Hölder Regularity of the Multifractional Stable Motion

Abstract

In this master thesis, we introduce a new multifractional stable motion, which we refer to as the Itô multifractional stable motion. The definition of the Itô multifractional stable motion is inspired by a relatively recently proposed alternative to the multifractional Brownian motion. The Itô multifractional stable motion is defined as

Y(t) = ∫_R(t-x)₊^H(x)-1/α - (-x)₊^H(x)-1/α dL(x).

Here (x)₊ = max(x, 0), α ∈ (0, 2), L is a standard symmetric α-stable Lévy process and finally, the multifractional parameter H is a jointly measurable stochastic process, adapted to the natural filtration generated by L, taking values in [H_-,H^-] ⊆ (0, 1). Under the assumption that H admits a deterministic modulus of continuity w and that H is strictly bounded from below by 1/α, it is proven that the uniform Hölder exponent ρ_Y^unif([a,b]) over a compact interval satisfies

 ρ_Y^unif([a,b]) ≥ min_t∈[a,b]H(t)-1/α.

Under the further assumption that w(h) log h → 0 as h ↓ 0, it is shown that Y is locally self-similar and that the pointwise Hölder exponent ρ_Y(t) satisfies

ρ_Y(t) ≤ H(t).