Convex hull of Brownian motion and Brownian bridge

TL;DR

This paper investigates the geometric properties of the convex hull formed by the union of planar Brownian motion and Brownian bridge trajectories, providing exact expectations for perimeter and area, and analyzing related probabilistic variables.

Contribution

It introduces explicit formulas for the expected perimeter and area of convex hulls involving Brownian motion and bridges, and extends results to multiple independent processes.

Findings

Exact expected perimeter and area of convex hulls are derived.

Probability density function of the maximum time for combined Brownian processes is explicitly characterized.

Results are generalized to multiple independent Brownian motions and bridges.

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.