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arXiv · 1510.08244

On convex hull and winding number of self similar processes

Youri Davydov
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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.

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Youri Davydov. 2015-10-28. On convex hull and winding number of self similar processes. https://arxiv.org/abs/1510.08244

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