Multifractional Stable Motion with Random Hurst Exponent

TL;DR

This paper introduces a nonstationary multifractional stable motion with a time-varying, possibly random Hurst exponent, extending fractional stable motion and precisely characterizing its local Hölder regularity.

Contribution

It defines a new class of multifractional stable processes with a random Hurst function and determines their local Hölder exponents exactly.

Findings

The process admits fractional stable motion as a tangent process.

The local Hölder exponent is explicitly linked to the Hurst function.

The model allows for less regular Hurst functions than previous definitions.

Abstract

The fractional stable motion is a prototypical stochastic process exhibiting both heavy tails and long-range dependence, parameterized via a stability index $\alpha$ and a Hurst exponent $H$. We consider a nonstationary extension where the Hurst exponent is a function of time, and potentially random. The construction admits the standard linear fractional stable motion as tangent process, and we exactly determine its local H\"older exponent in terms of the pointwise values of the Hurst function. This is in contrast to other definitions of multifractional processes, where the Hurst function needs to have additional regularity in time.