First passage time for Slepian process with linear barrier

Abstract

In this paper we extend results of L.A. Shepp by finding explicit formulas for the first passage probability $F_{a,b}(T|x)=\mathrm{Pr}(S(t)<a+bt\text{ for all }t\in [0,T]|S(0)=x)$, for all $T>0$, where $S(t)$ is a Gaussian process with mean 0 and covariance $\mathbb{E}S(t)S(t^{\prime })=\max \{0,1-|t-t^{\prime }|\}.$ We then extend the results to the case of piecewise-linear barriers and outline applications to change-point detection problems. Previously, explicit formulas for $F_{a,b}(T|x)$ were known only for the cases $b=0$ (constant barrier) or $T\le 1$ (short interval).

References in corpus (1)

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