Convex hull of Brownian motion and Brownian bridge
Abstract
In this article we study the convex hull spanned by the union of trajectories of a standard planar Brownian motion, and an independent standard planar Brownian bridge. We find exact values of the expectation of perimeter and area of such a convex hull. As an auxiliary result, that is of interest in its own right, we provide an explicit shape of the probability density function of a random variable that represents the time when combined maximum of a standard one-dimensional Brownian motion, and an independent standard one-dimensional Brownian bridge is attained. At the end, we generalize our results to the case of multiple independent standard planar Brownian motions and Brownian bridges.
Cite
@article{arxiv.2406.07079,
title = {Convex hull of Brownian motion and Brownian bridge},
author = {Stjepan Šebek},
journal= {arXiv preprint arXiv:2406.07079},
year = {2024}
}
Comments
13 pages. arXiv admin note: text overlap with arXiv:0912.0631 by other authors