Dini derivatives for Exchangeable Increment processes and applications

Abstract

Let $X$ be an exchangeable increment (EI) process whose sample paths are of infinite variation. We prove that, for any fixed $t$ almost surely, \[ \limsup_{h\to 0 \pm} (X_{t+h}-X_t)/h=\infty \quad\text{and}\quad \liminf_{h\to 0\pm} (X_{t+h}-X_t)/h=-\infty. \]This extends a celebrated result of Rogozin (1968) for L\'evy processes, and completes the known picture for finite-variation EI processes. Applications are numerous. For example, we deduce that both half-lines $(-\infty, 0)$ and $(0,\infty)$ are visited immediately for infinite variation EI processes (called upward and downward regularity). We also generalize the zero-one law of Millar (1977) for L\'evy processes by showing continuity of $X$ when it reaches its minimum in the infinite variation EI case; an analogous result for all EI processes links right and left continuity at the minimum with upward and downward regularity. We also consider results of Durrett, Iglehart and Miller (1977) on the weak convergence of conditioned Brownian bridges to the normalized Brownian excursion, and broadened to a subclass of L\'evy processes and EI processes by Uribe Bravo (2014) and Chaumont and Uribe Bravo (2015). We prove it here for all infinite variation EI processes. We furthermore obtain a description of the convex minorant for non-piecewise linear EI processes, the case of L\'evy processes given by Pitman and Uribe Bravo (2012). Our main tool to study the Dini derivatives is a change of measure for EI processes which extends the Esscher transform for L\'evy processes.