Stochastic Integral Equations Research Articles

Numerical solution for stochastic Volterra-Fredholm integral equations with delay arguments

We present a method for computing the stochastic operational matrix of integration to advance the study of stochastic Volterra-Fredholm integral equations (SVFIEs) based on delay arguments. First, the method evaluates the combined effects of the delay and its parameters on the accuracy improvement of the convergence rate. Our results can be applied to SVFIEs, with the operational delay matrices of the block pulse function simplified to algebraic ones. Numerical calculations were performed on a PC using Python 3 programs. Results also demonstrate the accuracy of approximate solutions; arithmetic operations are carried out without the need for derivation or integration.

Strong convergence analysis of spectral fractional diffusion equation driven by Gaussian noise with Hurst parameter less than [formula omitted]

We propose and analyze an efficient time discretization for the spectral fractional stochastic partial differential equation with Hurst parameter less than 12. By using variable substitution, the original equation is transformed into a system of equations, which includes a partial differential equation and a stochastic integral equation. Then the time discretization is composed of two parts. We discretize the partial differential equation in time by using a semi-implicit Euler scheme. Integration by parts is used to obtain the approximation of stochastic integral. We show the optimal error estimates of the time discretization in the strong sense. Finally, we provide some numerical examples to verify these theoretical results.

Stability Results for a Class of Fractional Itô–Doob Stochastic Integral Equations

In this paper, we study the Hyers–Ulam stability of Hadamard fractional Itô–Doob stochastic integral equations by using the Banach fixed point method and some mathematical inequalities. Finally, we exhibit three theoretical examples to apply our theory.

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.

Linear-quadratic stochastic volterra controls II. Optimal strategies and riccati-volterra equations

In this paper, we study linear-quadratic control problems for stochastic Volterra integral equations with singular and non-convolution-type coefficients. The weighting matrices in the cost functional are not assumed to be non-negative definite. From a new viewpoint, we formulate a framework of causal feedback strategies. The existence and the uniqueness of a causal feedback optimal strategy are characterized by means of the corresponding Riccati-Volterra equation. The causal feedback optimal strategy is explicitly written by a finite dimensional (matrix-valued) function which solves the Riccati-Volterra equation.

Correction to: Lagrange interpolation polynomials for solving nonlinear stochastic integral equations

Correction to: Lagrange interpolation polynomials for solving nonlinear stochastic integral equations

On the integration of Ito equations with a random or a W-symmetry

Symmetries can be used to integrate scalar Ito equation – or reduce systems of such equations – by the Kozlov substitution, i.e. passing to symmetry adapted coordinates. While the theory is well established for so called deterministic standard symmetries (the class originally studied by Kozlov), some points need clarification for so called random standard symmetries and W-symmetries. This paper is devoted to such clarification; in particular we note that the theory naturally calls, for these classes of symmetries, to also consider generalized Ito equations; and that while Kozlov theory is extended substantially unharmed for random standard symmetries, W-symmetries should be handled with great care, and cannot be used towards integration of stochastic equations, albeit they have different uses.

Existence and Uniqueness Theorem of Fuzzy Stochastic Ordinary Differential Equations

A fuzzy valued diffusion term, which in a fuzzy stochastic differential equation refers to one-dimensional Brownian motion, is defined by the meaning of the stochastic integral of a fuzzy process. In this paper, the existence and uniqueness theorem of fuzzy stochastic ordinary differential equations, based on the mean square convergence of the mathematical induction approximations to the associated stochastic integral equation, are stated and demonstrated.

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

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

Existence and stability results for the integrable solution of a singular stochastic fractional-order integral equation with delay

In this paper, we are concerning with the existence of the solution \( \V \in L_1([0,\tau],L_2(\Omega))\) of the singular stochastic fractional-order integral equation with delay \(\varrho(.) \), \[ \V(t) = B(t) t^{\alpha - 1} + \lambda ~ \I^{\beta} \G(t,\V(\varrho (t))), ~~~t\in (0,\tau], \] where \(B(t)\) is a given second order mean square stochastic process, \( \lambda \) is a parameter, \(\varrho (t) \leq t\), and \(\G(t,\V) \) is a measurable function in \(t \in (0,\tau]\) and satisfies Lipschitz condition on the second argument. %and \(x\in C_{1-\beta}([0,T],L_2(\Omega)) $ will be proved The Hyers-Ulam and generalized Hyers-Ulam-Rassias stability will be proved. Moreover, the continuous dependence of the solution on the process \(B(t)\) and \(\lambda\) will be studied. As applications, some nonlocal, weighted and nonlocal-weighted integral problems of stochastic fractional-order differential equations will be studied.

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.

Existence results for some stochastic functional integrodifferential systems driven by Rosenblatt process

Abstract This work investigates the existence and uniqueness of mild solutions to a class of stochastic integral differential equations with various time delay driven by the Rosenblatt process. We can obtain alternative conditions that guarantee mild solutions by using the resolvent operator in the Grimmer sense, stochastic analysis, fixed-point methods, and noncompact measures. We give an example to illustrate the theory.

N-Best kernel approximation in reproducing kernel Hilbert spaces

By making a seminal use of the maximum modulus principle of holomorphic functions we prove existence of n-best kernel approximation for a wide class of reproducing kernel Hilbert spaces of holomorphic functions in the unit disc, and for the corresponding class of Bochner type spaces of stochastic processes. This study thus generalizes the classical result of n-best rational approximation for the Hardy space and a recent result of n-best kernel approximation for the weighted Bergman spaces of the unit disc. The type of approximations have significant applications to signal and image processing and system identification, as well as to numerical solutions of the classical and the stochastic type integral and differential equations.

A Numerical Approximation of the Stochastic Ito-Volterra Integral Equation

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms and the solution are stochastic processes. Numerous studies have employed orthogonal polynomials, however most of them focus on deterministic rather than stochastic systems. This is the reason why in this study, we looked into a numerical solution for the stochastic Ito-Volterra integral equation using the explicit finite difference scheme and Bernstein polynomials as trial functions. The equidistant collocation procedure was used to calculate the unknown constant parameters in between and reach the desired approximation. The method was evaluated and contrasted with the Block Pulse method for approximate answers based on the aforementioned method, which were obtained and compared with others in the literature.

Lagrange interpolation polynomials for solving nonlinear stochastic integral equations

Lagrange interpolation polynomials for solving nonlinear stochastic integral equations

The Existence and Averaging Principle for a Class of Fractional Hadamard Itô–Doob Stochastic Integral Equations

In this paper, we investigate the existence and uniqueness properties pertaining to a class of fractional Hadamard Itô–Doob stochastic integral equations (FHIDSIE). Our study centers around the utilization of the Picard iteration technique (PIT), which not only establishes these fundamental properties but also unveils the remarkable averaging principle within FHIDSIE. To accomplish this, we harness powerful mathematical tools, including the Hölder and Gronwall inequalities.

Backward Doubly Stochastic Integral Equations of the Volterra Type and Some Related Problems

Backward Doubly Stochastic Integral Equations of the Volterra Type and Some Related Problems

A novel efficient technique for solving nonlinear stochastic Itô–Volterra integral equations

There is a growing need of stochastic integral equations (SIEs) to investigate the behavior of complex dynamical systems. Since real-world phenomena frequently dependent on noise sources, modeling them naturally necessitates the use of SIEs. As most SIEs cannot be solved explicitly, thus the behaviors of the studied systems are investigated using approximate solutions of their SIEs. Despite the fact that this problem has been soundly investigated and numerous methods have been presented, the practice demonstrated that obtaining satisfied approximations is not always guaranteed, necessitating the development of new effective techniques. This paper gives a new technique for solving nonlinear Itô–Volterra SIEs by reducing them to linear or nonlinear algebraic systems via the power of a combination of generalized Lagrange functions and Jacobi–Gauss collocation points. The accuracy and reliability of the new technique are evaluated and compared with the existing techniques. Moreover, sufficient conditions to make the estimate error tends to zero are given. The new technique shows surprisingly high efficiency over the existing techniques in terms of computational efficiency and approximation capability. The accuracy of the solution based on the new technique is much higher than that via the existing techniques. The required time of the new technique is much less than that of the existing techniques, where, in some circumstances, the existing techniques take more than 20 times as long as the new technique.

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.

Holonomic and Non-Holonomic Geometric Models Associated to the Gibbs–Helmholtz Equation

By replacing the internal energy with the free energy, as coordinates in a “space of observables”, we slightly modify (the known three) non-holonomic geometrizations from Udriste’s et al. work. The coefficients of the curvature tensor field, of the Ricci tensor field, and of the scalar curvature function still remain rational functions. In addition, we define and study a new holonomic Riemannian geometric model associated, in a canonical way, to the Gibbs–Helmholtz equation from Classical Thermodynamics. Using a specific coordinate system, we define a parameterized hypersurface in R4 as the “graph” of the entropy function. The main geometric invariants of this hypersurface are determined and some of their properties are derived. Using this geometrization, we characterize the equivalence between the Gibbs–Helmholtz entropy and the Boltzmann–Gibbs–Shannon, Tsallis, and Kaniadakis entropies, respectively, by means of three stochastic integral equations. We prove that some specific (infinite) families of normal probability distributions are solutions for these equations. This particular case offers a glimpse of the more general “equivalence problem” between classical entropy and statistical entropy.