Fractional Stochastic Differential Equations Research Articles

On Existence and Continuity Results of Solution for Multi-time Scale Fractional Stochastic Differential Equation

On Existence and Continuity Results of Solution for Multi-time Scale Fractional Stochastic Differential Equation

A distributed-order fractional stochastic differential equation driven by Lévy noise: Existence, uniqueness, and a fast EM scheme.

In this paper, we consider a distributed-order fractional stochastic differential equation driven by Lévy noise. We, first, prove the existence and uniqueness of the solution. A Euler-Maruyama (EM) scheme is constructed for the equation, and its strong convergence order is shown to be min{1-α∗,0.5}, where α∗ depends upon the weight function. Besides, we present a fast EM method and also the error analysis of the fast scheme. In addition, several numerical experiments are carried out to substantiate the mathematical analysis.

Non-instantaneous impulsive Hilfer–Katugampola fractional stochastic differential equations with fractional Brownian motion and Poisson jumps

The existence of solutions of non-instantaneous impulsive Hilfer–Katugampola fractional differential equations of order 1 / 2 < α < 1 and parameter 0 ≤ β ≤ 1 with fractional Brownian motion (fBm) and Poisson jumps is investigated in this paper. The required results are obtained based on fractional calculus, stochastic analysis, semigroups, and the fixed point theorem. In the end of the paper, an example is provided to illustrate the applicability of the theoretical results.

A Class of Fractional Stochastic Differential Equations with a Soft Wall

In this paper we are interested in fractional stochactic differential equations (SDEs) with a soft wall. What do we mean by such a type of equation? It has been established that SDE with reflection can be imagined as equations having a hard wall. Now, by introducing repulsion instead of reflection, one obtains an SDE with a soft wall. In contrast to the SDE with reflection, where the process cannot pass the hard wall, the soft wall is repulsive but not impenetrable. As the process crosses the soft wall boundary, it experiences the force of a chosen magnitude in the opposite direction. When the process is far from the wall, the force acts weakly. We find conditions under which SDE with a soft wall has a unique solution and construct an implicit Euler approximation with a rate of convergence for this equation. Using the example of the fractional Vasicek process with soft walls, we illustrate the dependence of the behaviour of the solution on the repulsion force.

Fractional discrete Temimi–Ansari method with singular and nonsingular operators: applications to electrical circuits

The goal of this article is to present a recently developed numerical approach for solving fractional stochastic differential equations with a singular Caputo kernel and a nonsingular Caputo–Fabrizio and Atangana–Baleanu (ABC) kernel. The proposed method is based on the discrete Temimi–Ansari method, which is combined with three different numerical schemes that are appropriate for the new fractional derivative operators. The proposed technique is used to investigate the effects of Gaussian white-noise and Gaussian colored-noise perturbations on the potential source and resistance in fractional stochastic electrical circuits. The proposed method’s robustness and efficiency were demonstrated by comparing its results to those of the stochastic Runge–Kutta method (SRK). The valuable point in this article is that the resulting numerical scheme is able to combine two powerful methods that can be extended into more complex stochastic models. The comparison of different fractional derivatives using Mathematica 12 software has been obtained and the simulation results demonstrate the merit of the contributed method.

Solving a nonlinear fractional SPDE with spatially inhomogeneous white noise

In this paper, we study the following nonlinear fractional stochastic partial differential equation where denotes the Markovian generator of a stable-like Feller process with variable order and is a measurable function. The forcing noise denoted by is a spatially inhomogeneous white noise. Under some assumptions on the catalytic measure of the inhomogeneous Brownian sheet , we study the moment bounds for the solution. As a byproduct, we prove that the solution is weakly full intermittent based on the moment estimates of the solution. We also study the Hölder regularity of the solution with respect to the temporal and spatial variables, respectively.

Limit behavior of the solution of Caputo-Hadamard fractional stochastic differential equations

In this letter, an averaging principle for Caputo-Hadamard fractional stochastic differential equations is established. It is showed the solution of the Caputo-Hadamard fractional stochastic differential equation converges to the solution of the averaged equation when the time scale parameter tends to zero. Compared to existing literatures, different estimates techniques are used to overcome the difficulties caused by the logarithmic term. It means that Khasminskii classical approach is extended to fractional stochastic differential equations of Caputo-Hadamard type.

S-asymptotically $ \omega $-periodic solutions in distribution for a class of stochastic fractional functional differential equations

&lt;abstract&gt;&lt;p&gt;In this paper, we introduce the concept of an S-asymptotically $ \omega $-periodic process in distribution for the first time, and by means of the successive approximation and the Banach contraction mapping principle, respectively, we obtain sufficient conditions for the existence and uniqueness of the S-asymptotically $ \omega $-periodic solutions in distribution for a class of stochastic fractional functional differential equations.&lt;/p&gt;&lt;/abstract&gt;

The S-asymptotically $ \omega $-periodic solutions for stochastic fractional differential equations with piecewise constant arguments

&lt;abstract&gt;&lt;p&gt;In this paper, two kinds of stochastic differential equations with piecewise constant arguments are investigated. Sufficient conditions for the existence of the square-mean S-asymptotically $ \omega $-periodic solutions of these two type equations are derived where $ \omega $ is an integer. Then, the global asymptotic stability for one of them is considered by using the comparative approach. In order to show the theoretical results, we give two examples.&lt;/p&gt;&lt;/abstract&gt;

Существование и единственность решений стохастических дробных дифференциальных уравнений в нескольких временных шкалах

A novel class of nonlinear stochastic fractional differential equations with delay and the Jumarie and Ito differentials is introduced in the paper. The aim of the study is to prove existence and uniqueness of solutions to these equations. The main results of the paper generalise some previous findings made for the non-delay and three-scale equations under additional restrictions on the fractional order of the Jumarie differentials, which are removed in our analysis. The techniques used in the paper are based on the properties of the singular integral operators in specially designed spaces of stochastic processes, the representation of delay equations as functional differential equations as well as Picard’s iterative method.

Stability analysis of new generalized mean-square stochastic fractional differential equations and their applications in technology

&lt;abstract&gt;&lt;p&gt;Stability theory has significant applications in technology, especially in control systems. On the other hand, the newly defined generalized mean-square stochastic fractional (GMSF) operators are particularly interesting in control theory and systems due to their various controllable parameters. Thus, the combined study of stability theory and GMSF operators becomes crucial. In this research work, we construct a new class of GMSF differential equations and provide a rigorous proof of the existence of their solutions. Furthermore, we investigate the stability of these solutions using the generalized Ulam-Hyers-Rassias stability criterion. Some examples are also provided to demonstrate the effectiveness of the proposed approach in solving fractional differential equations (FDEs) and evaluating their stability. The paper concludes by discussing potential applications of the proposed results in technology and outlining avenues for future research.&lt;/p&gt;&lt;/abstract&gt;

Fractional Stochastic Differential Equations Driven By G-Brownian Motion with Delays

Fractional Stochastic Differential Equations Driven By G-Brownian Motion with Delays

NUMERICAL SOLUTION OF PERSISTENT PROCESSES-BASED FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATIONS

This paper proposes the shifted Legendre polynomial approximations-based stochastic operational matrix of integration method to solve persistent processes-based fractional stochastic differential equations. The operational matrix of integration, stochastic operation matrix and fractional stochastic operational matrix of the shifted Legendre polynomials are derived. The stochastic differential equation is transformed into an algebraic system of [Formula: see text] equations by the operational matrices. For the proposed approach, a thorough discussion of the error analysis in [Formula: see text] norm is provided. The proposed method’s applicability, correctness, and accuracy are examined using a few numerical examples. Comparing the numerical examples to the other methods discussed in the literature demonstrates the solution’s effectiveness and attests to the solution’s high quality. The error analysis also reveals the method’s superiority. A more accurate solution is obtained, thus maintaining a minimum error.

Correction: Sobolev-type nonlinear Hilfer fractional stochastic differential equations with noninstantaneous impulsive

Correction: Sobolev-type nonlinear Hilfer fractional stochastic differential equations with noninstantaneous impulsive

Hyers-Ulam stability of Hadamard fractional stochastic differential equations

Hyers-Ulam stability of Hadamard fractional stochastic differential equations

Existence and controllability for conformable fractional stochastic differential equations with infinite delay via measures of noncompactness.

In this article, we consider conformable fractional stochastic differential equations (CFSDEs) driven by fBm with infinite delay via measures of noncompactness (MNC). As far as we know, there are few papers considering this issue. First, by virtue of a Mönch fixed point theorem and MNC, we explore the existence of solutions for CFSDEs. Subsequently, with the aid of Jensen inequality, Hölder inequality, stochastic analysis techniques, and semigroup theory, the controllability for this considered CFSDEs is investigated by employing a Mönch fixed point theorem. Thereafter, the controllability of CFSDEs with nonlocal conditions is discussed. Finally, the theoretical result is supported through an example.

An Euler-Maruyama Method for Variable Fractional Stochastic Differential Equations With Caputo Derivatives

An Euler-Maruyama Method for Variable Fractional Stochastic Differential Equations With Caputo Derivatives

On Averaging Principle for Caputo–Hadamard Fractional Stochastic Differential Pantograph Equation

In this paper, we studied an averaging principle for Caputo–Hadamard fractional stochastic differential pantograph equation (FSDPEs) driven by Brownian motion. In light of some suggestions, the solutions to FSDPEs can be approximated by solutions to averaged stochastic systems in the sense of mean square. We expand the classical Khasminskii approach to Caputo–Hadamard fractional stochastic equations by analyzing systems solutions before and after applying averaging principle. We provided an applied example that explains the desired results to us.

A New Result Concerning Nonlocal Controllability of Hilfer Fractional Stochastic Differential Equations via almost Sectorial Operators

This manuscript mainly focused on the nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators. The key ideas of the study are illustrated by using ideas from fractional calculus, the fixed point technique, and measures of noncompactness. Then, the authors establish new criteria for the mild existence of solutions and derive fundamental characteristics of the nonlocal controllability of a system. In addition, researchers offer theoretical and real-world examples to demonstrate the effectiveness and suitability of our suggested solutions.

Well-posedness and regularity for solutions of Caputo stochastic fractional delay differential equations

This paper is devoted to build the well-posedness and regularity for solutions of Caputo stochastic fractional delay differential equations (for short CSFDDE) of order α∈(12,1). Firstly, under local Lipschitz condition of coefficients, we show a result on the existence and uniqueness of solutions. Secondly, under global Lipschitz condition of coefficients, we show the continuous dependence of solutions on the initial values and on the fractional exponent α and the regularity in time for solutions is also derived. The main ingredient in the proof is to use a temporally weighted norm, Banach fixed point theorem and truncation procedure.