Certain Properties of Gaussian Processes and Their First‐Passage Times

Summary The class of Gauss‐Markov processes is characterized by their covariances. A functional equation is solved, giving the class of all Gauss–Markov processes with stationary transition probabilities. The notion of a conditionally Markov Gaussian process is introduced. A functional equation is solved which characterizes the class of stationary, conditionally Markov Gaussian processes. A class of non‐stationary conditionally Markov Gaussian processes is constructed. Using a transformation of the Wiener process, an explicit expression is obtained for the first‐passage‐time density function, from level x 0 ≠ 0 at time t 0 to the level zero at time t > t 0 , for all Gauss–Markov (G.M.) processes. For two particular classes of G.M. processes an explicit expression is obtained for the density of the first‐passage time, from any level x 0 at t 0 to any level x ≠ x 0 at time t > t 0 . Some of the results are extended to a class of Gaussian processes which are not Markovian.