On convex hull of d-dimensional fractional Brownian motion

TL;DR

This paper investigates whether the convex hull of a d-dimensional fractional Brownian motion contains the origin as an interior point, extending known properties from standard Brownian motion to fractional cases.

Contribution

It establishes an analogous property for the convex hull of d-dimensional fractional Brownian motion, extending classical results to fractional processes.

Findings

Convex hull of fractional Brownian motion contains 0 as an interior point

Extension of classical Brownian motion properties to fractional case

Provides theoretical insight into geometric properties of fractional Brownian motion

Abstract

It is well known that for standard Brownian motion $ \{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 contains 0 as an interior point for each $t > 0$ (see \cite{E}). The aim of this note is to state the analoguos property for $d$-dimensional fractional Brownian motion.