Small deviations in p-variation for stable processes

Abstract

Let $\{Z_t, t\geq 0\}$ be a strictly stable process on $\R$ with index $\alpha\in (0,2]$. We prove that for every $p > \alpha$, there exists $\gamma = \gamma (\alpha, p)$ and $\k = \k (\alpha, p)\in (0, +\infty)$ such that $$\lim_{\ee\downarrow 0}\ee^{\gamma}\log\pb\lcr ||Z||_{p}\leq \ee \rcr = - \k,$$ where $||Z||_{p}$ stands for the strong $p$-variation of $Z$ on $[0,1]$. The critical exponent $\gamma (\alpha, p)$ takes a different shape according as $|Z|$ is a subordinator and $p >1$, or not. The small ball constant $\k (\alpha, p)$ is explicitly computed when $p \leq 1$, and a lower bound on $\k (\alpha, p)$ is easily obtained in the general case. In the symmetric case and when $p > 2$, we can also give an upper bound on $\k (\alpha, p)$ in terms of the Brownian small ball constant under the $(1/p)$-H\"older 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.