Boundary crossing probabilities for $(q,d)$-Slepian-processes

Abstract

For $0<q<d$ fixed let $W^{[q,d]}=(W_t^{[q,d]}{)}_{t\in [q,d]}$ be a $(q,d)$-Slepian-process defined as centered, stationary Gaussian process with continuous sample paths and covariance \begin{align*} C_{W^{[q,d]}}(s,s+t) = (1-\frac{t}{q})^+, \quad q\leq s\leq s+t\leq d. \end{align*} Note that \begin{align*} \frac{1}{\sqrt{q}}(B_t-B_{t-q})_{t\in [q,d]}, \end{align*} where $B_t$ is standard Brownian motion, is a $(q,d)$-Slepian-process. In this paper we prove an analytical formula for the boundary crossing probability $\mathbb{P}\left(W_t^{[q,d]}>g(t)\text{for some }t\in [q,d]\right)$, $q<d\le 2q$, in the case $g$ 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.

Cited by in corpus (2)

• The joint distributions of running maximum of a Slepian processes
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