Stochastic Volterra Integro-differential Equations Research Articles

Convergence of the θ-Euler-Maruyama method for a class of stochastic Volterra integro-differential equations

This paper addresses the convergence analysis of the θ-Euler-Maruyama method for a class of stochastic Volterra integro-differential equations (SVIDEs). At first, we discuss the existence, uniqueness, boundedness and H¨older continuity of the theoretical solution. Subsequently, the strong convergence order of the θ-Euler-Maruyama approach for SVIDEs is shown. Finally, we provided numerical examples to illustrate the theoretical results.

Convergence and Stability of the Split-Step Theta Method for a Class of Stochastic Volterra Integro-Differential Equations Driven By Lévy Noise

Convergence and Stability of the Split-Step Theta Method for a Class of Stochastic Volterra Integro-Differential Equations Driven By Lévy Noise

Stability of the analytic solution and the partially truncated Euler–Maruyama method for a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients

In this paper, we develop the partially truncated Euler–Maruyama method of a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients. The mean-square exponential stability and the mean-square stability of the analytic solution and the partially truncated Euler–Maruyama method are investigated. Moreover, the theoretical results are illustrated by some numerical examples.

Backward Stochastic Volterra Integro-Differential Equations and Applications in Optimal Control Problems

In this article, a class of backward stochastic Volterra integro-differential equations (BSVIDEs) is introduced and studied. It is worthy mentioning that the proposed BSVIDEs cannot be covered by the existing backward stochastic Volterra integral equations (BSVIEs), and they also have the nice flow property such that Itô's formula becomes quite applicable. It is found that BSVIDEs can provide a neat sufficient condition for the solvability of BSVIEs with generator depending on the diagonal value of the solutions. As applications, the optimal control problems in terms of maximum principles and linear quadratic control problems of optimal control for forward stochastic Volterra integro-differential equations are investigated. In contrast with the BSVIEs in current literature, some interesting phenomena and advantages of BSVIDEs are revealed. (A corrected version is attached.)

Lipschitz continuity in the Hurst index of the solutions of fractional stochastic volterra integro-differential equations

The problem of investigating the continuity in the Hurst index arises naturally in statistical inferences related to fractional Brownian motion. In this paper, based on the techniques of the Malliavin calculus, we introduce a method to deal with this problem. We first provide an explicit bound on the difference between two non-smooth functionals of Malliavin differentiable random variables. Then, we apply the obtained bound to show Lipchitz continuity of fractional stochastic Volterra integro-differential equations and its additive functionals.

A pseudo-spectral scheme for variable order fractional stochastic Volterra integro-differential equations

<abstract><p>A spectral collocation method is proposed to solve variable order fractional stochastic Volterra integro-differential equations. The new technique relies on shifted fractional order Legendre orthogonal functions outputted by Legendre polynomials. The original equations are approximated using the shifted fractional order Legendre-Gauss-Radau collocation technique. The function describing the Brownian motion is discretized by means of Lagrange interpolation. The integral components are interpolated using Legendre-Gauss-Lobatto quadrature. The approach reveals superiority over other classical techniques, especially when treating problems with non-smooth solutions.</p></abstract>

Exponential stability of θ-EM method for nonlinear stochastic Volterra integro-differential equations

Mean square exponential stability of both exact solutions and the corresponding θ-EM method for stochastic Volterra integro-differential equations are investigated in this paper. For 12<θ≤1, we prove that both exact solutions and the corresponding θ-EM method for stochastic Volterra integro-differential equations are mean square exponentially stable under the Khasminskii-type conditions. If 0≤θ≤12, θ-EM method is mean square exponentially stable under the Khasminskii-type condition plus linear growth condition on f. By using Chebyshev inequality and Borel-Cantelli lemma, we can also prove that θ-EM method is almost surely exponentially stable. An example is provided to support our conclusions.

Numerical analysis of the balanced methods for stochastic Volterra integro-differential equations

Numerical analysis of the balanced methods for stochastic Volterra integro-differential equations

Approximate solution of stochastic Volterra integro-differential equations by using moving least squares scheme and spectral collocation method

In this paper, an attractive idea using moving least squares (MLS) and spectral collocation method is extended to estimate the solution of nonlinear stochastic Volterra integro-differential equations (NSVIDEs) that arise in mathematical modeling of natural systems in financial mathematics, physics, and engineering. An essential advantage of the proposed technique is that it does not require any preprocessing, such as mesh refinement. Another advantage that may be appealing to the readers of this article is that acceptable results can be obtained using a small number of points and basis functions, so the calculations are reduced. Applying the proposed scheme leads to the conversion of the problem into a system of algebraic equations. An error bound is presented to ensure the convergence and reliability of the method. Some illustrative examples are presented to reveal the efficiency and applicability of this technique.

Shifted fractional Legendre spectral collocation technique for solving fractional stochastic Volterra integro-differential equations

This paper presents a spectral collocation technique to solve fractional stochastic Volterra integro-differential equations (FSV-IDEs). The algorithm is based on shifted fractional order Legendre orthogonal functions generated by Legendre polynomials. The shifted fractional order Legendre–Gauss–Radau collocation (SFL-GR-C) method is developed for approximating the FSV-IDEs, with the objective of obtaining a system of algebraic equations. For computational purposes, the Brownian motion function W(x) is discretized by Lagrange interpolation, while the integral terms are interpolated by Legendre–Gauss–Lobatto quadrature. Numerical examples demonstrate the accuracy and applicability of the proposed technique, even when dealing with non-smooth solutions.

Strong super convergence of the balanced Euler method for a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients

In this paper, we propose the balanced Euler method of a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients. The moment boundedness is studied and the strong convergence is shown to be 1. Moreover, the theoretical results are illustrated by a numerical example.

Convergence of the balanced Euler method for a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients

In this paper, we propose the balanced Euler method of a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients. The moment boundedness and strong convergence are shown. Moreover, the theoretical results are illustrated by some numerical examples.

Mittag--Leffler Euler Integrator for a Stochastic Fractional Order Equation with Additive Noise

Motivated by fractional derivative models in viscoelasticity, a class of semilinear stochastic Volterra integro-differential equations, and their deterministic counterparts, are considered. A generalized exponential Euler method, named here as the Mittag-Leffler Euler integrator, is used for the temporal discretization, while the spatial discretization is performed by the spectral Galerkin method. The temporal rate of strong convergence is found to be (almost) twice compared to when the backward Euler method is used together with a convolution quadrature for time discretization. Numerical experiments that validate the theory are presented.

Theoretical and numerical analysis of a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients

In this paper, we consider the theoretical and numerical analysis of a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients. The existence, uniqueness and pth moment boundedness of the analytic solutions are investigated. Euler method is shown to be divergent in the strong mean square sense for super linear growth coefficients, so the truncated Euler-Maruyama method is presented and its moment boundedness and Lq-convergence are shown. Moreover, its pth moment boundedness and Lq-convergence (q∈[2,p) and p is a parameter in Khasminskii-type condition) rate are given under Local Lipschitz condition and Khasminskii-type condition. The theoretical results are illustrated by some numerical examples.

Theoretical and numerical analysis of the Euler–Maruyama method for generalized stochastic Volterra integro-differential equations

In this paper, we concern with the theoretical and numerical analysis of the generalized stochastic Volterra integro-differential equations (SVIDEs). The existence, uniqueness, boundedness and Hölder continuity of the analytic solutions for generalized SVIDEs are investigated. The Euler–Maruyama method for generalized SVIDEs is presented. The boundedness of the numerical solution is proved, and the strong convergence order is obtained. The theoretical results are illustrated by some numerical examples.

Strong Convergence of the Semi-Implicit Euler Method for a Kind of Stochastic Volterra Integro-Differential Equations

Strong Convergence of the Semi-Implicit Euler Method for a Kind of Stochastic Volterra Integro-Differential Equations

A spectral collocation method for stochastic Volterra integro-differential equations and its error analysis

Volterra integro-differential equations arise in the modeling of natural systems where the past influence the present and future, for example pollution, population growth, mechanical systems and financial market. Furthermore, as many real-world phenomena are subject to perturbations or random noise, it is natural to move from deterministic models to stochastic models. Generally exact solutions of such models are not available and numerical methods are used to obtain the approximate solutions. Therefore the efficiency and long-term behavior of approximate solutions for these systems is an important area of investigation. This paper presents a new numerical approach for the approximate solution of stochastic Volterra integro-differential (SVID) equations based on the Legendre-spectral collocation method. In order to fully use the properties of orthogonal polynomials, we use some function and a variable transformation to change the given SVID equation into a new equation, which is defined on the standard interval [-1,1]. For the evaluation of the integral term efficiently a Legendre–Gauss quadrature formula will be used. A rigorous error analysis of the proposed scheme will be provided under the assumption that the solution of the given SVID is sufficiently smooth. For the illustration of our theoretical results a number of numerical experiments will be performed.

Stochastic Volterra integro-differential equations driven by a fractional Brownian motion with delayed impulses

In this paper, the problem of existence of mild solutions for a stochastic Volterra integrodifferential equation with delayed impulses and driven by a fractional Brownian motion (Hurst parameter H ? (1/2,1)) is investigated. Here, we assume that the delayed impulses are linear and impulsive transients depend on not only their current but also historical states of the system. Utilizing the fixed point theorem combine with fractional power of operators and the semi-group theory, sufficient conditions that guarantee the existence and uniqueness of mild solutions for such equation are obtained. Finally, an example is presented to demonstrate the effectiveness of the proposed results.

The density of solutions to multifractional stochastic Volterra integro-differential equations

The aim of this paper is to study the density of solutions to stochastic Volterra integro-differential equations with multifractional noises. We prove the existence of the density of solutions and show that the density is bounded by lower and upper Gaussian bounds.

Weak Convergence of Finite Element Approximations of Linear Stochastic Evolution Equations with Additive Lévy Noise

We present an abstract framework to study weak convergence of numerical approximations of linear stochastic partial differential equations driven by additive Levy noise. We first derive a representation formula for the error which we then apply to study space-time discretizations of the stochastic heat equation, a Volterra-type integro-differential equation, and the wave equation as examples. For twice continuously differentiable test functions with bounded second derivative (with an additional condition on the second derivative for the wave equation) the weak rate of convergence is found to be twice the strong rate. The results extend earlier work by two of the authors, as we consider general square-integrable infinite-dimensional Levy processes and do not require boundedness of the test functions and their first derivative. Furthermore, the present framework is applicable to both hyperbolic and parabolic equations, and even to stochastic Volterra integro-differential equations.