On local regularity of multifractional Brownian motion : Hurst function estimation using a first difference increment estimator

Sammanfattning: Multifractional Brownian motion is a type of stochastic process with time-varying regularity. The main focus of this thesis is estimation of the pointwise regularity of such processes. The need for accurate estimation of the pointwise regularity is necessary as time series are assumed to be multifractional Brownian motion. Utilizing local properties of the multifractional Brownian motion, and a pointwise approximation method, this thesis proposes an estimator that overcomes the problem of separate estimation of a scaling constant known to be present in empirical data. It is shown by Monte Carlo simulation that the estimator manages to capture the behavior of the time-varying Hurst function for three cases of mBm processes. However, the pointwise estimates are volatile when considering a single trajectory. In an attempt to address this issue, a smoothing spline approach is applied. The smoothed single trajectory pointwise estimates shows better accuracy than the original pointwise estimates. For comparison purposes an estimator based on the Increment Ratio Statistic is introduced.