Boundary crossing probabilities for ()-Slepian-processes

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• 作者：Wolfgang Bischoff ; ^{wolfgang.bischoff@ku.de" class="auth_mail" title="E-mail the corresponding author} ; Andreas Gegg
• 关键词：60G15
• 刊名：Statistics & Probability Letters
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：118
• 期：Complete
• 页码：139-144
• 全文大小：388 K

文摘

For 0<q<d$0<q<d$ fixed let $W^{[q,d]}={\left(W_t^{[q,d]}\right)}_{t\in [q,d]}$ be a (q,d)$(q,d)$-Slepian-process defined as centered, stationary Gaussian process with continuous sample paths and covariance Note that where B_t$B_t$ is standard Brownian motion, is a (q,d)$(q,d)$-Slepian-process. In this paper we prove an analytical formula for the boundary crossing probability $\mathbb{P}(W_t^{[q,d]}>g\left(t\right)\text{for some}t\in [q,d])$, q<d≤2q$q<d\le 2q$, in the case g$g$ is a piecewise affine function. This formula can be used as approximation for the boundary crossing probability of an arbitrary boundary by approximating the boundary function by piecewise affine functions.