Fractional Brownian Motion

Introduction

Fractional Brownian motion (fBm) B^H = [, t ≥ 0} with Hurst index H ∈ (0, 1) is a very interesting stochastic object that has attracted increased attention due to its peculiar properties. On the one hand, this is a Gaussian random process with a fairly simple covariance function that provides the Hölder property of trajectories up to the order H. On the other hand, it is a generalization of the Wiener process, which corresponds to the value of the Hurst index H = 1/2. Finally, it is neither a process with independent increments, nor a Markov process, nor a semimartingale unless H = 1/2, and therefore it can be used to model quite complex real processes that demonstrate the phenomenon of memory, both long and short. Long memory corresponds to H > 1/2, while short memory is inherent in H < 1/2. The combination of these properties is useful in modeling the processes occurring in devices that provide cellular and other types of communication, in physical and biological systems and in finance and insurance. Thus, the fBm itself deserves special attention. We will not discuss all the aspects of fBm here, and recommend the books [BIA 08, KUB 17, MIS 08, MIS 17, MIS 18, NOU 12, NUA 03, SAM 06] for more detail concerning various fractional processes.

Now note that the absence of semimartingale and Markov properties always causes the study of the possibility of the approximation ...