On a class of backward stochastic Volterra integral equations

In this paper, under some restrictions of the time interval, we compare a class of backward stochastic Volterra integral equations with the corresponding simpler one; to be precise, we give the relations between their solutions under global and local Lipschitz conditions on their generator functions. Using these relations, it could be easier to study solutions of more complex equations, where coefficients in backward integrals could be treated as perturbations.     

Continuous-time dynamic risk measures by backward stochastic Volterra integral equations

Continuous-time dynamic convex and coherent risk measures are introduced. To obtain existence of such risk measures, backward stochastic Volterra integral equations (BSVIEs, for short) are studied. For such equations, notion of adapted M-solution is introduced, well-posedness is established, duality principles and comparison theorems are presented. Then a class of dynamic convex and coherent risk measures are identified as a component of the adapted M-solutions to certain BSVIEs.     

Backward stochastic Volterra integral equations on Markov chains

This paper studies Backward Stochastic Volterra Integral Equations (BSVIEs) driven by finite state, continuous time Markov chains. First, the existence and uniqueness of the solutions to two types of BSVIEs are established. Second, some scalar and vector comparison theorems are given. Finally, the applications of BSVIEs to a linear-quadratic optimal control problem and time-inconsistent coherent risk measures are presented.     

Power series solutions to Volterra integral equations

Power series type solutions are given for a wide class of linear and q-dimensional nonlinear Volterra equations on R^p. The basic assumption on the kernel K(x, y) is that K(x, xt) has a power series in x. For example, this holds for any analytic kernel.The kernel may be strongly singular, provided certain constants are finite. One and only one such power series solution exists. Its coefficients are given by a simple iterative formula. In many cases this may be solved explicitly. In particular an explicit formula for the resolvent is given.     

Linear Volterra backward stochastic integral equations

We present an explicit solution triplet $(Y,Z,K)$ to the backward stochastic Volterra integral equation (BSVIE) of linear type, driven by a Brownian motion and a compensated Poisson random measure. The process $Y$ is expressed by an integral whose kernel is explicitly given. The processes $Z$ and $K$ are expressed by Hida–Malliavin derivatives involving $Y$.     

New approach to optimal control of stochastic Volterra integral equations

ABSTRACT

We study optimal control of stochastic Volterra integral equations (SVIE) with jumps by using Hida-Malliavin calculus.

• We give conditions under which there exist unique solutions of such equations.

• Then we prove both a sufficient maximum principle (a verification theorem) and a necessary maximum principle via Hida-Malliavin calculus.

• As an application we solve a problem of optimal consumption from a cash flow modelled by an SVIE.

     

On global superconvergence of iterated collocation solutions to linear second-kind Volterra integral equations

It is well known that, in contrast to Fredholm integral equations, iterated collocation solutions (based on collocation at the Gauss points) to Volterra integral equations of the second kind exhibit optimal discrete superconvergence only at the mesh points. Here, we show that some degree of global superconvergence is possible on the entire interval.     

Spectral methods for weakly singular Volterra integral equations with smooth solutions

We propose and analyze a spectral Jacobi-collocation approximation for the linear Volterra integral equations (VIEs) of the second kind with weakly singular kernels. In this work, we consider the case when the underlying solutions of the VIEs are sufficiently smooth. In this case, we provide a rigorous error analysis for the proposed method, which shows that the numerical errors decay exponentially in the infinity norm and weighted Sobolev space norms. Numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence.     

Blow-up time for solutions to some nonlinear Volterra integral equations

The problem of the estimating of a blow-up time for solutions of Volterra nonlinear integral equation with convolution kernel is studied. New estimates, lower and upper, are found and, moreover, the procedure for the improvement of the lower estimate is presented. Main results are illustrated by examples. The new estimates are also compared with some earlier ones related to a shear band model.     

Lp solutions of backward stochastic differential equations with jumps

Given $p\in (1,2)$, we study ${\mathbb{L}}^p$ solutions of a multi-dimensional backward stochastic differential equation with jumps (BSDEJ) whose generator may not be Lipschitz continuous in $(y,z)$-variables. We show that such a BSDEJ with $p$-integrable terminal data admits a unique ${\mathbb{L}}^p$ solution by approximating the monotonic generator by a sequence of Lipschitz generators via convolution with mollifiers and using a stability result.     

A generalized existence theorem of backward doubly stochastic differential equations

In this paper, we deal with a class of one-dimensional backward doubly stochastic differential equations （BDSDEs）. We obtain a generalized comparison theorem and a generalized existence theorem of BDSDEs.     

On the solutions of the linear integral equations of Volterra type

Some boundaries about the solution of the linear Volterra integral equations of the form f(t)=1?K*f were obtained as |f(t)|?1, |f(t)|?2 and |f(t)|?4 in (J. Math. Anal. Appl. 1978; 64 :381–397; Int. J. Math. Math. Sci. 1982; 5 (1):123–131). The boundary of the solution function of an equation in this type was found as |f(t)|?2ⁿ in (Integr. Equ. Oper. Theory 2002; 43 :466–479), where t∈[0, ∞) and n is a natural number such that n?2. In (Math. Comp. 2006; 75 :1175–1199), it is shown that the boundary of the solution function of an equation in the same form can also be derived as that of (Integr. Equ. Oper. Theory 2002; 43 :466–479) under different conditions than those of (Integr. Equ. Oper. Theory 2002; 43 :466–479). In the present paper, the sufficient conditions for the boundedness of functions f, f′, f′′, …, f⁽ⁿ⁺³⁾, (n∈?) defined on the infinite interval [0, ∞) are given by our method, where f is the solution of the equation f(t)=1?K*f. Copyright © 2007 John Wiley & Sons, Ltd.     

The iterative correction method for Volterra integral equations

We show that the (n – 1)-fold application of an iterative correction technique to the iterated collocation solution corresponding to the one-point Gauss collocation solution for a Volterra integral equation of the second kind l6eads to a significant improvement in the precision of these approximations: the resulting rate of (global) convergence is .The work of first author has been supported by the Natural Sciences and Engineering Research Council of Canada (Research Grant OGP0009406).     

Tension spline collocation methods for singularly perturbed Volterra integro-differential and Volterra integral equations

We consider the numerical discretization of singularly perturbed Volterra integro-differential equations (VIDE)

(*)

and Volterra integral equations (VIE)

(**)

by tension spline collocation methods in certain tension spline spaces, where is a small parameter satisfying 0<1, and q₁, q₂, g and K are functions sufficiently smooth on their domains to ensure that Eqs. (*) and (**) posses a unique solution.We give an analysis of the global convergence properties of a new tension spline collocation solution for 0<1 for singularly perturbed VIDE and VIE; thus, extending the existing theory for =1 to the singularly perturbed case.     

Regularity of solutions to stochastic Volterra equations with infinite delay

In this article we give necessary and sufficient conditions providing regularity of solutions to stochastic Volterra equations with infinite delay on a -dimensional torus. The harmonic analysis techniques and stochastic integration in function spaces are used. The work applies to both the stochastic heat and wave equations.

     

Asymptotic periodicity of nonlinear discrete Volterra equations and applications

Sufficient conditions for the asymptotic periodicity of solutions of nonlinear discrete Volterra equations of Hammerstein type are obtained. Such results are applied to analyze the property of a class of numerical methods to preserve the asymptotic periodicity of the analytical solution of Volterra integral equations.     

The uniqueness of solutions of perturbed backward stochastic differential equations

We prove a result on the preservation of the pathwise uniqueness property for the adapted solution to backward stochastic differential equation under perturbations.