind Volterra integral equations(VIEs).It is proved that the quadrature DG and CG(QDG and QCG)methods obtained from the DG and CG methods by approximating the inner products by suitable numerical quadrature formulas,are equivalent to the DC and CC methods,respectively.In addition,the fully discretised DG and CG(FDG and FCG)methods are equivalent to the corresponding fully discretised DC and CC(FDC and FCC)methods.The convergence theories are established for DG and CG methods,and their semi-discretised(QDG and QCG)and fully discretized(FDG and FCG)versions.In particular,it is proved that the CG method for second-kind VIEs possesses a similar convergence to the DG method for first-kind VIEs.Numerical examples illustrate the theoretical results.

STRONG CONVERGENCE OF THE EULER-MARUYAMA METHOD FOR A CLASS OF STOCHASTIC VOLTERRA INTEGRAL EQUATIONS

In this paper,we consider the Euler-Maruyama method for a class of stochastic Volterra integral equations(SVIEs).It is known that the strong convergence order of the EulerMaruyama method is 12.However,the strong superconvergence order 1 can be obtained for a class of SVIEs if the kernelsσi(t,t)=0 for i=1 and 2;otherwise,the strong convergence order is 12.Moreover,the theoretical results are illustrated by some numerical examples.

Linear Volterra Integral Equations as the Limit of Discrete Systems

We consider the multidimensional abstract linear integral equation of Volterra type (1), as the limit of discrete Stieltjes-type systems and we prove results on the existence of continuous solutions. The functions x, α and f are Banach space-valued defined on a compact interval R of , R t is a subinterval of R depending on t ∈ R and (⋆) ∫ denotes either the Bochner-Lebesgue integral or the Henstock integral. The results presented here generalize those in [1] and are in the spirit of [3]. As a consequence of our approach, it is possible to study the properties of (1) by transferring the properties of the discrete systems. The Henstock integral setting enables us to consider highly oscillating functions.

Application of Homotopy Analysis Method for Solving Systems of Volterra Integral Equations

In this paper,we prove the convergence of homotopy analysis method(HAM).We also apply the homotopy analysis method to obtain approximate analytical solutions of systems of the second kind Volterra integral equations.The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions.It is shown that the solutions obtained by the homotopy-perturbation method(HPM)are only special cases of the HAM solutions.Several examples are given to illustrate the efficiency and implementation of the method.

Spectral methods for weakly singular Volterra integral equations with pantograph delays

In this paper, the convergence analysis of the Volterra integral equation of second kind with weakly singular kernel and pantograph delays is provided. We use some function transformations and variable transformations to change the equation into a new Volterra integral equation with pantograph delays defined on the interval [-1, 1], so that the Jacobi orthogonal polynomial theory can be applied conveniently. We provide a rigorous error analysis for the proposed method in the L∞-norm and the weighted L2-norm. Numerical examples are presented to complement the theoretical convergence results.

Linear Volterra Integral Equations

The Kurzweil-Henstock integral formalism is applied to establish the existence of solutions to the linear integral equations of Volterra-typewhere the functions are Banach-space valued. Special theorems on existence of solutions concerning the Lebesgu3 integral setting are obtained. These sharpen earlier results.

LOCAL GAUSSIAN-COLLOCATION SCHEME TO APPROXIMATE THE SOLUTION OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS USING VOLTERRA INTEGRAL EQUATIONS

This work describes an accurate and effective method for numerically solving a class of nonlinear fractional differential equations.To start the method,we equivalently convert these types of differential equations to nonlinear fractional Volterra integral equations of the second kind by integrating from both sides of them.Afterward,the solution of the mentioned Volterra integral equations can be estimated using the collocation method based on locally supported Gaussian functions.The local Gaussian-collocation scheme estimates the unknown function utilizing a small set of data instead of all points in the solution domain,so the proposed method uses much less computer memory and volume computing in comparison with global cases.We apply the composite non-uniform Gauss-Legendre quadrature formula to estimate singular-fractional integrals in the method.Because of the fact that the proposed scheme requires no cell structures on the domain,it is a meshless method.Furthermore,we obtain the error analysis of the proposed method and demon-strate that the convergence rate of the approach is arbitrarily high.Illustrative examples clearly show the reliability and efficiency of the new technique and confirm the theoretical error estimates.

STRONG CONVERGENCE OF THE EULER-MARUYAMA METHOD FOR NONLINEAR STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH TIME-DEPENDENT DELAY

We consider a nonlinear stochastic Volterra integral equation with time-dependent delay and the corresponding Euler-Maruyama method in this paper.Strong convergence rate(at fixed point)of the corresponding Euler-Maruyama method is obtained when coefficients f and g both satisfy local Lipschitz and linear growth conditions.An example is provided to interpret our conclusions.Our result generalizes and improves the conclusion in[J.Gao,H.Liang,S.Ma,Strong convergence of the semi-implicit Euler method for nonlinear stochastic Volterra integral equations with constant delay,Appl.Math.Comput.,348(2019)385-398.]