Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-29T04:24:32.638Z Has data issue: false hasContentIssue false
  • English
  • Français

First passage to a general threshold for a process corresponding to sampling at Poisson times

Published online by Cambridge University Press:  14 July 2016

Barry Belkin
Barry Belkin*
Affiliation:
Daniel H. Wagner, Associates, Paoli, Pennsylvania
Get access Rights & Permissions [Opens in a new window]

Extract

The problem of computing the distribution of the time of first passage to a constant threshold for special classes of stochastic processes has been the subject of considerable study. For example, Baxter and Donsker (1957) have considered the problem for processes with stationary, independent increments, Darling and Siegert (1953) for continuous Markov processes, Mehr and McFadden (1965) for Gauss-Markov processes, and Stone (1969) for semi-Markov processes. The results, however, generally express the first passage distribution in terms of transforms which can be inverted only in a relatively few special cases, such as in the classical case of the Weiner process and for certain stable and compound Poisson processes. For linear threshold functions and processes with non-negative interchangeable increments the first passage problem has been studied by Takács (1957) (an explicit result was obtained by Pyke (1959) in the special case of a Poisson process). Again in the case of a linear threshold, an explicit form for the first passage distribution was found by Slepian (1961) for the Weiner process. For the Ornstein-Uhlenbeck process and certain U-shaped thresholds the problem has recently been studied by Daniels (1969).

Type
Research Papers
Information
Journal of Applied Probability , Volume 8 , Issue 3 , September 1971 , pp. 573 - 588
Copyright
Copyright © Applied Probability Trust 1971 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barlow, R. E. and Proschan, F. (1965) Mathematical Theory of Reliability. John Wiley & Sons, Inc., New York.Google Scholar
Baxter, G. and Donsker, M. D. (1957) On the distribution of the supremum functional for processes with stationary independent increments. Trans. Amer. Math. Soc. 85, 73–87.CrossRefGoogle Scholar
Daniels, H. F. (1969) The minimum of a stationary Markov process superimposed on a U-shaped trend. J. Appl. Prob. 6, 399–408.CrossRefGoogle Scholar
Darling, D. A. and Siegert, A. J. F. (1953) The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624–632.CrossRefGoogle Scholar
Loane, E. P., Richardson, H. R. and Boylan, E. S. (1964) Theory of cumulative detection probability. Daniel H. Wagner, Associates Report to U.S. Naval Underwater Sound Laboratory, Chapter IV. (DDC No. AD 615 497).CrossRefGoogle Scholar
Loane, E. P., Curtis, E. C. and Richardson, H. R. (1966) Cumulative detection probability for continuous-parameter stochastic processes. Daniel H. Wagner, Associates Report to U. S. Naval Air Development Center, Appendix A. (DDC No. AD 803 585).Google Scholar
Mehr, C. B. and Mcfadden, J. A. (1965) Certain properties of Gaussian processes and their first-passage times. J. R. Statist. Soc. B 27, 505–522.Google Scholar
Pyke, R. (1959) The supremum and infimum of the Poisson process. Ann. Math. Statist. 30, 568–576.CrossRefGoogle Scholar
Riesz, F. and Sz.-Nagy, B. (1955) Functional Analysis. Frederick Ungar Publishing Company, New York.Google Scholar
Slepian, D. (1961) First passage time for a particular Gaussian process. Ann. Math. Statist. 32, 610–612.CrossRefGoogle Scholar
Stone, L. D. (1969) On the distribution of the supremum functional for semi-Markov processes with continuous state spaces. Ann. Math. Statist. 40, 844–853.CrossRefGoogle Scholar
Takács, L. (1957) On certain sojourn time problems in the theory of stochastic processes. Acta. Math. Acad. Sci. Hung. 8, 169–191.CrossRefGoogle Scholar
Takács, L. (1960) Stochastic Processes. John Wiley & Sons, Inc., New York.Google Scholar