论文标题
Lévy路径的凸面有多光滑？
How smooth can the convex hull of a Lévy path be?
论文作者
论文摘要
我们描述了$ c'$不断增加的莱维路径凸小小的衍生物$ c'$的增长率。由于凸小群是分段线性的，因此$ c'$可能在有限坡度的顶点时间$τ_s$ s $ s $ s = c'_ {τ_s} $或在$ 0 $的情况下表现出这种行为。虽然凸壳取决于整个路径，但我们表明衍生物$ c'$的局部波动仅取决于Lévy过程的小跳跃的精细结构，并且在所有时间范围内都是相同的。在吸引稳定过程的领域中，我们建立了鲜明的结果，基本上表征了$ c'$的连续性模量，直到亚属性因素。作为推论，我们在各种莱维过程中获得了$ 0 $ $ 0 $的增长率的新成果。
We describe the rate of growth of the derivative $C'$ of the convex minorant of a Lévy path at times where $C'$ increases continuously. Since the convex minorant is piecewise linear, $C'$ may exhibit such behaviour either at the vertex time $τ_s$ of finite slope $s=C'_{τ_s}$ 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 Lévy 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 $C'$ up to sub-logarithmic factors. As a corollary we obtain novel results for the growth rate at $0$ of meanders in a wide class of Lévy processes.
