Estimation of the local Hurst function of multifractional Brownian motion : A second difference increment ratio estimator

Abstract: In this thesis, a specific type of stochastic processes displaying time-dependent regularity is studied. Specifically, multifractional Brownian motion processes are examined. Due to their properties, these processes have gained interest in various fields of research. An important aspect when modeling using such processes are accurate estimates of the time-varying pointwise regularity. This thesis proposes a moving window ratio estimator using the distributional properties of the second difference increments of a discretized multifractional Brownian motion. The estimator captures the behaviour of the regularity on average. In an attempt to increase the accuracy of single trajectory pointwise estimates, a smoothing approach using nonlinear regression is employed. The proposed estimator is compared to an estimator based on the Increment Ratio Statistic.