Abstract and Applied Analysis

Volume 2014 (2014), Article ID 479195, 10 pages

http://dx.doi.org/10.1155/2014/479195

## An Averaging Principle for Stochastic Differential Delay Equations with Fractional Brownian Motion

Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, China

Received 15 November 2013; Accepted 19 December 2013; Published 22 January 2014

Academic Editor: Yaozhong Hu

Copyright © 2014 Yong Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An averaging principle for a class of stochastic differential delay equations (SDDEs) driven by fractional Brownian motion (fBm) with Hurst parameter in is considered, where stochastic integration is convolved as the path integrals. The solutions to the original SDDEs can be approximated by solutions to the corresponding averaged SDDEs in the sense of both convergence in mean square and in probability, respectively. Two examples are carried out to illustrate the proposed averaging principle.

#### 1. Introduction

The averaging principle in stochastic dynamical systems is often used, and it is useful and effective for exploring stochastic differential equations (SDEs) in many different fields [1–4]. In Gaussian random fluctuations case, this analytic technique has been developed by Stratonovich [5, 6] and Khasminskii [7, 8]. Shortly afterwards, researchers began to study the averaging principle of SDEs driven by Poisson noises [9, 10]. Recently, Zhu and his coworkers also investigated this averaging principle for a class of nonlinear systems with Poisson noises [11–13]. Instead of Poisson noises, Xu et al. established an averaging principle for SDEs with Lévy noise [14].

However, random fluctuations with long-range dependence, or correlated noises, are abundant, which may be modeled by fractional Brownian motion (fBm) with Hurst parameter in . Therefore, SDEs with fBm have played an increasingly significant role in various fields of applications, such as hydrology, queueing theory, and mathematical finance [15–18]. Against this background, Xu et al. presented an averaging principle for SDEs with fBm [19]. It should be noted that one assumes the future states will have nothing to do with the past states besides the present states of the systems, for most of the applications. However, under closer scrutiny, it becomes apparent that a more realistic model would include some of the past states of the systems. Fortunately, stochastic differential delay equations (SDDEs) give a mathematical formulation for such kinds of systems. For this reason, SDDEs have attracted more and more attentions except the averaging principle for SDDEs driven by fBm [20–22].

Therefore, we in the present paper will consider an averaging principle for the following SDDEs on of the form: where denotes a strictly positive time delay, are independent fBm with Hurst parameter in defined in a complete probability space , and the hereditary term, which is a measurable function, depends on the path , while is a continuous and smooth function. is a measurable function. represents the forward integral which is interpreted in Definition 1.

The simplified systems to (1) will be introduced as the so-called averaged systems, whose solutions will be applied to approximate the solutions of the original systems. And, we prove that the solutions of the original SDDEs converge to solutions of the corresponding averaged SDDEs in the sense of mean square and probability. In addition, the similar conclusion holds for SDDEs with fBm, where the stochastic differential or stochastic integral is of symmetric and backward types.

#### 2. Some Preliminaries

Since SDEs are interpreted via stochastic integrals, it is necessary to specify the integrals with respect to fBm [23–26].

Firstly, the process of fBm is Gaussian, and, hence we can develop a stochastic calculus of variation [16, 24, 26] (or Malliavin calculus) with respect to fBm.

Let be Borel measurable, . is given by Then we say that if If we equip with the inner product then becomes a separable Hilbert space.

Let be the set of smooth and cylindrical random variables of the form where (i.e., and all its partial derivatives are bounded), , and is a Hilbert space. The elements of may not be functions but distributions of negative order. Thanks to this reason, it is convenient to introduce the space of measurable function on satisfying And it is not difficult to show that is a Banach space with the norm .

The derivative operator of a smooth and cylindrical random variable is defined as the -valued random variable: Then, for any , the derivative operator is a closable operator from into . In addition, we denote as the iteration of the derivative operator for any integer . And the Sobolev space is the closure of with respect to the norm for any ( denotes the tensor product) Similarly, for a Hilbert space , we denote by the corresponding Sobolev space of -valued random variables. For any we denote by the subspace of formed by the elements such that .

By [23], we introduce the -derivative of :

Refer to [23, 26] to obtain more details.

Secondly, recall the definitions of the three types of pathwise integrals introduced by Russo and Vallois in [27].

*Definition 1. *Let be a stochastic process with integrable trajectories.(1)The symmetric integral of with respect to is defined as the limit in probability as tends to zero of
provided this limit exists.(2)The forward integral of with respect to is defined as the limit in probability as tends to zero of
provided this limit exists.(3)The backward integral of with respect to is defined as the limit in probability as tends to zero of
provided this limit exists.

At last, we introduce some auxiliary results.

Lemma 2. *Let be a stochastic process in the space and satisfy
**
By Remark 1 in [26] and Proposition in [23], then the symmetric integral coincides with the forward and backward integrals.*

*Definition 3. *The space of integrands is defined as the family of stochastic processes on , if , is -differentiable, the trace of exists, , , and
and for each sequence of partitions such that as ,
tend to 0 as , where , and .

*Remark 4. *If , then the three types of pathwise integrals exist and the following relations hold:
where denotes the Wick product.

We note that the result of Remark 4 also follows by Proposition in [23] and Lemma 2. The proof of Remark 4 can be found by Theorem in [23].

Lemma 5. *Let be a stochastic process in ; there exists a constant such that
*

*Lemma 6. Suppose that is a stochastic process in ; there exists a constant such that
The detailed proofs of Lemma 5 and Lemma 6 are in [19].*

*Remark 7. *In the same conditions with Lemmas 2, 5 and 6 and under Remark 4, we can get the symmetric, forward and backward integrals cases have the same conclusions as symmetric integral case in Lemma 6.

*3. An Averaging Principle for SDDEs with fBm*

*3.1. SDDEs Driven by fBm*

*One can define the integral with respect to fBm using a pathwise approach with . Indeed, if there is a stochastic process , whose trajectories are -Hölder continuous with , then the Riemann-Stieltjes integral exists for each trajectory [28].*

*Moreover, the pathwise Riemann-Stieltjes integral coincides with the symmetric integral in the Russo-Vallois sense which has been proved in [25, 29]. Therefore, in terms of Lemma 2 and [29], we can conclude that the pathwise Riemann-Stieltjes integral coincides with the three types of pathwise integrals in this paper.*

*In this section the forward integral of SDDEs with respect to fBm is firstly concerned. Then, the other two kinds of cases are given subsequently.*

*Now, considering the following assumptions on the coefficients of (1),(H1) is a measurable function which is differentiable in , and there exist some constants , and for every there exists such that the following properties hold:(C1), for all , for all ,(C2), for all , for all , for each ,(C3), for all , for all for each ;(H2) is a measurable function such that, for every and , depends only on . Moreover, there exists with and for all there exists such that(C4), for all , for all , , where , ,(C5), for all .*

*The existence and uniqueness of solutions of the SDDEs driven by fBm under the above assumptions (H1-H2) have been proved by Ferrante and Rovira in [30].*

*3.2. Main Results*

*In this section, we discuss a standard SDDE using an averaging principle.*

*The standard SDDE is defined as
where is a given -dimensional random variable as the initial condition, and the coefficients have the same conditions as in (1), and is a positive parameter with a fixed number.*

*Then, we introduce the averaged SDDE:
*

*Assume that (C1)–(C5) are satisfied for (20); besides the mappings , are measurable.*

*Moreover, we presume the coefficients meet the following additional inequalities:(C6)(C7)where , are positive bounded functions with .*

*Obviously, under similar conditions such as (19), (20) also has a unique solution . In the rest of the paper, we will consider the connections between the solution processes and .*

*Now we prove the following main theorems to show relationship between solution processes and .*

*Theorem 8. Assume that the original SDDE (19) and the averaged SDDE (20) both satisfy the assumptions (H1), (H2) and (C6), (C7). For a given arbitrarily small number , there exist , , and , such that, for any ,
*

*Proof. *By the above analysis, we start with
and employ the following inequality for and :
We obtain that
where denote the above terms, respectively.

Now we present some estimates for .

Firstly, we apply the inequality (23) to yield
where

By the Cauchy-Schwarz inequality for , we arrive at

According to condition (C5), the elementary inequality and , and taking expectation, we have
where denotes constant.

Then, for , using condition (C6), is a positive bounded function and taking expectation to yield
where denotes constant.

Putting and together, we reach

Now, taking expectation on we obtain
where

By Remark 7, Lemma 6, and (C1), we obtain
where , denote constants.

Due to condition (C7), we have
Then

Therefore, from the above discussions (30) and (35), we get

Now by the Gronwall-Bellman inequality, we obtain

Select , , such that, for all , we have
where
is a constant.

Consequently, given any number , we can select , such that, for every and for ,
This completes the proof.

*Theorem 9. Assume that the original SDDE (19) and the averaged SDDE (20) both satisfy the assumptions (H1), (H2) and (C6), (C7). Then for any number , there exist and , such that
*

*Proof. *On the basis of Theorem 8 and the Chebyshev-Markov inequality, for any given number , one can have
Let and the required result follows.

This completes the proof.

*Then we also can study the symmetric integral and backward integral of SDDEs with fBm. On the basis of (19) and (20), we can get the original SDDEs and the averaged SDDEswhere is the initial condition, and the coefficients satisfy the (C1)–(C5).*

*Theorem 10. Assume that the original SDDEs (43a), (44a) and the averaged SDDEs (43b), (44b) both satisfy the assumptions (H1), (H2) and (C6), (C7). For a given arbitrarily small number , there exist and :
And then for any number , we can get
*

*Proof. *The proof is similar to Theorems 8 and 9.

*Remark 11. *By Theorems 8, 9, and 10, that is to say, we get the same results for three types of pathwise integrals of SDDEs.

*4. Examples*

*4. Examples**Now we present two examples to demonstrate the procedure of the averaging principle. Define in Figures 1 and 2.*

*Example 1. *Consider the following SDDE driven by fBm:
with an initial condition , , where , are constants and is a fBm. Obviously,
Let
And define an averaged SDDE as
On , the linear SDDE driven by fBm becomes linear SDE
The explicit solution of this SDE is
where .

Repeating this procedure over the intervals , , and so forth, we can obtain the explicit solution.

Because all the conditions (C1)–(C7) are satisfied for function in SDDEs (19), (20), Theorems 8 and 9 hold. That is,
and as

Now we carry out the numerical simulation to get the solutions of (47) and (50) under the conditions of (a), , , , ,(b), , , , ,(c), , , , ,(d), , , , , respectively.

*Example 2. *Consider the following SDDE driven by fBm:
with an initial condition , , where , are constants and is a fBm. Obviously,
Let
And define a new averaged equation as:
On , the SDDEs driven by fBm become a linear SDE
The explicit solution of this SDE is
where .

Repeating this procedure over the intervals , , and so forth, we can obtain the explicit solution.

Because all the conditions (C1)–(C7) are satisfied for function in SDDEs (19) and (20), Theorems 8 and 9 hold. That is,
and as

Now we carry out the numerical simulation to get the solutions of (55) and (58) under the conditions of (a), , , , ,(b), , , , ,(c), , , , ,(d), , , , , respectively.

*Conflict of Interests*

*Conflict of Interests*

*The authors declare that there is no conflict of interests regarding the publication of this paper.*

*Acknowledgments*

*Acknowledgments*

*This work was supported by the NSF of China (Grant no. 11372247) and Shaanxi Province, Program for NCET, SRF for ROCS, SEM, NPU Foundation for Fundamental Research and Graduate Starting Seed Fund.*

*References*

*References*

- J. B. Roberts and P. D. Spanos, “Stochastic averaging: an approximate method of solving random vibration problems,”
*International Journal of Non-Linear Mechanics*, vol. 21, no. 2, pp. 111–134, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - W. Q. Zhu, “Recent developments and applications of the stochastic averaging method in random vibration,”
*ASME Applied Mechanics Reviews*, vol. 49, 10, pp. S72–S80, 1996. View at Publisher · View at Google Scholar · View at Scopus - N. Sri Namachchivaya and Y. K. Lin, “Application of stochastic averaging for nonlinear dynamical systems with high damping,”
*Probabilistic Engineering Mechanics*, vol. 3, pp. 185–196, 1988. View at Google Scholar - W. Q. Zhu, “Stochastic averaging methods in random vibration,”
*ASME Applied Mechanics Reviews*, vol. 41, no. 5, pp. 189–199, 1988. View at Publisher · View at Google Scholar - R. L. Stratonovich,
*Topics in the Theory of Random Noise*, Gordon and Breach, New York, NY, USA, 1967. - R. L. Stratonovich,
*Conditional Markov Processes and their Application to the Theory of Optimal Control*, Elsevier, New York, NY, USA, 1967. View at MathSciNet - R. Z. Khasminskii, “A limit theorem for the solution of differential equations with random right-hand sides,”
*Theory of Probability & Its Applications*, vol. 11, pp. 390–405, 1963. View at Publisher · View at Google Scholar - R. Z. Khasminskii, “Principle of averaging of parabolic and elliptic differential equations for Markov process with small diffusion,”
*Theory of Probability & Its Applications*, vol. 8, no. 1, pp. 1–21, 1963. View at Publisher · View at Google Scholar - I. M. Stoyanov and D. D. Bainov, “The averaging method for a class of stochastic differential equations,”
*Ukrainian Mathematical Journal*, vol. 26, no. 2, pp. 186–194, 1974. View at Publisher · View at Google Scholar · View at Scopus - V. G. Kolomiets and A. I. Mel'nikov, “Averaging of stochastic systems of integral-differential equations with Poisson noise,”
*Ukrainian Mathematical Journal*, vol. 43, no. 2, pp. 242–246, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - W. Q. Zhu, “Nonlinear stochastic dynamics and control in Hamiltonian formulation,”
*ASME Applied Mechanics Reviews*, vol. 59, no. 4, pp. 230–248, 2006. View at Publisher · View at Google Scholar - W. T. Jia, W. Q. Zhu, and Y. Xu, “Stochastic averaging of quasinon-integrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations,”
*International Journal of Non-Linear Mechanics*, vol. 51, pp. 45–53, 2012. View at Publisher · View at Google Scholar - Y. Zeng and W. Q. Zhu, “Stochastic averaging of quasi-nonintegrable-Hamiltonian systems under Poisson white noise excitation,”
*ASME Journal of Applied Mechanics*, vol. 78, no. 2, Article ID 021002, 11 pages, 2011. View at Publisher · View at Google Scholar · View at Scopus - Y. Xu, J. Duan, and W. Xu, “An averaging principle for stochastic dynamical systems with Lévy noise,”
*Physica D*, vol. 240, no. 17, pp. 1395–1401, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - W. E. Leland, M. S. Taqqu, W. Willinger, and D. V. Wilson, “On the self-similar nature of Ethernet traffic (extended version),”
*IEEE/ACM Transactions on Networking*, vol. 2, no. 1, pp. 1–15, 1994. View at Publisher · View at Google Scholar · View at Scopus - Y. Hu and B. Øksendal, “Fractional white noise calculus and applications to finance,”
*Infinite Dimensional Analysis, Quantum Probability and Related Topics*, vol. 6, no. 1, pp. 1–32, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Scheffer and F. R. Maciel, “The fractional Brownian motion as a model for an industrial airlift reactor,”
*Chemical Engineering Science*, vol. 56, no. 2, pp. 707–711, 2001. View at Publisher · View at Google Scholar · View at Scopus - N. Chakravarti and K. L. Sebastian, “Fractional Brownian motion models for polymers,”
*Chemical Physics Letters*, vol. 267, no. 1-2, pp. 9–13, 1997. View at Google Scholar · View at Scopus - Y. Xu and R. Guo, “Stochastic averaging principle for dynamical systems with fractional brownian motion,”
*AIMS Discrete and Continuous Dynamical Systems B*. In press. - T. Taniguchi, K. Liu, and A. Truman, “Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces,”
*Journal of Differential Equations*, vol. 181, no. 1, pp. 72–91, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - X. Mao, “Numerical solutions of stochastic functional differential equations,”
*LMS Journal of Computation and Mathematics*, vol. 6, pp. 141–161, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Tan and D. Lei, “The averaging method for stochastic differential delay equations under non-Lipschitz conditions,”
*Advances in Difference Equations*, vol. 2013, article 38, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - F. Biagini, Y. Hu, B. Øksendal, and T. Zhang,
*Stochastic Calculus for Fractional Brownian Motion and Applications*, Springer, London, UK, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - T. E. Duncan, Y. Hu, and B. Pasik-Duncan, “Stochastic calculus for fractional Brownian motion I. Theory,”
*SIAM Journal on Control and Optimization*, vol. 38, no. 2, pp. 582–612, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. S. Mishura,
*Stochastic Calculus for Fractional Brownian Motion and Related Processes*, vol. 1929 of*Lecture Notes in Mathematics*, Springer, Berlin, Germany, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - E. Alòs and D. Nualart, “Stochastic integration with respect to the fractional Brownian motion,”
*Stochastics and Stochastics Reports*, vol. 75, no. 3, pp. 129–152, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - F. Russo and P. Vallois, “Forward, backward and symmetric stochastic integration,”
*Probability Theory and Related Fields*, vol. 97, no. 3, pp. 403–421, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - L. C. Young, “An inequality of the Hölder type, connected with Stieltjes integration,”
*Acta Mathematica*, vol. 67, no. 1, pp. 251–282, 1936. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - D. Nualart and G. Via, “Stochastic integration with respect to fractional Brownian motion and applications,”
*Contemporary Mathematics*, vol. 336, pp. 3–39, 2003. View at Google Scholar - M. Ferrante and C. Rovira, “Convergence of delay differential equations driven by fractional Brownian motion,”
*Journal of Evolution Equations*, vol. 10, no. 4, pp. 761–783, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus

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