Neutral stochastic functional differential equations with additive perturbations
Introduction
It is well known that stochastic modelling including Gaussian white noise perturbations has played an important role in many areas of science and engineering for a long time. Having in mind that the Gaussian white noise is mathematically described as a formal derivative of a Brownian motion process, such stochastic modelling is based on various stochastic differential equations of the Ito type. Some of the most frequent and most important stochastic models used when dynamical systems depend not only on present and past states but involve derivatives with functionals, are described by very complex neutral stochastic functional differential equations. These equations were introduced by Kolmanovskii and Nosov [1], [2] to study the behavior of chemical engineering systems in which the physical and chemical processes were distinguished by their complexity, and to explore the theory of aeroelasticity in which aeroelastic efforts present an interaction between aerodynamic, elastic and inertial forces. Because of the complexity of the equations and the fact that it is almost impossible to solve them explicitly, the main interest in the field has often been directed to the existence, uniqueness and stability of their solutions. We refer the reader to more papers and books by Mao [3], [4], [5], for example, and the literature cited therein, as well as to [2], [6], among others.
Recall that perturbed stochastic differential equations are the topic of permanent interest of many authors, both theoretically and in applications in various areas of science and engineering. Mathematical stochastic models of complex phenomena under perturbations in analytical mechanics, control theory and population dynamics, for example, can be sometimes compared and approximated by appropriate unperturbed models of a simpler structure. Stochastic perturbations can be also used to stabilize an unstable dynamical system or to make a given stable system even more stable when it is already stable [7]. We highlight here Friedlin and Wentzell [8] studying stochastically perturbed dynamical systems, Khasminskii [9] who was one of the first to treat stochastic differential equations with small parameters and who stated the well-known averaging principle for stochastic differential equations of the Ito type, Stoyanov [10] who studied regularly perturbed stochastic differential equations, among other things. However, the essential motivation to the authors of the paper goes back to paper [11] by Stoyanov and Botev, where some special types of perturbations are independently considered for stochastic differential equations of the Ito type.
Our investigation is concerned with neutral stochastic functional differential equations containing deterministic and random perturbations dependent on a small parameter. Their solutions are compared with the solutions to the corresponding unperturbed equations so that, under some conditions, they become close in the th moment sense. This makes it possible for us to study some quantitative and qualitative properties of the solutions to the perturbed equations, by studying the solutions to the corresponding unperturbed equations. Similar problems are considered in papers [12], [13], [14], [15] by Janković and Jovanović for various types of perturbations and for more complex perturbed stochastic differential and integrodifferential equations. However, the technique used in the present paper is completely different and is conditioned by the past and present state spaces on which the neutral stochastic functional differential equations are defined.
Before stating the main results, we will briefly reproduce only the essential notations and definitions, which are necessary in our investigation. First, we assume that all random variables and processes considered here are defined on a complete probability space with a natural filtration generated by a standard d-dimensional Brownian motion , , i.e. . Let the Euclidean norm be denoted by and, for simplicity, for a matrix B, where is the transpose of a vector or matrix. Likewise, let be the operator norm for a matrix A.
For a given , let be the family of Borel measurable -valued functions , equipped with the normLet also be the family of Borel measurable bounded non-negative functions , such that (the weighing functions) and be the family of continuous bounded -valued stochastic processes , such that is -measurable for every s (we require that for ). The topic of our study is the following neutral stochastic functional differential equationwith the initial condition , whereand is an -valued stochastic process.
An -adapted process is said to be the solution to Eq. (1) if it satisfies the initial condition and the corresponding integral equation holds a.s., that is, for every ,Remember that in [2] Kolmanovskii and Nosov exposed the basic existence-and-uniqueness theorem: letfor all , where is a constant and . Let also the linear growth condition and the global Lipschitz condition (or in a weakened version, the local Lipschitz condition) in the second argument hold for and , that is, let there exist a constant such thatfor all , and . Then Eq. (1) has a unique a.s. continuous solution. Moreover, if there exists the pth moment for , , then (for more details see [2], [4], [6]).
Together with Eq. (2) we consider the following perturbed neutral stochastic functional differential equation in its integral form:where is a small parameter and are defined as , , respectively. Obviously, if we require that be close with , in some way, we can expect that the solutions x and are also close in some sense. The aim of our analysis is to study the closeness of the solutions in the -th moment sense, , whereand are defined as , respectively. Then, , could be understood as the perturbations for the perturbed equation (6), with respect to the unperturbed equation (2). Likewise, we introduce the following assumptions:
. There exist a positive integer m and a non-random value , such that , and. There exist a non-random value and non-negative bounded functions , and , defined on and dependent on , such that. We assume that the condition (3) holds for the functionals G and , while and the perturbations satisfy the linear growth condition (4) and the Lipschitz condition (5).
Note that the assumptions and imply that there exist unique a.s. continuous solutions and to Eqs. (2), (6), respectively, satisfying and , and that all the Lebesgue and Ito integrals employed further are well defined.
The paper is organized in the following way: In Section 2 we estimate globally the th moment closeness between the solutions and , which will be important in the description of our main results. Section 3 contains the main results, that is, the conditions under which the solutions are close in the th moment sense on finite time-intervals and on intervals whose length tends to infinity when tends to zero.
Section snippets
Preliminary results
In the sequel, with no special emphasis, the elementary inequality will be used several times:
We will also apply the inequality: For , and ,The proof holds straightforwardly by taking in the inequality , where , and [4, Lemma 4.1].
Likewise, the following integral inequality will be used: Lemma 1 Let and be non-negative and continuous functions on Gronwall–Bellman Lemma [16]
Main results
Having in mind that the perturbations are limited in the sense of the assumptions and , then, if we require that they tend to zero as , we could expect that , uniformly in . The following theorem contains conditions under which this conclusion is valid. Theorem 2 Let the conditions of Theorem 1 be satisfied and let , , , , tend to zero as tends to zero. Then, Proof Let us denote that
References (16)
Exponential stability in mean square of neutral stochastic differential functional equations
Syst. Control Lett.
(1995)- et al.
Perturbed stochastic hereditary differential equations with integral contractors
Comput. Math. Appl.
(2001) - et al.
On perturbed nonlinear Itô type stochastic integrodifferential equations
J. Math. Anal. Appl.
(2002) - V.B. Kolmanovskii, V.R. Nosov, Stability and Periodic Modes of Control Systems with Aftereffect. Nauka, Moscow,...
- et al.
Stability of Functional Differential Equations
(1986) - X. Mao, Stochastic Differential Equations and Applications, Horvood, Chichester, UK,...
Asymptotic properties of neutral stochastic differential delay equations
Stochast. Stochast. Rep.
(2000)Stochastic Functional Differential Equations
(1986)
Cited by (17)
Backward stochastic Volterra integral equations with additive perturbations
2015, Applied Mathematics and ComputationCitation Excerpt :The significant interest during the last several years has been directed to various complex perturbed stochastic differential and integrodifferential equations. We highlight here Friedlin and Wentzell [19], Khasminskii [20], Stoyanov [21], Stoyanov and Botev [22], Janković et al. [23–27] etc. In contrast to stochastic differential and integrodifferential equations, completely different procedures must applied to investigate the closeness between the solutions of perturbed and unperturbed BSDEs.
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2014, Applied Mathematics and ComputationCitation Excerpt :Here, we refer to Li and Li [9], Kubilius and Platen [10], Shaikhet [11], Gardon [12], Yuan and Mao [13], Mao et al. [14,15], Higham and Kloeden [16,17], Higham et al. [18] and Platen and Liberati [19] and references therein. Recently, motivated by the theory of aeroelasticity, a class of neutral stochastic equations has also received a great deal of attention and much work has been done on neutral stochastic equations, for example, [20–29]. In Section 2, we introduce some notations and hypotheses concerning Eq. (2); In Section 3, the existence and uniqueness of the solutions to Eq. (2) are investigated; In Section 4, we define the Euler approximate solution to NSDEwMSJs and show that the Euler approximate solution converge to the true solution under non-Lipschitz condition by applying some useful lemmas; In Section 5, we present two examples which illustrate the main results in this paper.
Almost sure exponential stability of solutions to highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler-Maruyama approximation
2013, Mathematical and Computer ModellingCitation Excerpt :Stochastic differential equations are widely known as a useful tool for describing those phenomena which are influenced by some random factors. Often the investigation of such phenomena requires more complex models based on (neutral) stochastic differential delay equations or (neutral) stochastic functional differential equations, which have been considered by many authors (see, for example [1–5]). Recently, highly nonlinear stochastic differential equations have attracted much attention because of their practical importance, especially in population dynamics, finance, and control, among others.
Perturbed backward stochastic differential equations
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Supported by Grant No. 144003 of MNTRS.