Stochastic Functional Differential Equations Research Articles

An approximate Taylor method for Stochastic Functional Differential Equations via polynomial condition

Abstract The subject of this paper is an analytic approximate method for a class of stochastic functional differential equations with coefficients that do not necessarily satisfy the Lipschitz condition nor linear growth condition but they satisfy some polynomial conditions. Also, equations from the observed class have unique solutions with bounded moments. Approximate equations are defined on partitions of the time interval and their drift and diffusion coefficients are Taylor approximations of the coefficients of the initial equation. Taylor approximations require Fréchet derivatives since the coefficients of the initial equation are functionals. The main results of this paper are the Lp and almost sure convergence of the sequence of the approximate solutions to the exact solution of the initial equation. An example that illustrates the theoretical results and contains the proof of the existence, uniqueness and moment boundedness of the approximate solution is displayed.

Stability analysis of theθ-method for hybrid neutral stochastic functional differential equations with jumps

• Few results on the stability of numerical methods for the neutral stochastic functional differential equations with Markovian switching and jumps, results obtained in this manuscript can enrich this area. • Illustrate the almost sure exponential stability and the mean-square exponential stability of the trivial solution. • Show the θ -method can preserve the almost sure exponential stability and the mean-square exponential stability of the trivial solution under some appropriate condition. • The almost sure exponential stability of the θ -method is obtained by the discrete semi-martingale convergence theorem without resorting to using the Borel-Cantelli lemma and the Chebyshev inequality. Few results seems to be known about the stability of numerical methods for the hybrid neutral stochastic functional differential equations with jumps (also known as the neutral stochastic functional differential equations with Markovian switching and jumps (NSFDEwMJs)). This paper mainly investigates the exponential stability (both the almost sure and the mean-square exponential stability) of the θ -method for NSFDEwMJs. Precisely, it is first illustrated that the trivial solution of the NSFDEwMJ is almost surely and mean-square exponentially stable. It is then shown that the θ -method can preserve the same conclusions of the trivial solution. Numerical examples are demonstrated to illustrate the obtained results.

Existence and Stability Results for Second-Order Neutral Stochastic Differential Equations With Random Impulses and Poisson Jumps

The objective of this paper is to investigate the existence and stability results of secondorder neutral stochastic functional differential equations (NSFDEs) in Hilbert space. Initially, we establish the existence results of mild solutions of the aforementioned system using the Banach contraction principle. The results are formulated using stochastic analysis techniques. In the later part, we investigate the stability results through the continuous dependence of solutions on initial conditions.

Feedback delay control of highly nonlinear stochastic functional differential equations with discrete‐time state observations

The purpose of this paper is to give the delay control based on discrete‐time state observations to stabilize highly nonlinear hybrid stochastic functional differential equations (SFDEs). It is considered that time lag generated by the controller in each discrete observation should be different. The new controlled hybrid SFDEs are affected the variable delay caused by the controller, the distributed delay, and the superlinear coefficients of the systems itself, which makes the problem handling more complicated. Then, a series of criteria for the exponential stability of the controlled SFDEs are obtained, and an upper bound for the discrete observation interval and variable delay is given. Finally, numerical example illustrates the proposed theoretical results.

Time-Optimal Control for Semilinear Stochastic Functional Differential Equations with Delays

The purpose of this paper is to find the time-optimal control to a target set for semilinear stochastic functional differential equations involving time delays or memories under general conditions on a target set and nonlinear terms even though the equations contain unbounded principal operators. Our research approach is to construct a fundamental solution for corresponding linear systems and establish variations of a constant formula of solutions for given stochastic equations. The existence result of time-optimal controls for one point target set governed by the given semilinear stochastic equation is also established.

Trotter-Kato approximations of stochastic neutral partial functional differential equations

Trotter-Kato approximations of stochastic neutral partial functional differential equations

On exponential stability in mean square of neutral stochastic functional differential equations

General neutral stochastic functional differential equations are considered. A novel approach to exponential stability of such equations is proposed. New criteria for the mean square exponential stability of neutral stochastic functional differential equations are given. A discussion and illustrative examples are presented.

Exponential Stability in Mean Square of Stochastic Functional Differential Equations with Infinite Delay

Exponential Stability in Mean Square of Stochastic Functional Differential Equations with Infinite Delay

Existence of p-mean Almost Periodic Mild Solution for Fractional Stochastic Neutral Functional Differential Equation

Existence of p-mean Almost Periodic Mild Solution for Fractional Stochastic Neutral Functional Differential Equation

Controllability of fractional neutral functional differential equations with infinite delay driven by fractional Brownian motion

Abstract In this work, we establish a controllability result for a class of fractional neutral stochastic functional differential equations with infinite delay driven by fractional Brownian motion. To attain our objective we adapt the argument of Lakhel & McKibben (2018, Stochastics 90, no. 3, 313–329) where the existence of mild solutions to such stochastic equations was studied. An example is provided to show the applicability of the theoretical result.

Partial Practical Exponential Stability of Neutral Stochastic Functional Differential Equations with Markovian Switching

In this paper, we investigate the partial practical exponential stability of neutral stochastic functional differential equations with Markovian switching. The main tool used to prove the results is the Lyapunov method. We analyze an illustrative example to show the applicability and interest of the main results.

Global Attracting Sets of Neutral Stochastic Functional Differential Equations Driven by Poisson Jumps

Global Attracting Sets of Neutral Stochastic Functional Differential Equations Driven by Poisson Jumps

Averaging of neutral stochastic partial functional differential equations involving delayed impulses

In this paper, we initiate a study on averaging principles for neutral stochastic partial functional differential equations (NSPFDEs) with delayed impulses. With the aid of inequality techniques, the semigroup approach and some technical transformations, we derive an averaging principle for a class of NSPFDEs involving delayed impulses. The results obtained ensure that one can focus on the averaged autonomous system without impulses in place of the original system, thus providing an effective way for reducing complexity. In the end, an example is worked out to demonstrate the theoretical results.

β -Stability in q -th Moment of Neutral Impulsive Stochastic Functional Differential Equations with Markovian Switching

In this paper, we investigate the β -stability in q-th moment for neutral impulsive stochastic functional differential equations with Markovian switching (NISFDEwMS). Moreover, β -stability in q-th moment is studied by using the Lyapunov techniques and a new Razumikhin-type theorem to prove our result. Finally, we check the main result by a numerical example.

Global attracting set, exponential stability and stability in distribution of SPDEs with jumps

A novel approach to the global attracting sets of mild solutions for stochastic functional partial differential equations driven by Lévy noise is presented. Consequently, some new sufficient conditions ensuring the existence of the global attracting sets of mild solutions for the considered equations are established. As applications, some new criteria for the exponential stability in mean square of the considered equations is obtained. Subsequently, by employing a weak convergence approach, we try to establish some stability conditions in distribution of the segment processes of mild solutions to stochastic delay partial differential equations with jumps under some weak conditions. Some known results are improved. Lastly, some examples are investigated to illustrate the theory.

The Convergence of exponential Euler method for weighted fractional stochastic equations

In this paper, we propose an exponential Euler method to approximate the solution of a stochastic functional differential equation driven by weighted fractional Brownian motion B{a,b} under some assumptions on a and b. We obtain also the convergence rate of the method to the true solution after proving an L2 -maximal bound for the stochastic ntegrals in this case.

Ulam-Hyers-Rassias Stability of Stochastic Functional Differential Equations via Fixed Point Methods

The Ulam-Hyers-Rassias stability for stochastic systems has been studied by many researchers using the Gronwall-type inequalities, but there is no research paper on the Ulam-Hyers-Rassias stability of stochastic functional differential equations via fixed point methods. The main goal of this paper is to investigate the Ulam-Hyers Stability (HUS) and Ulam-Hyers-Rassias Stability (HURS) of stochastic functional differential equations (SFDEs). Under the fixed point methods and the stochastic analysis techniques, the stability results for SFDE are investigated. We analyze two illustrative examples to show the validity of the results.

Time optimal control to a partial stochastic differential system with pseudo almost periodic coefficients

In this paper, we study the time optimal control of a class of partial stochastic impulsive functional differential systems with pseudo almost periodic coefficients in the α-norm functional spaces. Firstly, the existence of p-mean piecewise pseudo almost periodic mild solutions for the impulsive stochastic control systems is investigated by utilising stochastic analysis, analytic semigroup, the fixed-point techniques with fractional power arguments. Secondly, the existence of time optimal pairs of a system governed by partial stochastic impulsive functional differential equations is also obtained. Finally, two examples are provided to illustrate the time optimal control problems of parabolic control system with pseudo almost periodic coefficients and impulses.

Perturbed Impulsive Neutral Stochastic Functional Differential Equations

This paper studies the asymptotic behavior of the mild solution for a class of perturbed impulsive neutral stochastic functional differential equations driven by fractional Brownian motion in Hilbert space. We establish the conditions under which the mild solutions of perturbed impulsive neutral stochastic functional differential equation and the unperturbed one are close on finite time interval when the perturbation tends to zero. Moreover, we show the result holds on time interval whose length tends to infinity as the perturbation tends to zero. As an application, the asymptotic behavior of the mild solution for a class of perturbed impulsive neutral stochastic partial differential equations driven by fractional Brownian motion in Hilbert space is proposed to show the feasibility of the obtained result.

LaSalle-type theorems for stochastic functional differential equations with Markovian switching

In this article, we establish the LaSalle-type theorem for stochastic functional differential equations with Markovian switching (SFDEwMSs) under much weaken conditions. We would like to emphasize that we do not require the linear growth condition and the bounded moment condition on the solutions. Indeed, we allow the Lyapunov function operator could be dependent of time to cover a much wider class of SFDEwMSs. As a bonus, we obtain the criterion on the asymptotical stability and asymptotical boundedness for SFDEwMSs. For comparison and verification, we also present a specific example with much general coefficients.