Averaging Method for Neutral Stochastic Delay Differential Equations Driven by Fractional Brownian Motion

In this paper, we investigate the stochastic averaging method for neutral stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter H ∈ ð1/2, 1Þ. By using the linear operator theory and the pathwise approach, we show that the solutions of neutral stochastic delay differential equations converge to the solutions of the corresponding averaged stochastic delay differential equations. At last, an example is provided to illustrate the applications of the proposed results.


Introduction
Since Kolmogorov's work began in 1940 [1], the fractional Brownian motion fB H ðtÞ, t ≥ 0g with Hurst parameter H ∈ ð0, 1Þ has been studied by some authors [2,3]. B H ðtÞ is a zero mean Gaussian stochastic process with a covariance function determined by parameter H. It is selfsimilar and has the stationary increments and its increment process has long-range dependence when Hurst parameter 1/2 < H < 1, which makes B H ðtÞ a suitable candidate to model many complex phenomena in finance and other practical problems. When H ≠ 1/2, the B H ðtÞ neither is a semimartingale nor a Markov process. These properties mean that the analysis tools for the classical stochastic differential equation theory no longer work. The most obvious problem is how to define a proper notion of stochastic integral with respect to B H ðtÞ. Three main integration techniques with regard to B H ðtÞ have been researched, see, for example, [4][5][6][7][8] and the references therein.
In recent years, there has been much interest in a stochastic averaging method, which provides a powerful tool to approximate the original dynamical systems under random fluctuations by a simpler system. The first work, which was introduced by Khasminskii [9], investigated a stochastic averaging method for stochastic differential equations with Gaussian random fluctuations. Since then, the stochastic averaging method has been developed for many types of stochastic differential equations, see, e.g., [10][11][12][13][14][15].
Some phenomena of biological dynamics, engineering, and financial markets can be better understood when the effect of time delays is considered in the models, see, e.g., [16]. Hence, stochastic delay differential equations driven by fractional Brownian motion are proposed and have recently attracted great attention [17][18][19]. However, except [20] studied averaging method for stochastic delay differential equations of neutral type driven by G-Brownian motion, the averaging method for neutral stochastic delay differential equations is seldom considered. In this paper, our aim is to study the stochastic averaging method for neutral stochastic delay differential equations driven by fractional Brownian motion.

Model Description and Preliminaries
The aim of this section is to introduce the model and some preliminary lemmas.
In this paper, we will discuss the following neutral stochastic delay differential equations driven by fractional Brownian motion: with initial condition The following operator theory is useful to obtain our main results.
Lemma 2 (see [21]). Assume that D is a bounded linear operator on Banach space X and has an inverse bounded operator, for arbitrary ΔD : X ⟶ X, if kΔDk < 1/kD −1 k, then S = D + ΔD has a bounded inverse, and Proof. The proof processes are similar to Lemma 2.6 in [22], so we omit it here.
We note that Φ and Φ −1 are both linear operators. Then, the system (1) can be changed by the inverse transformation of Φ into the following: with initial condition u 0 = ðΦ −1 x 0 Þ. So, we can conclude that uðtÞ is a solution of system (5) if and only if ðΦ −1 uÞðtÞ is a solution of system (1). System (5) can be written in the integral form: where Ð t 0 gðs, xðsÞ, x, ðs − τðsÞÞÞdB H ðsÞ is a pathwise-defined integral which can be represented by fractional derivatives, see, e.g., Young et al. [4], Zähle [5], and Nualart and Răşcanu [7].
For h ∈ W α,1 0 , defining the left-side fractional derivative D α 0+ hðuÞ by Also denoted by For where Γð·Þ is the gamma function. Moreover, if h ∈ W α,1 0 and z ∈ W 1−α,∞ T , one can define the integral Ð t 0 hdz in the sense of Zähle [5] for all t ∈ ½0, T and it follows from Nualart and Răşcanu [7] that For our purpose, we adopt the following hypotheses on the coefficients.

Journal of Function Spaces
Let w ϵ ðtÞ denote the solution to the following averaged system: with w ϵ ð0Þ = u 0 . Then, y ϵ ðtÞ ≔ ðΦ −1 wÞ ϵ ðtÞ is the solution to the following averaged system: with y ϵ ð0Þ = x 0 . Now, from the above hypotheses, we are in conditions to demonstrate the relationship between x ϵ ðtÞ and y ϵ ðtÞ.

Main Results and Proofs
The main goal of this section is to use the averaging principle to investigate the neutral stochastic delay differential equations driven by fractional Brownian motion. Theorem 6 shows that the solution y ϵ ðtÞ of the averaged system (20) converges to solution x ϵ ðtÞ of the original system (17) in the sense of mean square.

Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.