Passages and Maxima for a Particular Gaussian Process

Abstract

Expressions for first and last passage probabilities, conditioned on both an initial and a subsequent value, for the Gaussian process with triangular covariance and mean zero, are derived. We use these to bound the passage probabilities of arbitrary functions, to derive formulas for the expectation of passage and excursion times, to prove the uniqueness of the maximum of the process in the interval $\lbrack 0, 1 \rbrack$, and to find a formula for the joint probability density function of the maximum and its instant of occurrence.