Stochastic Functional Differential Equations Research Articles

Discrete-time feedback stabilization for neutral stochastic functional differential equations driven by G-Lévy process

This paper mainly discusses the exponential stability of neutral stochastic functional differential equations with G-Lévy jump. For a given mean square exponential unstable neutral stochastic differential equations with G-Lévy jump, a discrete-time feedback control in the drift part is designed to realize feedback stability and the H∞ stable càdlàg solution of corresponding systems is obtained. The upper bound of state observation duration of the corresponding systems is derived by extending Mao′s method. Further up, the exponential stabilization conditions are established and some exponential stabilities of the solutions are proved by using the G-Lyapunov functional method. Moreover, an example is presented to verify the obtained results.

Existence and smoothness of the densities of stochastic functional differential equations with jumps

In this paper, we study the existence and smoothness of the densities of stochastic functional differential equations with jumps whose proof are based on recently well-developed lent particle method introduced by Bouleau and Denis.

Asymptotic behaviour analysis of hybrid neutral stochastic functional differential equations driven by Lévy noise

This paper focuses on the existence and uniqueness, the pth moment and almost sure stability with a general decay rate and the practical stability with general decay rate of a class of hybrid neutral stochastic functional differential equations driven by Lévy noise. Our crucial techniques include Lyapunov functions and the nonnegative semi-martingale convergence theorem. The conditions on the diffusion operator and neutral term are weaker than those in the related existing works. Examples are given to show the effectiveness of the obtained results.

Exponential Stability of Impulsive Neutral Stochastic Functional Differential Equations

This paper focuses on the problem of the pth moment and almost sure exponential stability of impulsive neutral stochastic functional differential equations (INSFDEs). Based on the Lyapunov function and average dwell time (ADT), two sufficient criteria for the exponential stability of INSFDEs are derived, which manifest that the result obtained in this paper is more convenient to be used than those Razumikhin conditions in former literature. Finally, two numerical examples and simulations are given to verify the validity of our result.

Delay‐dependent stability of highly nonlinear neutral stochastic functional differential equations

AbstractThis article focuses on the delay‐dependent stability of highly nonlinear hybrid neutral stochastic functional differential equations (NSFDEs). The delay dependent stability criteria for a class of highly nonlinear hybrid NSFDEs are derived via the Lyapunov functional. The stabilities discussed in this article include stability, asymptotically stability and exponential stability. A numerical example is given to illustrate the criteria established.

Random vortex dynamics via functional stochastic differential equations

In this paper, we present a novel, closed, three-dimensional random vortex dynamics system, which is equivalent to the Navier–Stokes equations for incompressible viscous fluid flows. The new random vortex dynamics system consists of a stochastic differential equation which is, in contrast with the two-dimensional random vortex dynamics equations, coupled to a finite-dimensional ordinary functional differential equation. This new random vortex system paves the way for devising new numerical schemes (random vortex methods) for solving three-dimensional incompressible fluid flow equations by Monte Carlo simulations. In order to derive the three-dimensional random vortex dynamics equations, we have developed two powerful tools: the first is the duality of the conditional distributions of a couple of Taylor diffusions, which provides a path space version of integration by parts; the second is a forward type Feynman–Kac formula representing solutions to nonlinear parabolic equations in terms of functional integration. These technical tools and the underlying ideas are likely to be useful in treating other nonlinear problems.

Stationary distribution of stochastic population dynamics with infinite delay

This paper examines the stationary distribution of the stochastic Lotka-Volterra model with infinite delay. Since the solutions of stochastic functional or delay differential equations depend on their history, they are non-Markov, which implies that traditional techniques based on the Markov property, cannot be applicable. This paper uses the variable substituting technique to obtain a higher-dimensional stochastic differential equation without delay. In this paper it is shown that the new equation satisfies the strong Feller property and strong Markov property, by which the invariant measure follows from the Krylov-Bogoliubov Theorem. Since distribution of the original model is actually the marginal distribution of the new equation, these results also show that there exists the stationary distribution for the original equation. Moreover, when the noise intensity is strongly dependent on the population size, by Hörmander's theorem, this paper also shows that the stationary distribution of this stochastic Lotka-Volterra system with infinite delay holds a C∞-smooth density.

Khasminskii-type theorem for a class of stochastic functional differential equations

Abstract This paper is concerned with the existence and uniqueness theorems for stochastic functional differential equations with Markovian switching and jump, where the linear growth condition is replaced by more general Khasminskii-type conditions in terms of a pair of Lyapunov-type functions.

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.

Pseudo almost automorphy of stochastic neutral partial functional differential equations with Lévy noise

ABSTRACT The paper introduces and studies the concepts of (Poisson) -pseudo almost automorphy of class r for stochastic processes. Under suitable conditions on the coefficients, we establish the existence, and uniqueness of mild solutions, which are -pseudo almost automorphic of class r in distribution for stochastic neutral partial functional differential equations with Lévy noise.

Analysis of stochastic neutral fractional functional differential equations

This work deals with the large deviation principle which studies the decay of probabilities of certain kind of extremely rare events. We consider stochastic neutral fractional functional differential equation with multiplicative noise and show large deviation principle for its solution processes in a suitable Polish space. The existence and uniqueness results are presented using the Picard iterative method, which is indeed essential for further analysis. The establishment of Freidlin–Wentzell type large deviation principle is solely based on the variational representation developed by Budhiraja and Dupuis in which the weak convergence technique is used to show the sufficient condition. Examples are provided to emphasize the theory.

Asymptotic behaviour analysis of stochastic functional differential equations with semi-Markovian switching signal

ABSTRACT In this paper, we develop the moment and almost-sure stability with a general decay rate for stochastic functional differential equations with semi-Markovian switching signal (SFDESMS). We first present a new equivalent concept of moment stability with general decay rate and its implication to almost-sure stability with general decay rate. Subsequently, using the present equivalent concept and implication relationship, we formulate some explicit sufficient criteria of the moment and almost sure stability with general decay rate for SFDESMS. In addition, we consider the application of our results to two special classes of SFDESMS and obtain the moment and almost-sure stability with general decay rate for two special classes of SFDESMS. At last, we illustrate the obtained theoretical results by numerical examples.

Stability of numerical solution to pantograph stochastic functional differential equations

The paper studies the convergence of the numerical solutions for pantograph stochastic functional differential equations which was proposed in Wu et al.(2022)[16]. We also show that the approximate solutions have the properties of almost surely polynomial stability and exponential stability.

Global Attractiveness and Quasi-Invariant Sets of Impulsive Neutral Stochastic Functional Differential Equations Driven by Tempered Fractional Brownian Motion

Global Attractiveness and Quasi-Invariant Sets of Impulsive Neutral Stochastic Functional Differential Equations Driven by Tempered Fractional Brownian Motion

Existence of relaxed optimal control for $G$-neutral stochastic functional differential equations with uncontrolled diffusion

Existence of relaxed optimal control for $G$-neutral stochastic functional differential equations with uncontrolled diffusion

Existence of solutions for fractional impulsive neutral functional differential equations driven by fractional Brownian motion

Abstract In this paper, we consider a class of fractional impulsive neutral stochastic functional differential equations with infinite delay driven by a fractional Brownian motion in a real separable Hilbert space. We prove the existence of mild solutions by using stochastic analysis and a fixed-point strategy.

Exponential ultimate boundedness and stability of stochastic differential equations with impulese

AbstractThe paper mainly studies globally pth moment exponentially ultimate boundedness and pth moment exponential stability of impulsive stochastic functional differential equations. By using the Lyapunov direct method of Razumikhin‐type condition and the principle of comparison, some sufficient conditions for globally pth moment exponentially ultimate boundedness and globally pth moment exponential stability are presented. Theorems require the linear coefficients of the upper bound of Lyapunov differential operators are time‐varying functions; this generalizes the previous results. When the time delay is not considered in the system, a unified criterion is given to achieve boundedness and stability when the system is disturbed by stabilizing impulse and destabilizing impulse. It shows that the stochastic differential equation may be unbounded or unstable, and it can be bounded or stable by adding appropriate impulsive perturbation. Finally, we use two examples to illustrate the validity of our results.

Asymptotic Bismut formulae for stochastic functional differential equations with infinite delay

Using Malliavin calculus, this paper establishes asymptotic Bismut formulae for stochastic functional differential equations with infinite delay. Both nondegenerate and degenerate diffusion coefficients are treated. In addition, combined with the corresponding exponential ergodicity, stabilization bounds for ∇ P t f \nabla P_{t}f as t → ∞ t\rightarrow \infty are derived.

Fast-slow-coupled stochastic functional differential equations

This paper focuses on two-time-scale coupled stochastic functional differential equations (SFDEs). The system under consideration has a slow component and a fast component. Both components depend on the segment process (an infinite dimension process) of the slow component. To overcome the difficulty due to the past dependence and the coupling of the segment process, such properties as the Hölder continuity and tightness on a space of continuous functions are investigated first for the segment process. In addition, it is also shown that the solution of a fixed-x equation depends continuously on the parameters. Then using the martingale problem formulation, an average principle is established by a direct averaging.

Stability for multi-linked stochastic delayed complex networks with stochastic hybrid impulses by Dupire Itô’s formula

In this paper, the mean-square exponential stability is investigated for multi-linked stochastic delayed complex networks with stochastic hybrid impulses. Distinct from the existing literature, we study the MSDCNs on the basis of the multi-linked stochastic functional differential equations that consider the impact of a certain past interval on the present. Moreover, the stochastic hybrid impulses we discuss possess stochastic impulsive moments and impulsive gain, which make the impulses fit better to the real-world demands for control. Also, a novel concept of average stochastic impulsive gain is proposed to measure the intensity of the stochastic hybrid impulses. By the use of Dupire Itô’s formula, based on Lyapunov method, graph theory and stochastic analysis techniques, two sufficient criteria for the mean-square exponential stability are derived, which are closely related to average stochastic impulsive gain, stochastic disturbance strength as well as the topological structure of the network itself. Finally, an application about neural networks is discussed and corresponding numerical example is presented to demonstrate the feasibility and effectiveness of the theoretical results.