Inverse Gaussian and its inverse process as the subordinators of fractional Brownian motion

In this paper we study the fractional Brownian motion (FBM) time changed by the inverse Gaussian (IG) process and its inverse, called the inverse to the inverse Gaussian (IIG) process. Some properties of the time-changed processes are similar to those of the classical FBM, such as long-range dependence. However, one can also observe different characteristics that are not satisfied by the FBM. We study the distributional properties of both subordinators, namely, IG and IIG processes, and also that of the FBM time changed by these subordinators. We establish also the connections between the subordinated processes considered and the continuous-time random-walk model. For the application part, we introduce the simulation procedures for both processes and discuss the estimation schemes for their parameters. The effectiveness of these methods is checked for simulated trajectories.

Figure 1

Relation between the process $G\left(t\right)$ and its inverse $H\left(t\right)$. Observe that constant periods of $H\left(t\right)$ occur accordingly to the jumps of the process $G\left(t\right)$.

Figure 2

Comparison of the TAMSD (blue star line) and ensemble average MSD (red solid line) for the process $X\left(t\right)$ for the case with $H=0.3$ and $\gamma =\delta =1$ (top) and $H=0.7$ and $\gamma =\delta =1$ (bottom).

Figure 3

Exemplary trajectories of the FBM time changed by the inverse Gaussian process for $\gamma =\delta =1$ and $H=0.3$ (top) and $H=0.7$ (bottom).

Figure 4

Exemplary trajectories of the FBM time changed by the inverse to the inverse Gaussian process for $\gamma =\delta =1$ and $H=0.3$ (top) and $H=0.7$ (bottom).