Method of Construction of the Stochastic Integral with Respect to Fractional Brownian Motion

Since the pioneering work of Hurst, and Mandelbrot, the fractional brownian motions have played and increasingly important role in many fields of application such as hydrology, economics and telecommunications. For every value of the Hurst index H ∈ (0,1) we define a stochastic integral with respect to fractional Brownian motion of index H. This process is called a (standard) fractional Brownian motion with Hurst parameter H. To simplify the presentation, it is always assumed that the fractional Brownian motion is 0 at t=0. If H = 1/2, then the corresponding fractional Brownian motion is the usual standard Brownian motion. If 1/2 < H < 1, Fractional Brownian motion (FBM) is neither a finite variation nor a semi-martingale. Consequently, the standard Ito calculus is not available for stochastic integrals with respect to FBM as an integrator if 1/2 < H < 1. The classic methods (Ito and Stiliege) are excluted. The most studied case is that where H is between 0 and 1/2. Several attempts to define the stochastic integral are made. But so far some difficulties subjust. We give in this paper, several construction methods. So for the construction, we will use other tools to deal with such situations.