First passage time for Slepian process with linear barrier

04/14/2019

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by   Jack Noonan, et al.

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In this paper we extend results of L.A. Shepp by finding explicit formulas for the first passage probability F_a,b(T | x)= Pr(S(t)<a+bt for all t∈[0,T] | S(0)=x), for all T>0, where S(t) is a Gaussian process with mean 0 and covariance E S(t)S(t')={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≤ 1 (short interval).