First Passage Time for a Particular Gaussian Process

Abstract

[We find an explicit formula for the first passage probability, Q_a(T | x) = P_r(S(t) 0, where S is the Gaussian process with mean zero and covariance ES(τ)S(t) = max (1 - |t - τ|, 0). Previously, Qa(T∣ x) was known only for T ≤ 1. In particular for T = n an integer and - < x < a < , Q_a(T x) = 1(x) _D (y_i - y_j+1 + a) dy_2 dy_n+1, where the integral is an n-fold integral on y2, ⋯, yn+1 over the region D given by D = {a - x < y_2 < y_1 < < y_n+1} and the determinant is of size (n + 1) (n + 1), 0 < i, j n, with y_0 0, y_1 a - x.]