Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-14T05:36:15.156Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonexponential asymptotics for the solutions of renewal equations, with applications

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Chuancun Yin  and
Junsheng Zhao
Chuancun Yin*
Affiliation:
Qufu Normal University
Junsheng Zhao*
Affiliation:
Qufu Normal University
*
∗Postal address: Department of Mathematics, Qufu Normal University, Qufu, 273165, Shandong, P. R. China.
∗Postal address: Department of Mathematics, Qufu Normal University, Qufu, 273165, Shandong, P. R. China.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Nonexponential asymptotics for solutions of two specific defective renewal equations are obtained. These include the special cases of asymptotics for a compound geometric distribution and the convolution of a compound geometric distribution with a distribution function. As applications of these results, we study the asymptotic behavior of the demographic birth rate of females, the perpetual put option in mathematics of finance, and the renewal function for terminating renewal processes.

Keywords

Asymptoticsdefective renewal equationdeterministic population modelcompound geometric distributionheavy-tailed distributionperpetual put optionrenewal function

MSC classification

Primary: 60K05: Renewal theory
Secondary: 60K10: Applications (reliability, demand theory, etc.) 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 43 , Issue 3 , September 2006 , pp. 815 - 824
Copyright
© Applied Probability Trust 2006 

References

Asmussen, S. (1998). A probabilistic look at the Wiener–Hopf equation. SIAM Rev. 40, 189–201.Google Scholar
Cai, J. and Garrido, J. (2002). Asymptotic forms and tails of convolutions of compound geometric distributions, with applications. In Recent Advances in Statistical Methods, ed. Chaubed, Y. P., Imperial College Press, London, pp. 114–131.Google Scholar
Cai, J. and Tang, Q. (2004). On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications. J. Appl. Prob. 41, 117–130.Google Scholar
Chover, J., Ney, P. and Wainger, S. (1973a). Degeneracy properties of subcritical branching processes. Ann. Prob. 1, 663–673.CrossRefGoogle Scholar
Chover, J., Ney, P. and Wainger, S. (1973b). Functions of probability measures. J. Anal. Math. 26, 255–302.Google Scholar
Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Prob. Theory Relat. Fields 72, 529–557.Google Scholar
Cline, D. B. H. (1987). Convolutions of distributions with exponential and subexponential tails. J. Austral. Math. Soc. A 43, 347–365.Google Scholar
Embrechts, P. and Goldie, C. M. (1980). On closure and factorization properties of subexponential and related distributions. J. Austral. Math. Soc. A 29, 243–256.CrossRefGoogle Scholar
Embrechts, P. and Goldie, C. M. (1982). On convolution tails. Stoch. Process. Appl. 13, 263–278.Google Scholar
Embrechts, P. and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1, 55–72.Google Scholar
Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335–347.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, New York.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Gerber, H. U. (1979). An Introduction to Mathematical Risk Theory. S. S. Huebner Foundation, University of Pennsylvania, Philadelphia.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (1998). Pricing perpetual options for Jump processes. N. Amer. Actuarial J. 2, 101–107.Google Scholar
Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132–141.Google Scholar
Klüppelberg, C. (1989). Subexponential distributions and characterizations of related classes. Prob. Theory Relat. Fields 82, 259–269.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, New York.Google Scholar
Teugels, J. L. (1975). The class of subexponential distributions. Ann. Prob. 3, 1001–1011.Google Scholar
Willmot, G., Cai, J. and Lin, X. S. (2001). Lundberg inequalities for renewal equations. Adv. Appl. Prob. 33, 674–689.Google Scholar