Nonlinear Dyn (2008) 53: 251–259 DOI 10.1007/s11071-007-9312-4 ORIGINAL PAPER Exact linearization of one dimensional Itô equations driven by fBm: Analytical and numerical solutions Gazanfer Unal · Ali Dinler Received: 13 November 2006 / Accepted: 9 November 2007 / Published online: 1 December 2007 © Springer Science+Business Media B.V. 2007 Abstract Necessary and sufficient conditions for the linearization of the one-dimensional Itô stochastic dif- ferential equations driven by fractional Brownian mo- tion (fBm) are given. Stochastic integrating factor has been introduced. A modified Milstein method has been developed to obtain numerical solutions. Analytical solutions have been compared with the numerical so- lutions for linearizable equations. Keywords Exact linearization · Stochastic integrating factor · Modified Milstein method 1 Introduction Fractional Brownian motion (fBm) has been intro- duced by Kolmogorov to study the turbulence in an in- compressible fluid flow [11]. Long range dependence and self-similarity properties of fBm announced itself in the Nile river level studies of Hurst [9]. The parame- ter H ∈ (0, 1) which characterizes fBm W H t (t ≥ 0) is called Hurst parameter. Mandelbrot and van Ness G. Unal ( ) Faculty of Commerce and Department of Mathematics, Yeditepe University, Kadikoy, Istanbul, Turkey e-mail: gazanferunal@gmail.com A. Dinler Faculty of Sciences, Istanbul Technical University, Maslak, Istanbul 34469, Turkey studied properties of the fBm in [12] (see Appendix for definition and some properties). They have also shown that fBm might find an application in finance. Standard Brownian motion becomes a special case of the fBm with the Hurst parameter H = 1/2. This prop- erty alone arises great interest in the literature. In con- trast with the standard Brownian motion, increments of W H t (t ≥ 0) are no longer statistically indepen- dent for nonoverlapping intervals of t . The correlation function C(s) becomes negative for H ∈ (0, 1/2) and it is known as antipersistent behavior [12]. This is also known as intermittency in the study of turbulent fluid flows [11]. The correlation between the increments of W H t (t ≥ 0) for nonoverlapping intervals become posi- tive for H ∈ (1/2, 1). It shows long-range dependence i.e. it becomes persistent. We now witness the foot prints of the fBm W H t (t ≥ 0) for H ∈ (1/2, 1) ranging from the characters of the solar activity to weather derivatives in mathe- matical finance [14]. An fBm W H t (t ≥ 0) is neither a semi-martingale nor a Markov process [1]. Therefore, a new stochastic calculus is needed for its treatment. Itô calculus for H ∈ (0, 1/2) has been given in [2] and for H ∈ (1/2, 1) in [1]. Recently, Bender [3] has de- veloped a new calculus which is valid for H ∈ (0, 1) (see also [5] for an alternative approach). Therefore, we rely on Bender’s Itô formula in this paper. Here, we consider one-dimensional, nonlinear Itô SDE (Stochastic Differential Equation) of the form