Stochastic Volterra Integral Equations Research Articles

Mean-square stability and convergence of a split-step theta method for stochastic Volterra integral equations

In this paper, a split-step θ method is introduced for solving stochastic Volterra integral equations with general smooth kernels. First, the mean-square boundedness and convergence properties of the numerical solution are analyzed. In particular, when the kernel function in the stochastic integral term satisfies a certain condition, the method can achieve superconvergence. Then, the mean-square stability of the method with respect to a convolution test equation is studied. A recurrence relation is found and mean-square stability regions are given by using root locus method. In particular, when the test equation degrades to the deterministic case, the new method with θ≥1∕2 is V0-stable (i.e., it can preserve the stability of the convolution test equation), which is superior to the stochastic θ method. Finally, some numerical experiments are given to verify the theoretical results.

Stochastic Volterra integral equations with jumps and the strong superconvergence of the Euler–Maruyama approximation

We study the existence and uniqueness of a solution to the stochastic Volterra integral equations (SVIEs) with jumps. Moreover, we apply the Euler–Maruyama approximation for SVIEs with jumps and investigate the boundedness and the convergence of the numerical solution. Furthermore, we show that the numerical solution strongly superconverges with order 1 if the diffusion and jump coefficients satisfy b(t,t)=0 and c(t,t,ξ)=0. Otherwise, the numerical solution is strongly convergent of order 12. The theoretical results are illustrated by numerical simulations and comprehensive examples.

Backward doubly stochastic Volterra integral equations and their applications

In this paper, we introduce a new class of equations called backward doubly stochastic Volterra integral equations (BDSVIEs, for short). First, the well-posedness of BDSVIEs in the sense of introduced M-solutions is established. Second, a comparison theorem of BDSVIEs is proved. As an application of the comparison theorem, we derive the existence of solutions of BDSVIEs with continuous coefficients. Third, a duality principle between linear forward doubly stochastic Volterra integral equations (FDSVIEs, for short) and BDSVIEs is obtained. Moreover, by virtue of the duality principle, a maximum principle of Pontryagin type is established for an optimal control problem of FDSVIEs.

Existence and smoothness of the density of the solution to fractional stochastic integral Volterra equations

We consider stochastic Volterra integral equations driven by a fractional Brownian motion with Hurst parameter . We first derive supremum norm estimates for the solution and its Malliavin derivative. We then show existence and smoothness of the density under suitable nondegeneracy conditions. This extends the results in Hu and Nualart [Differential equations driven by Hölder continuous functions of order greater than 1/2, Stoch. Anal. Appl. Abel Symp. 2 (2007), pp. 399–413] and Nualart and Saussereau [Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion, Stoch. Process. Appl. 119 (2009), pp. 391–409] where stochastic differential equations driven by fractional Brownian motion are considered. The proof uses a priori estimates for deterministic differential equations driven by a function in a suitable Sobolev space.

An iterative shifted Chebyshev method for nonlinear stochastic Itô–Volterra integral equations

In the present work, we propose an iterative method based on the shifted Chebyshev polynomials for the numerical investigation of the nonlinear stochastic Volterra integral equations. Moreover, the convergence of the proposed method is studied. The proposed iterative method is fast and easy to implement. Some numerical examples are given to show the applicability and efficiency of the presented method. The numerical results of the proposed method are compared with the reported results in the literature and with the exact solutions if they exist. The reported comparison confirm the applicability and high accuracy of the proposed method.

Lévy-driven stochastic Volterra integral equations with doubly singular kernels: existence, uniqueness, and a fast EM method

This paper considers Levy noise driven nonlinear stochastic Volterra integral equations with doubly weakly singular kernels, whose singular points include both s = 0 and s = t. The existence and uniqueness theorem of the true solution as well as the strong convergence rate of the Euler–Maruyama (EM) method are developed via establishing some fine estimates. Compared with the corresponding results by Wang (Statist Probab Lett 78: 1062–1071, 2008) and Zhang (J Differential Equations 244: 2226–2250, 2008), our results generalize the Gaussian noise case to the Levy noise case, relax the integrable limitations of singular kernels and establish an accurate convergence order. Moreover, based on the efficient sum-of-exponentials approximation, a fast EM method is presented to improve the low computational efficiency of the EM method. Specifically, when T ≫ 1, the computational complexity O(N2) and the storage O(N) of the EM method are reduced to $O(N\log N)$ and $O(\log N)$ respectively; while T ≈ 1, they are reduced to $O(N\log ^{2} N)$ and $O(\log ^{2} N)$ respectively, where T and N denote the terminal time and the total number of time steps respectively. Finally, the numerical example supports the theoretical results and explains the priority of the fast EM method.

Extended backward stochastic Volterra integral equations, Quasilinear parabolic equations, and Feynman–Kac formula

This paper is concerned with the relationship between backward stochastic Volterra integral equations (BSVIEs, for short) and a kind of non-local quasilinear (and possibly degenerate) parabolic equations. As a natural extension of BSVIEs, the extended BSVIEs (EBSVIEs, for short) are introduced and investigated. Under some mild conditions, the well-posedness of EBSVIEs is established and some regularity results of the adapted solution to EBSVIEs are obtained via Malliavin calculus. Then it is shown that a given function expressed in terms of the adapted solution to EBSVIEs uniquely solves a certain system of non-local parabolic equations, which generalizes the famous nonlinear Feynman–Kac formula in Pardoux–Peng [Backward stochastic differential equations and quasilinear parabolic partial differential equations, in Stochastic Partial Differential Equations and Their Applications (Springer, 1992), pp. 200–217].

Necessary conditions of Pontraygin’s type for general controlled stochastic Volterra integral equations

This article is addressed to giving a solution to a unsolved problem, i.e., to establish the necessary optimality conditions of Pontraygin’s type for controlled stochastic Volterra integral equations (SVIEs) when the control region is non-convex and the control variable enters into the diffusion. This problem has been open since [J. Yong, Stochastic Process Appl. 116 (2006) 779–795] obtained the analogue result for the case of convex control region. The key is to introduce a pair of suitable second-order adjoint processes (SOAPs). It is found that the usual way of using only one SOAP in the maximum condition for the classical setting of controlled stochastic differential equations does not work here.

Recursive Utility Processes, Dynamic Risk Measures and Quadratic Backward Stochastic Volterra Integral Equations

For an $\cF_T$-measurable payoff of a European type contingent claim, the recursive utility process/dynamic risk measure can be described by the adapted solution to a backward stochastic differential equation (BSDE). However, for an $\cF_T$-measurable stochastic process (called a position process, not necessarily $\dbF$-adapted), mimicking BSDE's approach will lead to a time-inconsistent recursive utility/dynamic risk measure. It is found that a more proper approach is to use the adapted solution to a backward stochastic Volterra integral equation (BSVIE). The corresponding notions are called equilibrium recursive utility and equilibrium dynamic risk measure, respectively. Motivated by this, the current paper is concerned with BSVIEs whose generators are allowed to have quadratic growth (in $Z(t,s)$). The existence and uniqueness for both the so-called adapted solutions and adapted M-solutions are established. A comparison theorem for adapted solutions to the so-called Type-I BSVIEs is established as well. As consequences of these results, some general continuous-time equilibrium dynamic risk measures and equilibrium recursive utility processes are constructed.

Theoretical Error Analysis of Solution for Two-Dimensional Stochastic Volterra Integral Equations by Haar Wavelet

The finding an efficient way to the approximate solutions of the stochastic integral equations is an essential requirement. In this paper we discuss the convergence analysis of the two-dimensional Haar wavelet functions (2D-HWFs) method for solve 2D linear stochastic Volterra integral equation. The illustrative examples are included to demonstrate the validity and applicability of this numerical method.

Symmetrical martingale solutions of backward doubly stochastic Volterra integral equations

This paper aims to study a new class of integral equations called backward doubly stochastic Volterra integral equations (BDSVIEs, for short). The notion of symmetrical martingale solutions (SM-solutions, for short) is introduced for BDSVIEs. And the existence and uniqueness theorem for BDSVIEs in the sense of SM-solutions is established.

Solvability of anticipated backward stochastic Volterra integral equations

In this paper, we focus on a new class of equations called the anticipated backward stochastic Volterra integral equations (ABSVIEs, for short). In this class of equations, the generator involves not only the present information of the solution but also its future one. Under the Lipschitz condition, we obtain the existence and uniqueness of the adapted M-solutions. Meanwhile, a comparison theorem is also proved.

Backward Stochastic Volterra Integral Equations With Non-Lipschitz Time Delayed Generators

In this note, we derive existence and uniqueness results for backward stochastic Volterra integral equations with time delayed generators under non-Lipschitz condition.

Quantum integral equations of Volterra type in terms of discrete-time normal martingale

In this paper, we aim to introduce a quantum linear stochastic Volterra integral equation of convolution type with operator-valued kernels in a nuclear topological algebra. We first establish the existence and uniqueness of the solutions and give the explicit expression of the solutions. Then we prove the continuity, continuous dependence on free terms and other properties of the solution.

A New Computational Method Based on Bernstein Operational Matrices for Solving Two-Dimensional Linear Stochastic Volterra Integral Equations

This paper is concerned with obtaining the approximate numerical solution of two-dimensional linear stochastic Volterra integral equation by using two-dimensional Bernstein polynomials as basis. Properties of these polynomials and operational matrix of integration together with the product operational matrix are utilized to transform the integral equation to a matrix equation which corresponds to a system of linear algebraic equations with unknown Bernstein coefficients. Some theorems are included to show the convergence and advantage of the proposed method. The numerical example illustrates the efficiency and accuracy of the method.

Well-posedness and EM approximations for non-Lipschitz stochastic fractional integro-differential equations

This paper considers the nonlinear stochastic fractional integro-differential equations (SFIDEs) under the non-Lipschitz conditions, which are general and include many stochastic (fractional) integro-differential equations discussed in literature. An important connection between SFIDEs and stochastic Volterra integral equations (SVIEs) is derived in detail by the Fubini theorem. Using the Euler–Maruyama (EM) approximation, we prove the existence, uniqueness and stability results of the solution to SFIDEs. Moreover, it is shown that the modified EM solution of SFIDEs shares strong first-order sharp convergence. The numerical examples are performed to show the accuracy and effectiveness of the numerical scheme and verify the correctness of our theoretical analysis. • The model covers many nonlinear stochastic (fractional) integro-differential models. • Well-posedness of the models is investigated under the non-Lipschitz condition. • The modified EM approximation of the models admits strong first-order convergence.

Backward stochastic Volterra integral equations—Representation of adapted solutions

For backward stochastic Volterra integral equations (BSVIEs, for short), under some mild conditions, the so-called adapted solutions or adapted M-solutions uniquely exist. However, satisfactory regularity of the solutions is difficult to obtain in general. Inspired by the decoupling idea of forward–backward stochastic differential equations, in this paper, for a class of BSVIEs, a representation of adapted M-solutions is established by means of the so-called representation partial differential equations and (forward) stochastic differential equations. Well-posedness of the representation partial differential equations are also proved in certain sense.

Numerical solution of linear stochastic Volterra integral equations via new basis functions

In this article, we use a new method based on orthogonal basis functions for the numerical solution of stochastic Volterra integral equations of the second kind (SVIE). By using this method, a SVIE can be reduced to a linear system of algebraic equations. Finally, to show the efficiency of the proposed method, we give two numerical examples.

Strong convergence of the semi-implicit Euler method for nonlinear stochastic Volterra integral equations with constant delay

This paper is mainly concerned with the strong convergence analysis of the semi-implicit Euler method for nonlinear stochastic Volterra integral equations (SVIEs) with constant delay. The solvability and the boundedness of the numerical solution are established. It is proved that the strong convergence order of the semi-implicit Euler method is 0.5 under Lipschitz conditions. Moreover, the strong superconvergence order is 1.0 if further, the kernel σ of the stochastic term satisfies σ(0)=σ(τ)=0. The theoretical results are illustrated by extensive numerical examples.

New approach to optimal control of stochastic Volterra integral equations

ABSTRACTWe 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.