FIRST-PASSAGE TIME FOR A PARTICULAR STATIONARY PERIODIC GAUSSIAN PROCESS

We find the first-passage probability that X(t) remains above a level a throughout a time interval of length T given X(O)= Xo for the particular stationary Gaussian process X with mean zero and (sawtooth) covariance p(7) = 1 - a 1 71 , 17 I- :51, with p(7 + 2) = p(7), -oo < 7 < 00. The desired probability is explicitly found as an infinite series of integrals of a two-dimensional Gaussian density over sectors. Simpler expressions are found for the case a = 0 and also for the unconditioned probability that X(t) be non-negative throughout [0, T]. Results of some numerical calculations are given.