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Hitting time distributions of single points for 1-dimensional generalized diffusion processes

Published online by Cambridge University Press:  22 January 2016

Makoto Yamazato
Makoto Yamazato*
Affiliation:
Department of Mathematics Nagoya Institute of Technology, Gokiso, Showa-ku Nagoya, 466, Japan
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In this paper, we will characterize the class of (conditional) hitting time distributions of single points of one dimensional generalized diffusion processes and give their tail behaviors in terms of speed measures of the generalized diffusion processes.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 119 , September 1990 , pp. 143 - 172
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1990

References

[1] Bondesson, L., Classes of infinitely divisible distributions and densities, Z. Wahrsch. Verw. Gebiete, 57 (1981), 39–71.Google Scholar
[2] Donoghue, W. F. Jr., Monotone matrix functions and analytic continuation, Springer, Berlin Heidelberg New York, 1974.Google Scholar
[3] Dym, H. and McKean, H. P., Gaussian processes, function theory, and the inverse spectral problem, Academic press, New York San Francisco London, 1976.Google Scholar
[4] Has’minskii, R. Z., Stochastic stability of differential equations, Sijthoff & Noordhoff, Alphen aan den Rijn Rockville, 1980.Google Scholar
[5] Ito, K. and McKean, H. P. Jr., Dffusion processes and their sample paths, second printing, Springer, Berlin Heidelberg New York, 1974.Google Scholar
[6] Kac, I. S. and Krein, M. G., On the spectral functions of the string, A. M. S. Translations Series 2, Vol. 103 (1974), 19–102, (original in Russian, “Mir”, Moscow (1968), 648–737).Google Scholar
[7] Kac, I. S. and Krein, M. G., Criteria for the discreteness of the spectrum of a singular string, Izv. Vyss. Ucebn. Zaved. Mat., 2 (1958), 136–153, (in Russian).Google Scholar
[8] Kasahara, Y., Spectral theory of generalized second order differential operators and its applications to Markov processes, Japan J. Math., 1 (1975), 67–84.Google Scholar
[9] Keilson, J., On the unimodality of passage time densities in birth-death processes, Statist. Neerlandica, 35 (1981), 49–55.Google Scholar
[10] Kent, J. T., Eigenvalue expansions for diffusion hitting times, Z. Wahrsch. Verw. Gebiete, 52 (1980), 309–319.Google Scholar
[11] Krein, M. G., On some classes of the effective determination of the densities of a non-homogeneous string from its spectral function, Dokl. Acad. Nauk SSSR 93 (1953), 617–620 (in Russian).Google Scholar
[12] Kotani, S. and Watanabe, S., Krein’s spectral theory of strings and generalized diffusion processes, Functional Analysis in Markov Processes (Fukushima, M., ed.), Lecture Notes in Math., 923 (1982), 235–259, Springer, Berlin Heidelberg New York.Google Scholar
[13] Seneta, E., Regularly varying functions, Lecture notes in Math., 58 (1976), Springer, Berlin Heidelberg New York.Google Scholar
[14] Stone, C., Limit theorems for random walks, birth and death processes, and diffusion processes, Illinois J. Math., 7 (1963), 638–660.CrossRefGoogle Scholar
[15] Widder, D. V., The Laplace Transform, Princeton Univ. Press, Princeton, 1972.Google Scholar
[16] Yamazato, M., Characterization of the class of upward first passage time distributions of birth and death processes and related results, J. Math. Soc. Japan, 40 (1988), 477–499.Google Scholar