Dini derivatives and regularity for exchangeable increment processes

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, \begin{equation*} \limsup _{h\to 0 \pm } \left (X_{t+h}-X_t\right )/h=\infty \quad \text {and}\quad \limsup _{h\to 0\pm } \left (X_{t+h}-X_t\right )/h=-\infty . \end{equation*} This extends a celebrated result of Rogozin for Lévy processes obtained in 1968 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 for Lévy 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 on the weak convergence of conditioned Brownian bridges to the normalized Brownian excursion considered in \cite{MR0436353} and broadened to a subclass of Lévy processes and EI processes by Chaumont and the second author. We prove it here for all infinite variation EI processes. We furthermore obtain a description of the convex minorant known for Lévy processes found in [Ann. Prob. 40 (2012), pp. 1636–1674] and extend it to non-piecewise linear EI processes. Our main tool to study the Dini derivatives is a change of measure for EI processes which extends the Esscher transform for Lévy processes.