Abstract
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 < <I>H</I> < 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 <<i> H</i> < 1. The classic methods (It<i>ô</i> 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.
Highlights
Stochastic calculus is the study of random phenomena depending on the time
We can define the Wiener integral staged with respect to fractional Brownian motions as follow: for a mbf (BtH )t≤0 one defines the integral of Wiener compared to the mbf for f ∈ ε by: n
The work which was to give meaning, to the integral of determined functions nists with respect to fractional Brownian motion is ahieved in a part by constructing classes of integrants for these deterministic functions which allows us to define the integral [16]
Summary
Stochastic calculus is the study of random phenomena depending on the time. As such, it is an extension of probability theorie [10]. Vallois [7] laid the first foundations for a stochastic calculus, generalizing the more classic ones of ito and tratonovich, one of the interests of which is that it makes it possible to give meaning to integrals against processses that are not not necessarily semi-martingales [14]. Diop Bou et al.: Method of Construction of the Stochastic Integral with Respect to Fractional Brownian Motion function being particularity simple. This is why, in folliwing, this process will be used to test the general results that we will establish [4]. Russo and Vallois Method wih makes it possible to give a meaning to integrals which are not necessarily semi-martingales
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