Convex Hulls of L\'evy Processes

TL;DR

This paper investigates the geometric properties of convex hulls formed by Le9vy processes, providing conditions for integrability, explicit mean formulas, and limit theorems, especially for symmetric b5-stable processes.

Contribution

It offers new explicit formulas for intrinsic volumes of convex hulls of Le9vy processes and establishes conditions for their geometric properties and limit behaviors.

Findings

Explicit formulas for the mean of intrinsic volumes in symmetric b5-stable cases

Conditions ensuring the origin lies inside the convex hull almost surely

Limit theorems for convex hulls with normal and stable process limits

Abstract

Let $X(t)$, $t\geq0$, be a L\'evy process in $\mathbb{R}^d$ starting at the origin. We study the closed convex hull $Z_s$ of $\{X(t): 0\leq t\leq s\}$. 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 L\'evy 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 L\'evy processes with normal and stable limits are also obtained.