Stochastic Integro-differential Equations Research Articles

Stepanov-like Pseudo S-Asymptotically (ω,c)-Periodic Solutions of a Class of Stochastic Integro-Differential Equations

The study of long-term behavior in stochastic systems is critical for understanding the dynamics of complex processes influenced by randomness. This paper addresses the existence and uniqueness of Stepanov-like pseudo S-asymptotically (ω,c)-periodic solutions for a class of stochastic integro-differential equations. These equations model systems where the interplay between deterministic and stochastic components dictates the overall dynamics, making periodic analysis essential. The problem addressed in this study is the lack of a comprehensive framework to describe the periodic behavior of such systems in noisy environments. To tackle this, we employ advanced techniques in stochastic analysis, fixed-point theorems and the properties of L1- and L2-convolution kernels to establish conditions for the existence and uniqueness of mild solutions under these extended periodicity settings. The methodology involves leveraging the decay properties of the operator kernels and the boundedness of stochastic integrals to ensure well-posedness. The major outputs of this study include novel results on the existence, uniqueness and stability of Stepanov-like pseudo S-asymptotically (ω,c)-periodic solutions, along with illustrative example demonstrating their applicability in real-world stochastic systems.

Optimal controls for multi-term fractional stochastic integro-differential equations with impulses and infinite delay

This paper aims to investigate a new class of functional fractional stochastic impulsive integro-differential equations with infinite delay in Hilbert space with two Riemann–Liouville fractional derivatives. To demonstrate the existence of mild solutions for the considered systems, we use the fixed point approach, stochastic analysis, fractional calculus and ( α , β , λ ) resolvent family. Moreover, by utilising the Marzur lemma, we derived the optimal control results for the proposed problem. Finally, we provide an example to illustrate the obtained results.

Existence Results and Trajectory Controllability of Conformable Hilfer Fractional Neutral Stochastic Integro-differential Equations

This manuscript commits to analyzing the existence and uniqueness of mild solution and trajectory controllability of conformable Hilfer fractional neutral stochastic integro-differential system with infinite delay through Lipschitz, growth conditions and properties of semigroup theory, controllability theory, stochastic analysis techniques, and Banach fixed point theorem with example. This manuscript mainly focuses on fractional system provide the mathematical foundation to understand and control complex, real-world systems with memory, and randomness.

Existence and exponential stability in pth moment of non-autonomous stochastic integro-differential equations

This study is concerned with the existence and exponential stability of solutions of non-autonomous neutral stochastic integro-differential equations with delay; the linear part of this equation is dependent on time and generates a linear evolution system. The obtained results are applied to some neutral stochastic integro-differential equations. These kinds of equations arise in systems related to couple oscillators in a noisy environment or in viscoelastic materials under random or stochastic influences. Finally, we provide an example to illustrate the results.

On solvability and optimal controls for impulsive stochastic integrodifferential varying-coefficient model

This article concentrates in analyzing optimal controls for stochastic integrodifferential equation (SIDE) in Hilbert space. Necessary parameters are imposed to demonstrate the system that follows a unique variation of parameter formula using Leray Schauder Alternative. Subsequently, the existence of optimal control is investigated for the considered Lagrange control problem. The theoretical example with the mechanical example of ethanol fuelled engine are discussed to validate the results obtained.

Finite dimensional approximation to fractional stochastic integro-differential equations with non-instantaneous impulses

This manuscript proposes a class of fractional stochastic integro-differential equations (FSIDEs) with non-instantaneous impulses in an arbitrary separable Hilbert space. We use a projection scheme of increasing sequence of finite dimensional subspaces and projection operators to define approximations. In order to demonstrate the existence and convergence of approximate solutions, we utilize stochastic analysis theory, fractional calculus, theory of fractional cosine family of linear operators, and fixed point approach. Furthermore, we examine the convergence of the Faedo–Galerkin (F–G) approximate solutions to the mild solution of our given problem. Finally, a concrete example involving a partial differential equation is provided to validate the main abstract results.

On behaviours for the solution to a certain third-order stochastic integro-differential equation with time delay

In the present paper, Lyapunov functional (LF) is employed to discuss the continuability and boundedness of solutions for a third-order non-autonomous stochastic integro-differential equation (SIDE) with time delay. The third-order differential equation is ablated to a system of first-order differential equations together with its equivalent quadratic function to derive a suitable downright LF and then we study the behaviour of the solutions. A numerical example is considered to support our results. Moreover, we use the Euler-Maruyama method to get an approximate numerical solution for the considered system. The obtained result complements some recent ones in the literature.

Hyers-Ulam stability result for Hilfer fractional integrodifferential stochastic equations with fractional noises and non-instantaneous impulses

Hyers-Ulam stability result for Hilfer fractional integrodifferential stochastic equations with fractional noises and non-instantaneous impulses

Attractive solutions for Hilfer fractional neutral stochastic integro-differential equations with almost sectorial operators

<abstract><p>This paper studies the integro-differential equations of Hilfer fractional (HF) neutral stochastic evolution on an infinite interval with almost sectorial operators and their attractive solutions. We use semigroup theory, stochastic analysis, compactness methods, and the measure of noncompactness (MNC) as the foundation for our methodologies. We establish the existence and attractivity theorems for mild solutions by considering the fact that the almost sectorial operator is both compact and noncompact. Example that highlight the key findings are also provided.</p></abstract>

Approximate controllability of non-instantaneous impulsive stochastic integrodifferential equations driven by Rosenblatt process via resolvent operators

In this work, we investigate the existence of a mild solution and the approximate controllability of non-instantaneous impulsive stochastic integrodifferential equations driven by the Rosenblatt process in Hilbert space with the Hurst parameter \(\mathsf{H} \in (1/2, 1)\). We achieve the result using the semigroup theory of bounded linear operators, Grimmer's resolvent operator theory, and stochastic analysis. Using Krasnoselskii's and Schauder's fixed point theorems, we demonstrate the existence of mild solutions and the approximate controllability of the system. Finally, an example shows the potential for significant results.

Trajectory controllability of neutral stochastic integrodifferential equations with mixed fractional Brownian motion

This work focuses on the existence and trajectory (T-)controllability of mixed fractional Brownian motion (fBm) with the Hurst index and neutral stochastic integrodifferential equations (NSIDEs) with deviating argument and fBm. Stochastic integrodifferential equations (SIDEs) are solved in Hilbert space using stochastic analysis, the resolvent operator, and Krasnoselskii's fixed point theorem (KFPT). Furthermore, providing adequate assumptions, the T-controllability of the considered system is organised by using extended Gronwall's inequality. We demonstrate the theoretical insights and numerical simulations are included which is unique and makes this work more interesting. The obtained results generalise existing results from [Chalishajar, D. N., George, R. K., & Nandakumaran, A. K. (2010). Trajectory controllability of nonlinear integro-differential system. Journal of Franklin Institute, 347(7), 1065–1075.; Durga, N., Muthukumar, P., & Malik, M. (2022). Trajectory controllability of Hilfer fractional neutral stochastic differential equation with deviated argument and mixed fractional Brownian motion. Optimisation, 1–27.; Muslim, M., & George, R. K. (2019). Trajectory controllability of the nonlinear systems governed by fractional differential equations. Differential Equations and Dynamical Systems, 27, 529–537.; Dhayal, R., Malik, M., & Abbas, S. (2021). Approximate and trajectory controllability of fractional stochastic differential equation with non-instantaneous impulses and Poisson jumps. Asian Journal of Control, 23(6), 2669–2680.].

Optimal Control for Neutral Stochastic Integrodifferential Equations with Infinite Delay Driven by Poisson Jumps and Rosenblatt Process

In this paper, we investigate the optimal control problems for a class of neutral stochastic integrodifferential equations (NSIDEs) with infinite delay driven by Poisson jumps and the Rosenblat process in Hilbert space involving concrete-fading memory-phase space, in which we define the advanced phase space for infinite delay for the stochastic process. First, we introduce conditions that ensure the existence and uniqueness of mild solutions using stochastic analysis theory, successive approximation, and Grimmer’s resolvent operator theory. Next, we prove exponential stability, which includes mean square exponential stability, and this especially includes the exponential stability of solutions and their maps. Following that, we discuss the existence requirements of an optimal pair of systems governed by stochastic partial integrodifferential equations with infinite delay. Then, we explore examples that illustrate the potential of the main result, mainly in the heat equation, filter system, traffic signal light systems, and the biological processes in the human body. We conclude with a numerical simulation of the system studied. This work is a unique combination of the theory with practical examples and a numerical simulation.

Optimal control of conformable fractional neutral stochastic integrodifferential systems with infinite delay

The main concern of the manuscript deals with the optimal control problem of conformable fractional neutral stochastic integrodifferential systems with infinite delay. This study is motivated by SAR and RAR systems advantage of providing control over many factors such as power, frequency, phase, polarization, incidence angle, spatial resolution and swath width, all of which are important when designing and operating a radar system. Initially, we investigate the existence of mild solutions for the conformable fractional stochastic integrodifferential equations with infinite delay using stochastic analysis techniques and Banach fixed point theorem. In the later part we establish the existence of mild solutions of the conformable fractional neutral stochastic integrodifferential system with infinite time delay. Furthermore, the existence of optimal control of the corresponding Lagrange optimal control problem is investigated. An example is provided to illustrate the applications of the obtained results. We explain the limitation we have with the existence software, and developed a numerical scheme to justify the theory.

On stability of nonlocal neutral stochastic integro differential equations with random impulses and Poisson jumps

This article aims to examine the existence and Hyers-Ulam stability of non-local random impulsive neutral stochastic integrodifferential delayed equations with Poisson jumps. Initially, we prove the existence of mild solutions to the equations by using the Banach fixed point theorem. Then, we investigate stability via the continuous dependence of solutions on the initial value. Next, we study the Hyers-Ulam stability results under the Lipschitz condition on a bounded and closed interval. Finally, we give an illustrative example of our main result.

Additive dynamic models for correcting numerical model outputs

Numerical air quality models are pivotal for the prediction and assessment of air pollution, but numerical model outputs may be systematically biased. An additive dynamic model is proposed to correct large-scale raw model outputs using data from other sources, including readings collected at ground monitoring networks and weather outputs from other numerical models. An additive partially linear model specification is employed for the nonlinear relationships between air pollutants and covariates. In addition, a multi-resolution basis function approximate is proposed to capture the different small-scale variations of biases, and a discretized stochastic integro-differential equation is constructed to characterize the dynamic evolution of the random coefficients at each spatial resolution. An expectation-maximization algorithm is developed for parameter estimation and a multi-resolution ensemble-based scheme is embedded to accelerate the computation. For statistical inference, a conditional simulation technique is applied to quantify the uncertainty of parameter estimates and bias correction results. The proposed approach is used to correct the biased raw outputs of PM2.5 from the Community Multiscale Air Quality (CMAQ) system for China's Beijing-Tianjin-Hebei region. Our method improves the root mean squared error and continuous rank probability score by 43.70% and 34.76%, respectively. Compared to other statistical methods under different metrics, our model has advantages in both correction accuracy and computational efficiency.

Fractional neutral stochastic integrodifferential equations with Caputo fractional derivative: Rosenblatt process, Poisson jumps and Optimal control

The objective of this paper is to investigate the existence of mild solutions and optimal controls for a class of fractional neutral stochastic integrodifferential equations driven by Rosenblatt process and Poisson jumps in Hilbert spaces. First we establish a new set of sufficient conditions for the existence of mild solutions of the aforementioned fractional systems by using the successive approximation approach. The results are formulated and proved by using the fractional calculus, solution operator and stochastic analysis techniques. The existence of optimal control pairs of system governed by fractional neutral stochastic differential equations driven by Rosenblatt process and poisson jumps is also been presented. An example is provided to illustrate the theory.

Weighted fractional stochastic integro-differential equation with infinite delay

In this paper, we consider a weighted fractional stochastic integro-differential equation with infinite delay and nonzero initial values involving a Riemann–Liouville fractional derivative of order 1/2<alpha <1. The existence of a mild solution is investigated using fractional calculus, stochastic analysis, and the fixed point theorem. An example is also provided to illustrate the obtained result.

Wellposedness and controllability results of stochastic integrodifferential equations with noninstantaneous impulses and Rosenblatt process

The purpose of this work is to investigate a novel class of noninstantaneous impulsive stochastic integrodifferential equations (SIDEs) driven by Brownian motion and Rosenblatt process. We construct a new set of adequate assumptions for the existence and uniqueness of mild solutions using stochastic analysis, analytic semigroup theory, integral equation theory, and a fixed point methodology. Additionally, we study the asymptotic behavior of mild solutions and provide stochastic system controllability results. Finally, we include an example to illustrate the application of our main findings.

Solvability and optimal controls of neutral stochastic integro-differential equations driven by fractional Brownian motion

In this article, the authors set up an optimal control of neutral stochastic integro-differential equations (NSIDEs) driven by fractional Brownian motion (fBm) in a Hilbert space by using Grimmer resolvent operators. Sufficient conditions for mild solutions are formulated and proved by using the Banach contraction mapping principle and stochastic analytic techniques. We have extended the problem in [Issaka et al. (2020) Results on nonlocal stochastic integro-differential equations are driven by a fractional Brownian motion. Open Mathematics, 18(1), 1097–1112] to NSIDEs driven by fBm and have used modified techniques to make them compatible with optimal controls of stochastic integro-differential systems. In addition, the optimal control of the proposed problem is presented using Balder's theorem. Such optimal control of NSIDEs with fBm is widely used in automatic control, aircraft and air traffic control, electrical networks, wavelet expansions, etc. Finally, an example illustrates the potential of the main results.

A Meshless Method for the Numerical Solution of Fractional Stochastic Integro-Differential Equations Based on the Moving Least Square Approach

A Meshless Method for the Numerical Solution of Fractional Stochastic Integro-Differential Equations Based on the Moving Least Square Approach