How smooth can the convex hull of a Lévy path be?

Abstract

We describe the rate of growth of the derivative $C^{\prime }$ of the convex minorant of a Lévy path at times where $C^{\prime }$ increases continuously. Since the convex minorant is piecewise linear, $C^{\prime }$ may exhibit such behaviour either at the vertex time ${\mathit{\tau }}_s$ of finite slope $s=C_{{\mathit{\tau }}_s}^{\prime }$ or at time 0 where the slope is $-\mathrm{\infty }$. While the convex hull depends on the entire path, we show that the local fluctuations of the derivative $C^{\prime }$ 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 $C^{\prime }$ 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.