Dynamics of stocks prices based in the Black & Scholes equation and nonlinear stochastic differentials equations

The effect of non-linear terms on Itô–Langevin diffusion model with aim to analyze the change generated on long-tail distribution of the probability density of the returns and volatilities and long range memory is investigated. In particular, whether the model still obeys to stylized facts obeyed by the financial markets that is, the behavior of the long-tail distribution of the returns and volatilities and the correspondent scale law. More specifically, whether after inclusion of nonlinear terms in the model, it still satisfies to the inverse cubic law obeyed for quasi all markets and therefore, if the model still obey to the aspects or statistical regularities obeyed by the financial markets.