On the convex hull of symmetric stable processes

TL;DR

This paper derives formulas for mean mixed volumes and expected intrinsic volumes of the convex hull of symmetric alpha-stable processes, and analyzes the asymptotic behavior of their volume as time approaches zero.

Contribution

It provides new explicit formulas for mean mixed volumes and expected intrinsic volumes of convex hulls of symmetric stable processes, and studies their small-time asymptotics.

Findings

Derived formulas for mean mixed volumes of convex hulls

Calculated expected intrinsic volumes of Z_t

Analyzed volume asymptotics as t approaches zero

Abstract

Let alpha \in (1, 2] and X be an R^d-valued alpha-stable process with independent and symmetric components starting in 0. We consider the closure S_t of the path described by X on the interval [0, t] and its convex hull Z_t. The first result of this paper provides a formula for certain mean mixed volumes of Z_t and in particular for the expected first intrinsic volume of Z_t. The second result deals with the asymptotics of the expected volume of the stable sausage Z_t+B (where B is an arbitrary convex body with interior points) as t \to 0.