Convex Hulls of Lévy Processes

Abstract

Let $X(t)$, $t\ge 0$, be a Lévy process in ${\mathbb{R}}^d$ starting at the origin. We study the closed convex hull $Z_s$ of $\{X(t):0\le t\le s\}$. In particular, we provide conditions for the integrability of the intrinsic volumes of the random set $Z_s$ and find explicit expressions for their means in the case of symmetric $\alpha$-stable Lévy processes. If the process is symmetric and each its one-dimensional projection is non-atomic, we establish that the origin a.s. belongs to the interior of $Z_s$ for all $s>0$. Limit theorems for the convex hull of Lévy processes with normal and stable limits are also obtained.

11 pages