In this paper, we derive explicit formulas for the first-passage probabilities of the process S(t) = W(t) − W(t + 1), where W(t) is the Brownian motion, for linear and piece-wise linear barriers on arbitrary intervals [0,T]. Previously, explicit formulas for the first-passage probabilities of this process were known only for the cases of a constant barrier or T ≤ 1. The first-passage probabilities results are used to derive explicit formulas for the power of a familiar test for change-point detection in the Wiener process.

中文翻译：

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具有线性和分段线性势垒的Slepian过程的首次通过时间

在本文中，我们为过程S（t）= W（t）-W（t + 1）的第一遍概率导出了明确的公式，其中W（t）是布朗运动，线性和分段运动任意间隔[0，T ]上的线性障碍。以前，对于该方法的第一通道概率明确的公式只对于恒定屏障或的情况下，公知的Ť ≤1。第一通道概率结果用于导出明确的公式为熟悉测试的功率变化-维纳过程中的点检测。