Fractional Brownian motion: Difference iterative forecasting models

Forecasting non-stationary stochastic time series represents a rather complex problem. The reason is that such temporal series are not only self-similar but also exhibit a Long-Range Dependence (LRD). As it is known, the Fractional Brown Motion (FBM) can generate a non-stationary stochastic time series with self-similarity and LRD. In this study we investigate the properties of the LRD for identification of self-similarity and the LRD of non-stationary stochastic series by Hurst exponent. Parameter estimation is proposed for Stochastic differential Equation (SDE) of FBM based on Maximum Likelihood Estimation (MLE), and proves the convergence of MLE. The SDE is discretized.The difference equation constructed is the prediction model of the iterative format based on FBM. Monte Carlo simulation is applied to check the validity and accuracy of parameter estimation. We also give a practical example to demonstrate the appropriateness of the predictive model.