Neutral stochastic functional differential equations with additive perturbations

Abstract

The paper deals with the solution to the neutral stochastic functional differential equation whose coefficients depend on small perturbations, by comparing it with the solution to the corresponding unperturbed equation of the equal type. We give conditions under which these solutions are close in the $(2m)$th mean, on finite time-intervals and on intervals whose length tends to infinity as small perturbations tend to zero.

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 $(2m)$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 $(\Omega \text{,}\mathcal{F}\text{,}\{{\mathcal{F}}_t{\}}_{t⩾0}\text{,}P)$ with a natural filtration $\{{\mathcal{F}}_t{\}}_{t⩾0}$ generated by a standard d-dimensional Brownian motion $w=\{w(t)\text{,}t⩾0\}$, $w(t)=(w_1(t)\text{,}{w}_2(t)\text{,}\dots \text{,}{w}_d(t){)}^T$, i.e. ${\mathcal{F}}_t=\sigma \{w(s)\text{,}0⩽s⩽t\}$. Let the Euclidean norm be denoted by $|·|$ and, for simplicity, $\mathit{trace}[B^{T}B]=|B{|}^2$ for a matrix B, where $B^T$ is the transpose of a vector or matrix. Likewise, let $\Vert A\Vert =\sup \{|\mathit{Ax}|:|x|=1\text{,}x\in R^n\}$ be the operator norm for a matrix A.

For a given $\tau >0$, let $L^2([-\tau \text{,}0]\text{;}{R}^n)$ be the family of Borel measurable $R^n$-valued functions $\phi (s)\text{,}-\tau ⩽s⩽0$, equipped with the norm

$$\Vert \phi {\Vert }_{L^2}={\left({\int }_{-\tau }^0|\phi (s){|}^2\mathit{ds}\right)}^{1/2}<\infty \text{.}$$

Let also $\mathcal{W}([-\tau \text{,}0]\text{;}{R}_{+})$ be the family of Borel measurable bounded non-negative functions $\eta (s)\text{,}-\tau ⩽s⩽0$, such that ${\int }_{-\tau }^0\eta (s)\mathit{ds}=1$ (the weighing functions) and ${\mathcal{GB}}_{{\mathcal{F}}_0}([-\tau \text{,}0]\text{;}{R}^n)$ be the family of continuous bounded $R^n$-valued stochastic processes $\xi =\{\xi (s)\text{,}-\tau ⩽s⩽0\}$, such that $\xi (s)$ is ${\mathcal{F}}_0$-measurable for every s (we require that ${\mathcal{F}}_s={\mathcal{F}}_0$ for $-\tau ⩽s⩽0$). The topic of our study is the following neutral stochastic functional differential equation

$$d[x(t)-G(x_t)]=[f(t\text{,}x(t))+g(t\text{,}{x}_t)]\mathit{dt}+\sigma (t\text{,}{x}_t)\mathit{dw}(t)\text{,}t⩾0$$

with the initial condition $x_0=\xi =\{\xi (s)\text{,}-\tau ⩽s⩽0\}\in {\mathcal{GB}}_{{\mathcal{F}}_0}([-\tau \text{,}0]\text{;}{R}^n)$, where

$$\begin{array}{cc}\hfill & G:L^2([-\tau \text{,}0]\text{;}{R}^n)\to R^n\text{,}f:R_{+}\times R^n\to R^n\text{,}\hfill \\ \hfill & g:R_{+}\times L^2([-\tau \text{,}0]\text{;}{R}^n)\to R^n\text{,}\sigma :R_{+}\times L^2([-\tau \text{,}0]\text{;}{R}^n)\to R^n\times R^d\hfill \end{array}$$

and $x_t=\{x(t+s)\text{,}-\tau ⩽s⩽0\}$ is an $L^2([-\tau \text{,}0]\text{;}{R}^n)$-valued stochastic process.

An ${\mathcal{F}}_t$-adapted process $x=\{x(t)\text{,}-\tau ⩽t⩽\infty \}$ 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 $t⩾0$,

$$x(t)-G(x_t)=\xi (0)-G(x_0)+{\int }_0^t[f(s\text{,}x(s))+g(s\text{,}{x}_s)]\mathit{ds}+{\int }_0^t\sigma (s\text{,}{x}_s)\mathit{dw}(s)\text{a.s.}$$

Remember that in [2] Kolmanovskii and Nosov exposed the basic existence-and-uniqueness theorem: let

$$|G(\phi )-G(\varphi ){|}^2⩽k{\int }_{-\tau }^0\eta (s)|\phi (s)-\varphi (s){|}^2\mathit{ds}$$

for all $\phi \text{,}\varphi \in L^2([-\tau \text{,}0]\text{;}{R}^n)$, where $k\in (0\text{,}1)$ is a constant and $\eta (·)\in$ $\mathcal{W}([-\tau \text{,}0]\text{;}{R}_{+})$. 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 $f\text{,}g$ and $\sigma$, that is, let there exist a constant $L>0$ such that

$$|f(t\text{,}x)|⩽L(1+|x|)\text{,}|g(t\text{,}\phi )|+|\sigma (t\text{,}\phi )|⩽L(1+\Vert \phi {\Vert }_{L^2})\text{,}$$

$$|f(t\text{,}x)-f(t\text{,}y)|⩽L|x-y|\text{,}$$

$$|g(t\text{,}\phi )-g(t\text{,}\varphi )|+|\sigma (t\text{,}\phi )-\sigma (t\text{,}\varphi )|⩽L\Vert \phi -\varphi {\Vert }_{L^2}$$

for all $t⩾0$, $x\text{,}y\in R^n$ and $\phi \text{,}\varphi \in L^2([-\tau \text{,}0]\text{;}{R}^n)$. Then Eq. (1) has a unique a.s. continuous solution. Moreover, if there exists the pth moment for $\xi$, $p>0$, then $E{\sup }_{-\tau ⩽t<\infty }|x(t\text{;}\xi ){|}^p<\infty$ (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:

$$x^{\epsilon }(t)-\tilde{G}(x_t^{\epsilon })={\xi }^{\epsilon }(0)-\stackrel{\sim }{G}(x_0^{\epsilon }\text{,}\epsilon )+{\int }_0^t[\tilde{f}(s\text{,}{x}^{\epsilon }(s)\text{,}\epsilon )+\tilde{g}(s\text{,}{x}_s^{\epsilon }\text{,}\epsilon )]\mathit{ds}+{\int }_0^t\tilde{\sigma }(s\text{,}{x}_s^{\epsilon }\text{,}\epsilon )\mathit{dw}(s)\text{,}t⩾0\text{,}$$

where $\epsilon \in (0\text{,}1)$ is a small parameter and ${\xi }^{\epsilon }\text{,}\stackrel{\sim }{G}\text{,}\tilde{f}\text{,}\tilde{g}\text{,}\tilde{\sigma }$ are defined as $\xi$, $G\text{,}f\text{,}g\text{,}\sigma$, respectively. Obviously, if we require that ${\xi }^{\epsilon }\text{,}\stackrel{\sim }{G}\text{,}\tilde{f}\text{,}\tilde{g}\text{,}\tilde{\sigma }$ be close with $\xi$, $G\text{,}f\text{,}g\text{,}\sigma$ in some way, we can expect that the solutions x and $x^{\epsilon }$ are also close in some sense. The aim of our analysis is to study the closeness of the solutions in the $(2m)$-th moment sense, $m\in N$, where

$$\begin{array}{cc}\hfill & \stackrel{\sim }{G}(\phi \text{,}\epsilon )=G(\phi )+\alpha (\phi \text{,}\epsilon )\text{,}\tilde{f}(t\text{,}x\text{,}\epsilon )=f(t\text{,}x)+\beta (t\text{,}x\text{,}\epsilon )\text{,}\hfill \\ \hfill & \tilde{g}(t\text{,}\phi \text{,}\epsilon )=g(t\text{,}\phi )+\gamma (t\text{,}\phi \text{,}\epsilon )\text{,}\tilde{\sigma }(t\text{,}\phi \text{,}\epsilon )=\sigma (t\text{,}\phi )+\delta (t\text{,}\phi \text{,}\epsilon )\hfill \end{array}$$

and $\alpha \text{,}\beta \text{,}\gamma \text{,}\delta$ are defined as $G\text{,}f\text{,}g\text{,}\sigma$, respectively. Then, ${\xi }^{\epsilon }$, $\alpha \text{,}\beta \text{,}\gamma \text{,}\delta$ could be understood as the perturbations for the perturbed equation (6), with respect to the unperturbed equation (2). Likewise, we introduce the following assumptions:

${\mathcal{A}}_1$. There exist a positive integer m and a non-random value ${\delta }_0(\epsilon )$, such that $E{\sup }_{t\in [-\tau \text{,}0]}|\xi (t){|}^{2m}<\infty$, $E{\sup }_{t\in [-\tau \text{,}0]}|{\xi }^{\epsilon }(t){|}^{2m}<\infty$ and

$$E\underset{t\in [-\tau \text{,}0]}{\sup }|{\xi }^{\epsilon }(t)-\xi (t){|}^{2m}⩽{\delta }_0(\epsilon )\text{.}$$

${\mathcal{A}}_2$. There exist a non-random value $\overline{\alpha }(\epsilon )$ and non-negative bounded functions $\overline{\beta }(·)$, $\overline{\gamma }(·)$ and $\overline{\delta }(·)$, defined on $[0\text{,}T]$ and dependent on $\epsilon$, such that

$$\begin{array}{cc}\hfill & \underset{\phi \in L_2}{\sup }|\alpha (\phi \text{,}\epsilon )|⩽\overline{\alpha }(\epsilon )\text{,}\underset{x\in R}{\sup }|\beta (t\text{,}x\text{,}\epsilon )|⩽\overline{\beta }(t\text{,}\epsilon )\text{,}\hfill \\ \hfill & \underset{\phi \in L_2}{\sup }|\gamma (t\text{,}\phi \text{,}\epsilon )|⩽\overline{\gamma }(t\text{,}\epsilon )\text{,}\underset{\phi \in L_2}{\sup }|\delta (t\text{,}\phi \text{,}\epsilon )|⩽\overline{\delta }(t\text{,}\epsilon )\text{.}\hfill \end{array}$$

${\mathcal{A}}_3$. We assume that the condition (3) holds for the functionals G and $\alpha$, while $f\text{,}g\text{,}\sigma$ and the perturbations $\beta \text{,}\gamma \text{,}\delta$ satisfy the linear growth condition (4) and the Lipschitz condition (5).

Note that the assumptions ${\mathcal{A}}_1$ and ${\mathcal{A}}_3$ imply that there exist unique a.s. continuous solutions $x(t)$ and $x^{\epsilon }(t)$ to Eqs. (2), (6), respectively, satisfying $E{\sup }_{t\in [-\tau \text{,}T]}|x(t){|}^{2m}<\infty$ and $E{\sup }_{t\in [-\tau \text{,}T]}|x^{\epsilon }(t){|}^{2m}<\infty$, 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 $(2m)$th moment closeness between the solutions $x(t)$ and $x^{\epsilon }(t)$, 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 $(2m)$th moment sense on finite time-intervals $[-\tau \text{,}T]$ and on intervals $[-\tau \text{,}{T}^{\epsilon }]$ whose length tends to infinity when $\epsilon$ tends to zero.