On convex hull and winding number of self similar processes

It is well known that for a standard Brownian motion (BM) $ {B(t), ;t \geq 0}$withvaluesin$$\mathbb{R}$^d$,itsconvexhull$ V(t)=\conv {{,B(s),;s \leq t }$withprobability$1$foreach$t > 0$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 $+\mathrm{\infty }\mathrm{.}$ 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.