Convex Hulls of Lévy Processes

2015/12/22 by Molchanov, Ilya, Wespi, Florian · 1 citation
#52A22 #60D05 #60G51 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1512.07015

Abstract

Let X(t), t≥0, be a Lévy process in ℝ^d starting at the origin. We study the closed convex hull Z_s of \X(t): 0≤ t≤ s\. In particular, we provide conditions for the integrability of the intrinsic volumes of the random set Z_s and find explicit expressions for their means in the case of symmetric α-stable Lévy processes. If the process is symmetric and each its one-dimensional projection is non-atomic, we establish that the origin a.s. belongs to the interior of Z_s for all s>0. Limit theorems for the convex hull of Lévy processes with normal and stable limits are also obtained.

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• Improved bounds on the inradius and inverse inradius process of the convex hull of planar Brownian motion

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