Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-24T04:23:48.433Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic distributions for the Ornstein-Uhlenbeck process

Published online by Cambridge University Press:  14 July 2016

John A. Beekman
John A. Beekman*
Affiliation:
Ball State University, Muncie, Indiana
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper gives the asymptotic distributions, as the time period grows infinite, of the first exit times above a fixed constant and from upper and lower constant boundaries for the Ornstein-Uhlenbeck stochastic process. The results of a large amount of numerical analysis illustrate the asymptotic forms.

Keywords

ORNSTEIN-UHLENBECK PROCESSRANDOM WALKFIRST EXIT TIMES
Type
Research Papers
Information
Journal of Applied Probability , Volume 12 , Issue 1 , March 1975 , pp. 107 - 114
Copyright
Copyright © Applied Probability Trust 1975 

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

[1] Bateman Manuscript Project (1953) Higher Transcendental Functions. Vol. 2, McGraw-Hill, New York.Google Scholar
[2] Beekman, J. A. (1967) Gaussian Markov processes and a boundary value problem. Trans. Amer. Math. Soc. 126, 29–42.Google Scholar
[3] Bellman, R. and Harris, T. (1951) Recurrence times for the Ehrenfest model. Pacific J. Math. 1, 179–193.Google Scholar
[4] Breiman, L. (1966) First exit times from a square root boundary. Proc. Fifth Berkeley Symp. Math. Statist. Prob. 2, Part 2, 9–16, University of California Press, Berkeley.Google Scholar
[5] Darling, D. A. and Siegert, A. J. F. (1953) The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624–639.Google Scholar
[6] Durbin, J. (1971) Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test. J. Appl. Prob. 8, 431–453.Google Scholar
[7] Fortet, R. (1943) Les fonctions aléatoires du type de Markoff associées à certaines equations lineaires aux derivées partielles du type parabolique. J. Math. Pures Appl. 22, 177–243.Google Scholar
[8] Mandl, P. (1968) Analytic Treatment of One-Dimensional Markov Processes. Springer-Verlag, Prague.Google Scholar
[9] Marcus, M. B. (1972) Upper bounds for the asymptotic maxima of continuous Gaussian processes. Ann. Math. Statist. 43, 522–533.Google Scholar
[10] 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
[11] Newell, G. F. (1962) Asymptotic extreme value distributions for one dimensional diffusion processes. J. Math. Mech. 11, 481–496.Google Scholar
[12] Orey, S. (1970) Growth rate of Gaussian processes with stationary increments. Bull. Amer. Math. Soc. 76, 609–611.Google Scholar
[13] Orey, S. (1972) Growth rate of certain Gaussian processes. Proc. Sixth Berkeley Symp. Math. Statist. Prob. University of California Press, Berkeley.Google Scholar
[14] Pickands, J. III (1967) Maxima of stationary Gaussian processes. Z. Wahrscheinlichkeitsth. 7, 190–223.Google Scholar
[15] Qualls, C. and Watanabe, H. (1972) Asymptotic properties of Gaussian processes. Ann. Math. Statist. 43, 580–596.Google Scholar
[16] Siegert, A. J. F. (1951) On the first passage time probability problem. Phys. Rev. 81, 617–623.Google Scholar
[17] Sweet, A. L. and Hardin, J. C. (1970) Solutions for some diffusion processes with two barriers. J. Appl. Prob. 7, 423–431.Google Scholar
[18] Wang, M. C. and Uhlenbeck, G. E. (1945) On the theory of Brownian motion, II. Rev. Mod. Phys. 17, 323–342.Google Scholar
[19] Whittaker, E. T. and Watson, G. N. (1935) A Course of Modern Analysis. Cambridge University Press, Cambridge.Google Scholar