Radon-Nikodym derivatives, passages and maxima for a Gaussian process with particular covariance and mean

Abstract

We find expressions for the R–N derivative of the stationary Gaussian process with the particular covariance and mean, respectively, R(t, s) = max(1 – |t – s|, 0) and m(t)= aR(t, D), 0 ≦ D ≦ 1, within the time interval [0, 1]. We use these results, and a lemma on multiple reflections of the Wiener process, to find formulae for the probabilities of first passage time and maxima in [0, 1], and bounds on the former within [– 1, 1]. While previous work dealt extensively with the zero mean process, mean functions, as defined here, appear in signal detection and parameter estimation problems under the hypothesis that a rectangular signal centered at t = D is present in an observed process.