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

For $0<q< d$ fixed let $W^{[q,d]}=(W^{[q,d]}_t)_{t\in {[q,d]}}$ 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 $\mathbb{P}\left(W^{[q,d]}_t > g(t) \; \text{for some } t\in[q,d]\right)$, $q< d\leq 2q$, 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.