How smooth can the convex hull of a L\'evy path be?
arXiv preprint arXiv:2206.09928, 2022•arxiv.org
We describe the rate of growth of the derivative $ C'$ of the convex minorant of a L\'evy path
at times where $ C'$ increases continuously. Since the convex minorant is piecewise linear,
$ C'$ may exhibit such behaviour either at the vertex time $\tau_s $ of finite slope $ s= C'_
{\tau_s} $ or at time $0 $ where the slope is $-\infty $. While the convex hull depends on the
entire path, we show that the local fluctuations of the derivative $ C'$ depend only on the fine
structure of the small jumps of the L\'evy process and are the same for all time horizons. In�…
at times where $ C'$ increases continuously. Since the convex minorant is piecewise linear,
$ C'$ may exhibit such behaviour either at the vertex time $\tau_s $ of finite slope $ s= C'_
{\tau_s} $ or at time $0 $ where the slope is $-\infty $. While the convex hull depends on the
entire path, we show that the local fluctuations of the derivative $ C'$ depend only on the fine
structure of the small jumps of the L\'evy process and are the same for all time horizons. In�…
We describe the rate of growth of the derivative of the convex minorant of a L\'evy path at times where increases continuously. Since the convex minorant is piecewise linear, may exhibit such behaviour either at the vertex time of finite slope or at time where the slope is . While the convex hull depends on the entire path, we show that the local fluctuations of the derivative depend only on the fine structure of the small jumps of the L\'evy process and are the same for all time horizons. In the domain of attraction of a stable process, we establish sharp results essentially characterising the modulus of continuity of up to sub-logarithmic factors. As a corollary we obtain novel results for the growth rate at of meanders in a wide class of L\'evy processes.
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