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Passage-time moments for continuous non-negative stochastic processes and applications

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

M. Menshikov  and
R. J. Williams
M. Menshikov*
Affiliation:
Moscow State University
R. J. Williams*
Affiliation:
University of California at San Diego
*
∗ Postal address: Laboratory of Large Random Systems, Moscow State University, Moscow 119899, Russia. Research supported in part by a grant from the International Science Foundation (SOROS).
∗∗ Postal address: Department of Mathematics, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0112, USA. Research supported in part by NSF Grant GER 9023335.
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Abstract

We give criteria for the finiteness or infiniteness of the passage-time moments for continuous non-negative stochastic processes in terms of sub/supermartingale inequalities for powers of these processes. We apply these results to one-dimensional diffusions and also reflected Brownian motion in a wedge. The discrete-time analogue of this problem was studied previously by Lamperti and more recently by Aspandiiarov, Iasnogorodski and Menshikov [2]. Our results are continuous analogues of those in [2], but our proofs are direct and do not rely on approximation by discrete-time processes.

Keywords

PASSAGE-TIME MOMENTSSUB/SUPERMARTINGALE INEQUALITIESDIFFUSIONSREFLECTED BROWNIAN MOTION IN A WEDGE

MSC classification

Primary: 60G44: Martingales with continuous parameter 60J60: Diffusion processes
Type
General Applied Probablity
Information
Advances in Applied Probability , Volume 28 , Issue 3 , September 1996 , pp. 747 - 762
Copyright
Copyright © Applied Probability Trust 1996 

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References

[1] Aspandiiarov, S. and Iasnogorodski, R. (1994) Tails of passage-times and an application to stochastic processes with boundary reflection in wedges. Preprint. .Google Scholar
[2] Aspandiiarov, S., Iasnogorodski, R. and Menshikov, M. (1994) Passage-time moments for non-negative stochastic processes and an application to reflected random walks in a quadrant. Preprint. .Google Scholar
[3] Athreya, K. B. and Kurtz, T. G. (1970) A generalization of Dynkin's identity and some applications. Ann. Prob. 1, 570–579.Google Scholar
[4] Burkholder, D. L. (1977) Exit times of Brownian motion, harmonic majorization and Hardy spaces. Adv. Math. 26, 182–205.Google Scholar
[5] Burkholder, D. L. and Gundy, R. F. (1970) Extrapolation and interpolation of quasi-linear operators on martingales. Acta Math. 124, 249–304.Google Scholar
[6] Carmona, R. and Klein, A. (1983) Exponential moments for hitting times of uniformly ergodic Markov processes. Ann. Prob. 11, 648–655.Google Scholar
[7] Chung, K. L. and Williams, R. J. (1990) Introduction to Stochastic Integration. Birkhäuser, Boston.Google Scholar
[8] Hajek, B. (1982) Hitting-time and occupation-time bounds implied by drift analysis with applictions. Adv. Appl. Prob. 14, 502–525.Google Scholar
[9] Lamperti, J. (1963) Criteria for stochastic processes II: passage-time moments. J. Math. Anal. Appl. 7, 127–145.Google Scholar
[10] Masoliver, J. and Llosa, J. (1990) First-passage-time noninteger moments for some diffusion and dichotomous processes. Phys. Rev. A 41, 5357–5361.Google Scholar
[11] Meyn, S. P. and Tweedie, R. L. (1993) Markov Chains and Stochastic Stability. Springer, Berlin.Google Scholar
[12] Touminen, P. and Tweedie, R. L. (1994) Subgeometric rates of convergence of f-ergodic Markov chains. Adv. Appl. Prob. 26, 775–798.CrossRefGoogle Scholar
[13] Varadhan, S. R. S. and Williams, R. J. (1985) Reflected Brownian motion in a wedge. Commun. Pure Appl. Math. 38, 405–443.Google Scholar
[14] Weiss, G. H., Havlin, S. and Matan, O. (1989) Properties of noninteger moments in a first passage time problem. J. Stat. Phys. 55, 435–439.Google Scholar
[15] Yamazato, M. (1988) Characterization of the class of upward first passage time distributions of birth and death processes and related results. J. Math. Soc. Japan. 40, 477–499.CrossRefGoogle Scholar
[16] Yamazato, M. (1990) Hitting time distribution of single points for 1-dimensional generalized diffusion processes. Nagoya Math. J. 119, 143–172.Google Scholar