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Sojourns and extremes of a diffusion process on a fixed interval

Published online by Cambridge University Press:  01 July 2016

Simeon M. Berman
Simeon M. Berman*
Affiliation:
New York University
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Abstract

Let X(t), , be an Ito diffusion process on the real line. For u > 0 and t > 0, let Lt(u) be the Lebesgue measure of the set . Limit theorems are obtained for (i) the distribution of Lt(u) for u → ∞and fixed t, and (ii) the tail of the distribution of the random variable max[0, t]X(s). The conditions on the process are stated in terms of the drift and diffusion coefficients. These conditions imply the existence of a stationary distribution for the process.

Keywords

DIFFUSION PROCESSLIMIT DISTRIBUTIONSOJOURNMAXIMUM VALUE
Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 4 , December 1982 , pp. 811 - 832
Copyright
Copyright © Applied Probability Trust 1982 

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References

[1] Berman, S. M. (1970) Excursions above high levels for stationary Gaussian processes. Pacific J. Math. 36, 63–79.CrossRefGoogle Scholar
[2] Berman, S. M. (1971) Maxima and high level excursions of stationary Gaussian processes. Trans. Amer. Math. Soc. 160, 65–85.CrossRefGoogle Scholar
[3] Berman, S. M. (1972) A class of limiting distributions of high level excursions of Gaussian processes. Z. Wahrscheinlichkeitsth. 21, 121–134.CrossRefGoogle Scholar
[4] Berman, S. M. (1972) Maximum and high level excursion of a Gaussian process with stationary increments. Ann. Math. Statist. 43, 1247–1266.CrossRefGoogle Scholar
[5] Berman, S. M. (1973) Excursions of stationary Gaussian processes above high moving barriers. Ann. Prob. 1, 365–387.CrossRefGoogle Scholar
[6] Berman, S. M. (1974) Sojourns and extremes of Gaussian processes. Ann. Prob. 2, 999–1026; Correction 8 (1980), 999.CrossRefGoogle Scholar
[7] Berman, S. M. (1982) Sojourns and extremes of stationary processes. Ann. Prob. 10, 1–46.CrossRefGoogle Scholar
[8] Doob, J. L. (1955) Martingales and one-dimensional diffusion. Trans. Amer. Math. Soc. 78, 168–208.CrossRefGoogle Scholar
[9] De Haan, L. (1970) On Regular Variation and its Application to the Weak Convergence of Sample Extremes. Mathematisch Centrum, Amsterdam.Google Scholar
[10] Ito, K. and McKean, H. P. (1965) Diffusion Processes and their Sample Paths. Springer-Verlag, New York.Google Scholar
[11] Malmquist, S. (1954) On certain confidence contours for distribution functions. Ann. Math. Statist. 25, 523–533.CrossRefGoogle Scholar
[12] Pickands, J. (1969) Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145, 75–86.Google Scholar
[13] Skorokhod, A. V. (1965) Studies in the Theory of Random Processes. Addison-Wesley, Reading, Ma.Google Scholar
[14] Stroock, D. W. and Varadhan, S. R. S. (1979) Multidimensional Diffusion Processes. Springer-Verlag, New York.Google Scholar