Fractional Brownian motion with two-variable Hurst exponent

Abstract

Abstract In this paper, we consider some experimental and analytical expositions of the time dependency of Hurst exponent. It is our motivation to introduce a new kind of fBm with a two-variable and time-dependent Hurst exponent (fBm-H) by an analysis in the standard deviation of some financial time series. We prove the existence of such processes and also give some simulations of sample paths of some kinds of these processes. Moreover, some important properties such as continuity, stationary increments are identified as well as self-similarity feature in a new manner. Finally, we prove the strong convergency of the Maximum Likelihood Estimation (MLE) of the parameters of the Black–Scholes (BS) model with Generalized Mixed Fractional Brownian Motion (GMFBM) noises. We use that to model the rate of a stock price. The results return that by increasing the number of fBm’s summing in the GMFBM, the model will be more precisely to the real values of the rate of the price data.