Discriminating between scaled and fractional Brownian motion via p-variation statistics

Abstract

In recent years the anomalous diffusion processes have found various applications including polymers. The most common anomalous diffusion processes are continuous time random walk and fractional Brownian motion. However in the literature one can find different systems like fractional Levy stable motion or scaled Brownian motion. A most common tool for anomalous diffusion behavior recognition is the time-averaged mean square displacement. However, this statistics has the same behavior for fractional and scaled Brownian motions. Therefore there is need to apply different approach in order to discriminate between those two anomalous diffusion processes. One of the possibility is the p-variation statistics which demonstrates different behavior for fractional and scaled Brownian motions. In this paper we prove the formula for p-variation of scaled Brownian motion and compare it to the known formula of p-variation for fractional Brownian motion. Moreover, we show how to apply them to real trajectories analysis in order to recognize the proper anomalous diffusion model. The theoretical results are supported by simulated trajectories.