First-passage time for a particular stationary periodic Gaussian process
- 1 March 1976
- journal article
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 13 (1) , 27-38
- https://doi.org/10.2307/3212662
Abstract
We find the first-passage probability that X(t) remains above a level a throughout a time interval of length T given X(0) = x0 for the particular stationary Gaussian process X with mean zero and (sawtooth) covariance P(τ) = 1 – α | τ |, | τ | ≦ 1, with ρ(τ + 2) = ρ(τ), – ∞ < τ < ∞. 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.Keywords
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