Procesos de Hurts y movimientos brownianos fraccionales en mercados fractales

The independence assumption of the Brownian motion used in the Black-Scholes equation and the valuation of diverse derivatives is revised in this paper. The results of (R/S) method from Fractals Theory, which are used for getting the Hurst’ coefficient, show the existence of long memory in some variables from Mexico and the United States. Fractional Brownian motion (FBM) is a stochastic process more general than the traditional Brownian motion. FBM includes independent and no independent processes. Financial ideas and concepts, like martingales, conditional expected values and the Itô´s Lemma, which are used in traditional Brownian motion, are reproduced with FMB and new and more general mathematical tools built in a Hilbert space are developed as a support to these processes. The paper shows their use in a Black-Scholes fractional equation, the valuation of derivatives, a general equation for bonds and the term structure of a Vasicek´s model for persistent financial series. The generalization includes the deduction of the Black-Scholes equation with the use of the H-J-B method in the problem of a stochastic consumer, in which the variable is modeled as FBM process and its volatility is modeled as a second FBM