On convex hull and winding number of self similar processes

TL;DR

This paper demonstrates that properties like convex hulls containing the origin and infinite winding numbers, known for Brownian motion, extend to a broader class of self-similar processes including fractional Brownian motions and stable Levy processes.

Contribution

It generalizes key geometric and topological properties from Brownian motion to a wider class of self-similar stochastic processes.

Findings

Convex hulls of these processes contain the origin as an interior point.

Winding numbers of these processes are infinite.

Properties hold for fractional Brownian motions and stable Levy processes.

Abstract

It is well known that for a standard Brownian motion (BM) $ \{B(t), \;t \geq 0\}$ with values in $\mathbb{R}^d$, its convex hull $ V(t)=\conv \{\{\,B(s),\;s \leq t \}$ with probability $1$ for each $t > 0$ contains $0$ as an interior point (see Evans (1985)). We also know that the winding number of a typical path of a $2$-dimensional BM is equal to $+\infty.$ The aim of this article is to show that these properties aren't specifically "Brownian", but hold for a much larger class of $d$-dimensional self similar processes. This class contains in particular $d$-dimensional fractional Brownian motions and (concerning convex hulls) strictly stable Levy processes.