Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-06T21:40:33.939Z Has data issue: false hasContentIssue false
  • English
  • Français

On moments and tail behaviors of storage processes

Part of: Stochastic processes Special processes Markov processes Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Arturo Kohatsu-Higa  and
Makoto Yamazato
Arturo Kohatsu-Higa*
Affiliation:
Universitat Pompeu Fabra
Makoto Yamazato*
Affiliation:
University of the Ryukyus
*
∗Postal address: Department of Economics, Universitat Pompeu Fabra, Ramón Trias Fargas, 25–27, 08005 Barcelona, Spain. Email address: [email protected]
∗∗Postal address: Department of Mathematics, Faculty of Science, University of the Ryukyus, Senbaru1, Nishihara-cho, Okinawa, Japan 903-0213.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the existence of moments and the tail behavior of the densities of storage processes. We give sufficient conditions for existence and nonexistence of moments using the integrability conditions of submultiplicative functions with respect to Lévy measures. We then study the asymptotical behavior of the tails of these processes using the concave or convex envelope of the release rate function.

Keywords

Dam processstorage processsubordinatorsubmultiplicative functionsubexponentiality

MSC classification

Primary: 60G52: Stable processes 60H10: Stochastic ordinary differential equations 60J75: Jump processes 60K10: Applications (reliability, demand theory, etc.) 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 4 , September 2003 , pp. 1069 - 1086
Copyright
Copyright © Applied Probability Trust 2003 

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

Asmussen, S. (1998). Subexponential asymptotics for stochastic processes: extremal behaviour, stationary distributions and first passage probabilities. Ann. Appl. Prob. 8, 354–374.Google Scholar
Brockwell, P. J., Resnick, S. J., and Tweedie, R. L. (1982). Storage processes with general release rule and additive inputs. Adv. Appl. Prob. 14, 392–433.Google Scholar
Chistyakov, V. P. (1964). A theorem on sums of independent positive random variables and its applications to branching processes. Theory Prob. Appl. 9, 640–648.CrossRefGoogle Scholar
Deshmukh, S. and Pliska, S. (1980). Optimal consumption and exploration of nonrenewable resources under uncertainty. Econometrica 48, 177–200.CrossRefGoogle Scholar
Dharmadhicari, S., and Joag-dev, K. (1988). Unimodality, Convexity and Applications. Academic Press, San Diego, CA.Google Scholar
Embrechts, P., Goldie, C. M., and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335–347.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Grigoriu, M., and Samorodnitsky, G. (2002). Tails of solutions of certain nonlinear stochastic differential equations driven by heavy tailed Lévy motions. In Proc. 2nd MaPhySto Conf. Lévy Process. Theory Appl., MaPhySto, University of Aarhus, pp. 202–205.Google Scholar
Kendall, D. G. (1957). Some problems in the theory of dams. J. R. Statist. Soc. B 19, 207–212.Google Scholar
Rosinski, J. and Samorodnitsky, G. (1993). Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Prob. 21, 996–1014.Google Scholar
Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Sato, K., and Yamazato, M. (1984). Operator-self-decomposable distributions as limit distributions of processes of Ornstein—Uhlenbeck type. Stoch. Process. Appl. 17, 73–100.Google Scholar
Sigman, K., and Yao, D. (1994). Finite moments for inventory processes. Ann. Appl. Prob. 4, 765–778.CrossRefGoogle Scholar