Stochastic Volterra Integral Equations Research Articles

Well-posedness and regularity of mean-field backward doubly stochastic Volterra integral equations and applications to dynamic risk measures

In this paper, the theory of mean-field backward doubly stochastic Volterra integral equations (MF-BDSVIEs) is studied. First, we derive the well-posedness of M-solutions to MF-BDSVIEs, and prove the comparison theorem for such a type of equations. Furthermore, the regularity result of the M-solution for MF-BDSVIEs is established by virtue of Malliavin calculus. Finally, as an application of the comparison theorem, we obtain the properties of dynamic risk measures governed by MF-BDSVIEs.

Spike Variations for Stochastic Volterra Integral Equations

Spike Variations for Stochastic Volterra Integral Equations

Numerical approximation of nonlinear stochastic Volterra integral equation based on Walsh function

Numerical approximation of nonlinear stochastic Volterra integral equation based on Walsh function

On Ulam type of stability for stochastic integral equations with Volterra noise

Abstract This paper concerns the existence, uniqueness and stability of solutions of stochastic Volterra integral equations perturbed by some random processes. The obtained results extend, generalize and enrich the theory of stochastic Volterra integral equations in literature. Lastly, for illustration, we give an example that agrees with the theoretical analysis.

Cubature Method for Stochastic Volterra Integral Equations

Cubature Method for Stochastic Volterra Integral Equations

Numerical Solution of Nonlinear Backward Stochastic Volterra Integral Equations

This work uses the collocation approximation method to solve a specific type of backward stochastic Volterra integral equations (BSVIEs). Using Newton’s method, BSVIEs can be solved using block pulse functions and the corresponding stochastic operational matrix of integration. We present examples to illustrate the estimate analysis and to demonstrate the convergence of the two approximating sequences separately. To measure their accuracy, we compare the solutions with values of exact and approximative solutions at a few selected locations using a specified absolute error. We also propose an efficient method for solving a triangular linear algebraic problem using a single integral equation. To confirm the effectiveness of our method, we conduct numerical experiments with issues from real-world applications.

Numerical solution of stochastic Volterra integral equations based on uniform Haar wavelets by using direct method

Nowadays, wavelets have played a crucial role in approximating the solution of a wide range of problems arising in science and engineering. Wavelets have been used in numerous areas of applied mathematics as diverse as signal analysis, statistics, computer-aided geometric, image processing and numerical analysis. The uniform Haar wavelets are most widely used wavelets for solving stochastic integral equations, outperforming Legendre, Daubechies, etc. The uniform Haar wavelets are generated from pairs of piecewise constant functions and can be simply integrated. Furthermore, the uniform Haar functions are orthogonal and form a good transform basis. In this paper, we have proposed an efficient numerical method to solve a class of stochastic linear Volterra integral equations (SLVIEs) based on the uniform Haar wavelets. The aim of this paper is to apply numerical method to obtain approximate solution of SLVIEs. By using this method, the SLVIEs can be reduced to a linear system of algebraic equations. Also, the error analysis of this method is performed. Finally, some numerical examples are given to demonstrate pertinent features of this method. Furthermore, we compare obtained results from this method with the achieved results from relevant studies.

Linear-Quadratic Optimal Controls for Stochastic Volterra Integral Equations: Causal State Feedback and Path-Dependent Riccati Equations

Linear-Quadratic Optimal Controls for Stochastic Volterra Integral Equations: Causal State Feedback and Path-Dependent Riccati Equations

Anticipated Backward Stochastic Volterra Integral Equations with Jumps and Applications to Dynamic Risk Measures

Anticipated Backward Stochastic Volterra Integral Equations with Jumps and Applications to Dynamic Risk Measures

Dynamic risk measures via backward doubly stochastic Volterra integral equations with jumps

In this article, we study dynamic risk measures by means of backward doubly stochastic Volterra integral equations (BDSVIEs, for short) with jumps. We establish the well-posedness of BDSVIEs with jumps in the sense of M-solution and prove a comparison theorem of BDSVIEs with jumps. Finally, we study properties of dynamic risk measures induced by BDSVIEs with jumps. Our results extend the well-posedness and the comparison theorem of BDSVIEs without jumps to the setting with jumps, and extend dynamic risk measures induced by BSDEs, BDSDEs, and BSVIEs to the case of BDSVIEs with jumps.

On the Maximum Principle for Optimal Control Problems of Stochastic Volterra Integral Equations with Delay

In this paper, we prove both necessary and sufficient maximum principles for infinite horizon discounted control problems of stochastic Volterra integral equations with finite delay and a convex control domain. The corresponding adjoint equation is a novel class of infinite horizon anticipated backward stochastic Volterra integral equations. Our results can be applied to discounted control problems of stochastic delay differential equations and fractional stochastic delay differential equations. As an example, we consider a stochastic linear-quadratic regulator problem for a delayed fractional system. Based on the maximum principle, we prove the existence and uniqueness of the optimal control for this concrete example and obtain a new type of explicit Gaussian state-feedback representation formula for the optimal control.

Singular stochastic Volterra integral equations with Mittag–Leffler kernels: well-posedness and strong convergence of θ-Maruyama method

This paper focusses on the singular stochastic Volterra integral equations with Mittag–Leffler kernels. Some qualitative properties of the solution are given under local Lipschitz condition, which include uniqueness and existence, boundedness of pth moments, Hölder continuity and continuous dependence on the initial value. The θ-Maruyama method is proposed for solving the equations. The strong convergence results of this method are obtained under global Lipschitz condition and local Lipschitz condition, respectively. Some numerical examples are given to verify the theoretical results.

Fast $$\theta $$-Maruyama scheme for stochastic Volterra integral equations of convolution type: mean-square stability and strong convergence analysis

Fast $$\theta $$-Maruyama scheme for stochastic Volterra integral equations of convolution type: mean-square stability and strong convergence analysis

Approximations for adapted M-solutions of type-II backward stochastic Volterra integral equations

In this paper, we study a class of Type-II backward stochastic Volterra integral equations (BSVIEs). For the adapted M-solutions, we obtain two approximation results, namely, a BSDE approximation and a numerical approximation. The BSDE approximation means that the solution of a finite system of backward stochastic differential equations (BSDEs) converges to the adapted M-solution of the original equation. As a consequence of the BSDE approximation, we obtain an estimate for the L2-time regularity of the adapted M-solutions of Type-II BSVIEs. For the numerical approximation, we provide a backward Euler-Maruyama scheme, and show that the scheme converges in the strong L2-sense with the convergence speed of order 1/2. These results hold true without any differentiability conditions for the coefficients.

Multi-Patch Epidemic Models with General Exposed and Infectious Periods

We study multi-patch epidemic models where individuals may migrate from one patch to another in either of the susceptible, exposed/latent, infectious and recovered states. We assume that infections occur both locally with a rate that depends on the patch as well as “from distance” from all the other patches. The migration processes among the patches in either of the four states are assumed to be Markovian, and independent of the exposed and infectious periods. These periods have general distributions, and are not affected by the possible migrations of the individuals. The infection “from distance” aspect introduces a new formulation of the infection process, which, together with the migration processes, brings technical challenges in proving the functional limit theorems. Generalizing the methods in Pang and Pardoux [Ann. Appl. Probab. 32 (2022) 1615–1665], we establish a functional law of large number (FLLN) and a function central limit theorem (FCLT) for the susceptible, exposed/latent, infectious and recovered processes. In the FLLN, the limit is determined by a set of Volterra integral equations. In the special case of deterministic exposed and infectious periods, the limit becomes a system of ODEs with delays. In the FCLT, the limit is given by a set of stochastic Volterra integral equations driven by a sum of independent Brownian motions and continuous Gaussian processes with an explicit covariance structure.

Mean field backward doubly stochastic Volterra integral equations and their applications

This paper deals with a class of mean field backward doubly stochastic Volterra integral equations (MFBDSVIEs, for short). Firstly, we prove the existence and uniqueness results in the sense of M-solutions for MFBDSVIEs as well as a comparison theorem is obtained. Based on comparison theorem, we will study the existence of minimal solution with linear growth condition. Secondly, two duality principles between linear mean field forward doubly stochastic Volterra integral equations (MFFDSVIEs, for short) and MFBDSVIEs are derived. As an application for the duality principles, we show a maximum principle of Pontryagin type for an optimal control problem of MFFDSVIEs.

Variation of constants formulae for forward and backward stochastic Volterra integral equations

In this paper, we provide variation of constants formulae for linear (forward) stochastic Volterra integral equations (SVIEs, for short) and linear Type-II backward stochastic Volterra integral equations (BSVIEs, for short) in the usual Itô's framework. For these purposes, we define suitable classes of stochastic Volterra kernels and introduce new notions of the products of adapted L2-processes. Observing the algebraic properties of the products, we obtain the variation of constants formulae by means of the corresponding resolvent. Our framework includes SVIEs with singular kernels such as fractional stochastic differential equations. Also, our results can be applied to general classes of SVIEs and BSVIEs with infinitely many iterated stochastic integrals. The duality principle between generalized SVIEs and generalized BSVIEs is also proved.

Functional central limit theorems for epidemic models with varying infectivity

In this paper, we prove a functional central limit theorem (FCLT) for a stochastic epidemic model with varying infectivity and general infectious periods recently introduced in R. Forien et al. [Epidemic models with varying infectivity, SIAM J. Appl. Math. 81 (2021), pp. 1893–1930]. The infectivity process (total force of infection at each time) is composed of the independent infectivity random functions of each infectious individual, which starts at the time of infection. These infectivity random functions induce the infectious periods (as well as exposed, recovered or immune periods in full generality), whose probability distributions can be very general. The epidemic model includes the generalized non–Markovian SIR, SEIR, SIS, SIRS models with infection-age dependent infectivity. In the FCLTs for the generalized SIR and SEIR models, the limits of the diffusion-scaled fluctuations of the infectivity and susceptible processes are a unique solution to a two-dimensional Gaussian-driven stochastic Volterra integral equations, and then given these solutions, the limits for the infected (exposed/infectious) and recovered processes are Gaussian processes expressed in terms of the solutions to those stochastic Volterra integral equations. We also present the FCLTs for the generalized SIS and SIRS models.

Some Results on Backward Stochastic Differential Equations of Fractional Order

In this article, we deal with fractional stochastic differential equations, so-called Caputo type fractional backward stochastic differential equations (Caputo fBSDEs, for short), and study the well-posedness of an adapted solution to Caputo fBSDEs of order \(\alpha \in (\frac{1}{2},1)\) whose coefficients satisfy a Lipschitz condition. A novelty of the article is that we introduce a new weighted norm in the square integrable measurable function space that is useful for proving a fundamental lemma and its well-posedness. For this class of systems, we then show the coincidence between the notion of stochastic Volterra integral equation and the mild solution.

Fast Euler–Maruyama method for weakly singular stochastic Volterra integral equations with variable exponent

Fast Euler–Maruyama method for weakly singular stochastic Volterra integral equations with variable exponent