可交换增量过程的Dini导数及其应用
Dini derivatives for Exchangeable Increment processes and applications
摘要：
设$X$是一个可交换增量过程，其样本路径为无穷变差.本文证明了对任何固定的$t$几乎必定有\[ \limsup_{h\to 0 \pm} (X_{t+h}-X_t)/h=\infty \quad\text{and}\quad \liminf_{h\to 0\pm} (X_{t+h}-X_t)/h=-\infty. \]这推广了Rogozin（1968）关于Lévy过程的一个著名结果，完善了有限变差EI过程的已知图象.应用程序很多。例如，我们推导出两个半线$(-\infty, 0)$和$(0,\infty)$立即访问无限变化EI过程（称为向上和向下正则性）。我们还推广了Millar（1977）关于Lévy过程的零一律，通过证明在无限变差EI情形下当$X$达到其最小值时它是连续的;对于所有EI过程，一个类似的结果将最小值处的左右连续性与上下正则性联系起来。我们还考虑了Durrett，Iglehart和Miller（1977）关于条件布朗桥弱收敛到归一化布朗漂移的结果，并由Uribe Bravo（2014）和Chaumont和Uribe Bravo（2015）扩展到Lévy过程和EI过程的一个子类。我们在这里证明它的所有无限变化EI过程。我们进一步得到了非分段线性EI过程的凸次优的描述，Pitman和Uribe Bravo（2012）给出了Lévy过程的情况。我们研究Dini导数的主要工具是EI过程的测度变换，它推广了Lévy过程的Esscher变换.
英文摘要：
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 Lévy 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 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 (1977) on the weak convergence of conditioned Brownian bridges to the normalized Brownian excursion, and broadened to a subclass of Lévy 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 Lévy 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 Lévy processes.
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