First passage time for Slepian process with linear barrier

TL;DR

This paper derives explicit formulas for the first passage probability of a Gaussian Slepian process with linear barriers, extending previous results and applying to change-point detection.

Contribution

It provides new explicit formulas for the first passage probability of the Slepian process with linear barriers, generalizing prior work limited to constant barriers or short intervals.

Findings

Explicit formulas for first passage probabilities with linear barriers

Extension of results to piecewise-linear barriers

Applications to change-point detection problems

Abstract

In this paper we extend results of L.A. Shepp by finding explicit formulas for the first passage probability $F_{a,b}(T\, |\, x)={\rm Pr}(S(t)<a+bt \text{ for all } t\in[0,T]\,\, | \,\,S(0)=x)$, for all $T>0$, where $S(t)$ is a Gaussian process with mean 0 and covariance $\mathbb{E} S(t)S(t')=\max\{0,1-|t-t'|\}\,.$ 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 $T\leq 1$ (short interval).