论文标题
Lévy路径的凸面何时光滑？
When is the convex hull of a Lévy path smooth?
论文作者
论文摘要
我们根据其过渡定律表征了一维lévy过程的类别，这些过程的图具有连续的（平面）凸面。我们表明，这种现象是通过一系列无限变化的过程表现出来的，并且略微取决于零量度的行为。我们引入了一类强烈侵蚀的Lévy过程，其DINI衍生物在所有局部轨迹的最小值中都消失了，这些轨迹的所有轨迹都会对所有线性漂移进行所有扰动，并证明这些精确是具有光滑凸面的过程。我们研究凸船体的平滑度如何破坏和构建示例，这些示例表现出各种平滑/非平滑行为。最后，我们猜测，无限的变化lévy过程要么被严重侵蚀或突然，因此维登的命中率猜想所暗示。在有限变化的情况下，我们以莱维度量来表征船体的平滑度。
We characterise, in terms of their transition laws, the class of one-dimensional Lévy processes whose graph has a continuously differentiable (planar) convex hull. We show that this phenomenon is exhibited by a broad class of infinite variation Lévy processes and depends subtly on the behaviour of the Lévy measure at zero. We introduce a class of strongly eroded Lévy processes, whose Dini derivatives vanish at every local minimum of the trajectory for all perturbations with a linear drift, and prove that these are precisely the processes with smooth convex hulls. We study how the smoothness of the convex hull can break and construct examples exhibiting a variety of smooth/non-smooth behaviours. Finally, we conjecture that an infinite variation Lévy process is either strongly eroded or abrupt, a claim implied by Vigon's point-hitting conjecture. In the finite variation case, we characterise the points of smoothness of the hull in terms of the Lévy measure.
