Stochastic Integro-differential Equations Research Articles

SPECTRAL GALERKIN METHOD FOR STOCHASTIC SPACE-TIME FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION

This work is devoted to deal with a stochastic space-time fractional integro-differential equation in the Hilbert space $ L^{2}(0,1) $, by studing its spatial approximation. Precisely, we use the spectral Galerkin method to prove that the spatial approximation converges strongly (i.e. in the space $ L^{p}(\Omega, L^{2}(0,1)) $), by imposing only a regularity condition on the initial value.

Existence results for some partial stochastic integrodifferential equations with nonlocal conditions in Hilbert spaces

Existence results for some partial stochastic integrodifferential equations with nonlocal conditions in Hilbert spaces

Numerical solution of the stochastic neural field equation with applications to working memory

The main goal of the present work is to investigate the effect of noise in some neural fields, used to simulate working memory processes. The underlying mathematical model is a stochastic integro-differential equation. In order to approximate this equation we apply a numerical scheme which uses the Galerkin method for the space discretization. In this way we obtain a system of stochastic differential equations, which are then approximated in two different ways, using the Euler–Maruyama and the Itô–Taylor methods. We apply this numerical scheme to explain how a population of cortical neurons may encode in its firing pattern simultaneously the nature and time of sequential stimulus events. Numerical examples are presented and their results are discussed.

Optimal controls problems for some impulsive stochastic integro-differential equations with state-dependent delay

ABSTRACT In this paper, optimal control problems for a class of stochastic functional integral-differential equations in Hilbert spaces are investigated. First, the existence of mild solutions is investigated using stochastic analysis theory, fixed point theorems, and Grimmer's resolvent operator theory. Following that, the existence requirements of optimal pairs of systems governed by stochastic partial integro-differential equations with infinite delay are discussed. The results are achieved by the use of a combination of Lipschitz and Carathéodory conditions. At the end of the paper, an illustration is supplied to help highlight the key findings.

DYNAMIC UTILITY AND RELATED NONLINEAR SPDES DRIVEN BY LÉVY NOISE

In this work, we study a class of consistent dynamic utilities in a incomplete financial market including jumps. First, we show that the dynamic utility is solution of a non-linear second-order stochastic partial integro-differential equation (SPIDE). Second, a complete study of the primal and the dual problems, allows us, firstly, to establish a connection between the utility-SPIDE and two SDEs satisfied by the optimal processes. Based on this connection, stochastic flow technics for SDEs and characteristic method, the SPIDE is completely solved and monotony and concavity properties of the solution are proved.

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>

Optimal Controls for Neutral Stochastic Integrodifferential Equations with Infinite Time Delay and Deviated Argument Driven by Rosenblatt Process

Optimal Controls for Neutral Stochastic Integrodifferential Equations with Infinite Time Delay and Deviated Argument Driven by Rosenblatt Process

Trajectory Controllability of Stochastic Integro-Differential Equation with State Dependent Delay Driven by Mixed Brownian Motion Suffered by Non-Instantaneous Impulses

Trajectory Controllability of Stochastic Integro-Differential Equation with State Dependent Delay Driven by Mixed Brownian Motion Suffered by Non-Instantaneous Impulses

Atangana–Baleanu time-fractional stochastic integro-differential equation

We study the Atangana–Baleanu time-fractional stochastic integro-differential equation. Using Banach fixed point theory, we prove existence and uniqueness of mild solution. In addition, upper second moment growth estimate of the mild solution is proved and the result gives a precise exponential growth. We also give some specific examples and illustrate their growth moment bounds with graphs.

Uniqueness and Stability Results on Non-local Stochastic Random Impulsive Integro-Differential Equations

The paper is concerned with stochastic random impulsive integro-differential equations with non-local conditions. The sufficient conditions guarantees uniqueness of mild solution derived using Banach fixed point theorem. Stability of the solution is derived by incorporating Banach fixed point theorem with certain inequality techniques.

Asymptotic behavior of solutions of some stochastic integro-differential equations with infinite delay

In this paper, a new concept of square-mean compact asymptotically almost automorphy for stochastic processes is introduced and a composition theorem is also proved. And then, this notion is applied to investigate existence and uniqueness theorems for squaremean compact asymptotically almost automorphic mild solutions to an abstract stochastic integro-differential equation with infinite delay. Moreover, an example is given to illustrate our main results.

Null controllability of Hilfer fractional stochastic integrodifferential equations with noninstantaneous impulsive and Poisson jump

Abstract In this paper, we investigate the sufficient conditions for null controllability of noninstantaneous impulsive Hilfer fractional stochastic integrodifferential system with the Rosenblatt process and Poisson jump. The required results are obtained based on fractional calculus, stochastic analysis, and Sadovskii’s fixed point theorem. Finally, an example is given to illustrate the obtained results.

Variations of the solution to a fourth order time-fractional stochastic partial integro-differential equation

Variations of the solution to a fourth order time-fractional stochastic partial integro-differential equation

Approximate Controllability of Second Order Neutral Stochastic Integro Differential Equations with Impulses Driven By Fractional Brownian Motion

Approximate Controllability of Second Order Neutral Stochastic Integro Differential Equations with Impulses Driven By Fractional Brownian Motion

Controllability of Neutral Impulsive Stochastic Integrodifferential Equations Driven by a Rosenblatt Process and Unbounded Delay

In this manuscript, we establish the controllability of neutral impulsive stochastic integrodifferential equations driven by a Rosenblatt process with infinite delay in separable Hilbert space. The controllability results is obtained by using fixed-point technique and via resolvent operator.

Reduced-order models for coupled dynamical systems: Data-driven methods and the Koopman operator.

Providing efficient and accurate parameterizations for model reduction is a key goal in many areas of science and technology. Here, we present a strong link between data-driven and theoretical approaches to achieving this goal. Formal perturbation expansions of the Koopman operator allow us to derive general stochastic parameterizations of weakly coupled dynamical systems. Such parameterizations yield a set of stochastic integrodifferential equations with explicit noise and memory kernel formulas to describe the effects of unresolved variables. We show that the perturbation expansions involved need not be truncated when the coupling is additive. The unwieldy integrodifferential equations can be recast as a simpler multilevel Markovian model, and we establish an intuitive connection with a generalized Langevin equation. This connection helps setting up a parallelism between the top-down, equation-based methodology herein and the well-established empirical model reduction (EMR) methodology that has been shown to provide efficient dynamical closures to partially observed systems. Hence, our findings, on the one hand, support the physical basis and robustness of the EMR methodology and, on the other hand, illustrate the practical relevance of the perturbative expansion used for deriving the parameterizations.

Controllability for nonlocal stochastic integrodifferential evolution equations with the lack of compactness

In this work, we study the controllability of a class of stochastic integrodifferential equations with noncompact semigroups and nonlocal condition on Hilbert spaces.We suppose that the linear part admits a resolvent operator in the sense of Grimmer. Our main results are obtained by utilizing stochastic analysis theory, Kuratowski measure of noncompactness and Mönch fixed point Theorem. As a result, we obtain a generalization of a host of important results in the literature, without assuming the compactness of the resolvent operator. Finally, we provide an example to illustrate the efficiency of our results.

Construction of operational matrices based on linear cardinal B-spline functions for solving fractional stochastic integro-differential equation

The main purpose of this paper is to develop a new method based on operational matrices of the linear cardinal B-spline (LCB-S) functions to numerically solve of the fractional stochastic integro-differential (FSI-D) equations. To reach this aim, LCB-S functions are introduced and their properties are considered, briefly. Then, the operational matrices based on LCB-S functions are constructed, for the first time, including the fractional Riemann-Liouville integral operational matrix, the stochastic integral operational matrix, and the integer integral operational matrix. The main characteristic of the new scheme is to convert the FSI-D equation into a linear system of algebraic equations which can be easily solved by applying a suitable method. Also, the convergence analysis and error estimate of the proposed method are studied and an upper bound of error is obtained. Numerical experiments are provided to show the potential and efficiency of the new method. Finally, some numerical results, for various values of perturbation in the parameters of the main problem are presented which can indicate the stability of the suggested method.

Spatial Moduli of Non-Differentiability for Time-Fractional SPIDEs and Their Gradient

High order and fractional PDEs have become prominent in theory and in modeling many phenomena. In this paper, we study spatial moduli of non-differentiability for the fourth order time fractional stochastic partial integro-differential equations (SPIDEs) and their gradient, driven by space-time white noise. We use the underlying explicit kernels and spectral/harmonic analysis, yielding spatial moduli of non-differentiability for time fractional SPIDEs and their gradient. On one hand, this work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. On the other hand, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of time fractional SPIDEs and their gradient.

Graph neural fields: A framework for spatiotemporal dynamical models on the human connectome.

Tools from the field of graph signal processing, in particular the graph Laplacian operator, have recently been successfully applied to the investigation of structure-function relationships in the human brain. The eigenvectors of the human connectome graph Laplacian, dubbed "connectome harmonics", have been shown to relate to the functionally relevant resting-state networks. Whole-brain modelling of brain activity combines structural connectivity with local dynamical models to provide insight into the large-scale functional organization of the human brain. In this study, we employ the graph Laplacian and its properties to define and implement a large class of neural activity models directly on the human connectome. These models, consisting of systems of stochastic integrodifferential equations on graphs, are dubbed graph neural fields, in analogy with the well-established continuous neural fields. We obtain analytic predictions for harmonic and temporal power spectra, as well as functional connectivity and coherence matrices, of graph neural fields, with a technique dubbed CHAOSS (shorthand for Connectome-Harmonic Analysis Of Spatiotemporal Spectra). Combining graph neural fields with appropriate observation models allows for estimating model parameters from experimental data as obtained from electroencephalography (EEG), magnetoencephalography (MEG), or functional magnetic resonance imaging (fMRI). As an example application, we study a stochastic Wilson-Cowan graph neural field model on a high-resolution connectome graph constructed from diffusion tensor imaging (DTI) and structural MRI data. We show that the model equilibrium fluctuations can reproduce the empirically observed harmonic power spectrum of resting-state fMRI data, and predict its functional connectivity, with a high level of detail. Graph neural fields natively allow the inclusion of important features of cortical anatomy and fast computations of observable quantities for comparison with multimodal empirical data. They thus appear particularly suitable for modelling whole-brain activity at mesoscopic scales, and opening new potential avenues for connectome-graph-based investigations of structure-function relationships.