On convex hull and winding number of self similar 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.