Indicator fractional stable motions

TL;DR

This paper introduces indicator fractional stable motions, a new family of symmetric alpha-stable processes that complement local time fractional stable motions, extending the understanding of stable processes with different Hurst parameters.

Contribution

The paper defines and studies indicator fractional stable motions, a novel family of symmetric alpha-stable processes with Hurst parameters between 0 and 1/2, complementing existing local time fractional stable motions.

Findings

Indicator fractional stable motions exist for 0<H<1/2.

These processes are symmetric alpha-stable and extend the family of stable motions.

They complement local time fractional stable motions for 1/2<H<1.

Abstract

Using the framework of random walks in random scenery, Cohen and Samorodnitsky (2006) introduced a family of symmetric $\alpha$-stable motions called local time fractional stable motions. When $\alpha=2$, these processes are precisely fractional Brownian motions with $1/2<H<1$. Motivated by random walks in alternating scenery, we find a "complementary" family of symmetric $\alpha$-stable motions which we call indicator fractional stable motions. These processes are complementary to local time fractional stable motions in that when $\alpha=2$, one gets fractional Brownian motions with $0<H<1/2$.