The stochastic nonlinear model based on Itô diffusion is proposed as a mathematical model for price dynamics of financial markets. We study this model with relation to concrete stylised facts about financial markets. We investigate the behavior of the long tail distribution of the volatilities and verify the inverse power law behavior which is obeyed for some financial markets. Furthermore, we obtain the behavior of the long range memory and obtain that it follows to a distinct behavior of other stochastic models that are used as models for the finances. Furthermore, we have made an analysis by using Fokker-Planck equation independent on time with the aim of obtaining the cumulative probability distribution of volatilities P ( g ) , however, the probability density found does not exhibit the cubic inverse law.
Keywords: Itô calculus; price dynamics; stochastic differential equation.