Neutral Stochastic Equations Research Articles

Solving stochastic equations with unbounded nonlinear perturbations

This paper is interested in semilinear stochastic equations having unbounded nonlinear perturbations in the deterministic part and/or in the random part. Moreover, the linear part of these equations is governed by a not necessarily analytic semigroup. The main difficulty with these equations is how to define the concept of mild solutions due to the chosen type of unbounded perturbations. To overcome this problem, we first proved a regularity property of the stochastic convolution with respect to the domain of ‘admissible’ unbounded linear operators (not necessarily closed or closable). This is done using Yosida extensions of such unbounded linear operators. After proving the well-posedness of these equations, we also establish the Feller property for the corresponding transition semigroups. Several examples like heat equations and Schrödinger equations with nonlocal perturbations terms are given. Finally, we give an application to a general class of semilinear neutral stochastic equations.

Continuity and approximation properties of solutions to fractional neutral stochastic functional differential equations with non-Lipschitz coefficients

ABSTRACT This paper aims to investigate a fractional neutral stochastic functional differential equation (FNSFDE) with non-Lipschitz coefficients. Under the assumptions, we first establish the continuity of the solution in the fractional order of the equation. Furthermore, an Euler-Maruyama (EM) approximation is constructed and then we obtain the strong convergence of the numerical scheme. Specially, if the non-Lipschitz conditions are replaced with the Lipschitz conditions, we shall get a definite convergence rate, which is related to the fractional order of the equation. Finally, we consider the averaging principle for the fractional neutral stochastic equation, which provides us with an easy way to study the properties of the equation.

New results on exponential stability in mean square of neutral stochastic equations with delays

This work addresses the exponential stability in mean square of general neutral stochastic differential equations with delays. A simple approach to the exponential stability of the neutral stochastic differential equations with delays is proposed. Novel criteria for the exponential stability in mean square of such equations are presented. A comparison to existing results and an illustrative example are given.

Existence results for second-order neutral stochastic equations driven by Rosenblatt process

In this paper we consider a class of second-order impulsive stochastic functional differential equations driven simultaneously by a Rosenblatt process and a standard Brownian motion in a Hilbert space. We prove an existence and uniqueness result under non-Lipschitz condition which is weaker than Lipschitz one and we establish some conditions ensuring the controllability for the mild solution by means of the Banach fixed point principle. At the end we provide a practical example in order to illustrate the viability of our result. <br><br> Розглянуто клас імпульсних стохастичних функціонально-диференціальних рівнянь другого порядку, які керуються процесом Розенблата і стандартним броунівський рухом у ґільбертовому просторі одночасно за умови, яка є слабкішою за умови Ліпшица. Також встановлено умови керованості для помірного розв'язку за допомоги принципу Банаха про нерухому точку. Наведено приклад з практики, що ілюструє отримані результати.

An effective averaging theory for fractional neutral stochastic equations of order [formula omitted] with Poisson jumps

This paper, focusing on the fractional neutral stochastic differential equations (FNSDEs) in the Euclidean space Rn, successfully provides the first evidence of a new fractional averaging theorem via rigorous mathematical deductions. With the help of integration by part, the fractional term is handled simply and ingeniously. Based on this new idea, we show that the mild solutions of two fractional systems before and after averaging are equivalent in mean square sense. The study here gives a general approach to come up with the Khasminskii averaging principle for FNSDEs.

Approximate Controllability of Impulsive Neutral Stochastic Differential Equations Driven by Poisson Jumps

This work studies the approximate controllability of a class of impulsive neutral stochastic differential equations with infinite delay and Poisson jumps involving generalized Caputo fractional derivative under the condition that the corresponding linear system is approximately controllable. Utilizing the fixed point theory and sectorial operator theory, the existence of the mild solution of the impulsive neutral stochastic equation is established imposing weaker regularity on nonlinear terms. A set of sufficient conditions establishing controllability results is derived with the help of stochastic analysis and fractional calculus. Finally, an example is provided to illustrate the obtained abstract result.

Retarded Neutral Stochastic Equations Driven by Multiplicative Fractional Brownian Motion

In this article, we develop an existence and uniqueness theory of pathwise mild solutions for a class of stochastic neutral partial functional differential equations that are driven by an infinite-dimensional multiplicative fractional noise. Our existence and uniqueness result is divided into two parts: first we establish the existence and uniqueness of a local solution, which proof rests upon the Banach fixed point theorem; then the existence of a unique global solution is based on certain regularity properties of the local solution as well as a suitable way of extending the local solution to the entire interval in finitely many steps.

Existence and Stability Results for Second-Order Stochastic Equations Driven by Fractional Brownian Motion

Fractional Brownian motion has been widely used to model a number of phenomena in diverse fields of science and engineering. In this article, we investigate the existence, uniqueness and stability of mild solutions for a class of second-order nonautonomous neutral stochastic evolution equations with infinite delay driven by fractional Brownian motion (fBm) with Hurst parameter H ∈ (1/2, 1) in Hilbert spaces. More precisely, using semigroup theory and successive approximation approach, we establish a set of sufficient conditions for obtaining the required result under the assumption that coefficients satisfy non-Lipschitz condition with Lipschitz condition being considered as a special case. Further, the result is deduced to study the second-order autonomous neutral stochastic equations with fBm. The results generalize and improve some known results. Finally, as an application, stochastic wave equation with infinite delay driven by fractional Brownian motion is provided to illustrate the obtained theory.