Nonlinear Dyn (2012) 67:2719–2726 DOI 10.1007/s11071-011-0183-3 ORIGINAL PAPER Solving nonlinear stochastic differential equations with fractional Brownian motion using reducibility approach Caibin Zeng · Qigui Yang · Yang Quan Chen Received: 12 April 2011 / Accepted: 26 July 2011 / Published online: 1 September 2011 © Springer Science+Business Media B.V. 2011 Abstract This paper presents some sufficient and necessary conditions for reducing the nonlinear sto- chastic differential equations (SDEs) with fractional Brownian motion (fBm) to the linear SDEs. The explicit solution of the reduced equation is computed by its integral equation or the variation of param- eters technique. Two illustrative examples are pro- vided to demonstrate the applicability of the proposed approach. Keywords Stochastic differential equation · Fractional Brownian motion · Reducibility · Itô formula 1 Introduction The fractional Brownian motion (fBm) is a self-similar centered Gaussian process with stationary increments C. Zeng · Q. Yang School of Sciences, South China University of Technology, Guangzhou 510640, P.R. China C. Zeng e-mail: zeng.cb@mail.scut.edu.cn Q. Yang e-mail: qgyang@scut.edu.cn Y.Q. Chen ( ) Center for Self-Organizing and Intelligent Systems, Electrical and Computer Engineering Department, Utah State University, Logan, UT 84322-4160, USA e-mail: yangQuan.chen@usu.edu and its variance equals t 2H , where 0 <H< 1 is called the Hurst parameter. This process was originally intro- duced by Kolmogorov in his study of turbulence [1]. Subsequently, many other applications have been sug- gested (for example, [2–5]). For this reason, and also because fBm contains the fundamental classical Brow- nian motion as a special case when H = 1/2, there has been an increasing interest in the research activity re- lated to fBm (refer to the books [6, 7], and the refer- ences cited therein). It was proved in [8] that fBm is not semimartin- gale if H ∈ (0, 1/2) ∪ (1/2, 1). Therefore, the beau- tiful classical theory of stochastic analysis [9] is not applicable to stochastic differential equations (SDEs) driven by fBm with H  = 1/2. In order to obtain good mathematical models based on fBm it is necessary to have a stochastic calculus for such processes. Sev- eral contributions in the literature have been already devoted to stochastic integration and differentiation with respect to fBm ([10–17], and the references cited therein). Then one can construct SDEs with fBm in the modeling of many situations, such as economic data [18], biology [19], turbulence [20], and medicine [21]. The theory on existence and uniqueness of solu- tions of SDEs with fBm has been discussed (see the book [7] for a review). Now it is natural and interesting to consider the question of which SDEs with fBm can be solved. When can solutions be represented explic- itly in term of ordinary and stochastic integrals. The present paper is to answer this question by the notion