Small deviations in p-variation for stable processes

Abstract

Let $\{Z_t,t\ge 0\}$ be a strictly stable process on $\mathbb{R}$ with index $\hat{I}\pm \in (0,2]$. We prove that for every $p>\hat{I}\pm$, there exists ${\hat{I}}^3={\hat{I}}^3(\hat{I}\pm ,p)$ and $\k = \k (α, p)\in (0, +\infty)$ such that $$\lim_{\ee\downarrow 0}\ee^γ\log\pb\lcr ||Z||_{p}\leq \ee \rcr = - \k,$$ where $||Z|{|}_p$ stands for the strong $p$-variation of $Z$ on $[0,1]$. The critical exponent ${\hat{I}}^3(\hat{I}\pm ,p)$ takes a different shape according as $|Z|$ is a subordinator and $p>1$, or not. The small ball constant $\k (α, p)$ is explicitly computed when $p\le 1$, and a lower bound on $\k (α, p)$ is easily obtained in the general case. In the symmetric case and when $p>2$, we can also give an upper bound on $\k (α, p)$ in terms of the Brownian small ball constant under the $(1/p)$-Hölder semi-norm. Along the way, we remark that the positive random variable $||Z|{|}_p^p$ is not necessarily stable when $p>1$, which gives a negative answer to an old question of P.~E.~Greenwood.

16 pages. Submitted