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First-passage time for a particular stationary periodic Gaussian process

Published online by Cambridge University Press:  14 July 2016

L. A. Shepp  and
D. Slepian
L. A. Shepp
Affiliation:
Bell Laboratories, Murray Hill, New Jersey
D. Slepian*
Affiliation:
Bell Laboratories, Murray Hill, New Jersey
*
*Also of the University of Hawaii.
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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

FIRST PASSAGELEVEL-CROSSING PROBABILITYGAUSSIAN PERIODIC PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 13 , Issue 1 , March 1976 , pp. 27 - 38
Copyright
Copyright © Applied Probability Trust 1976 

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References

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