Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T13:36:20.348Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-homogeneous time convolutions, renewal processes and age-dependent mean number of motorcar accidents

Published online by Cambridge University Press:  22 August 2014

Fulvio Gismondi ,
Jacques Janssen  and
Raimondo Manca
Fulvio Gismondi
Affiliation:
University Guglielmo Marconi, Via Plinio 44, 00193 Roma, Italy
Jacques Janssen
Affiliation:
Solvay Business School ULB, 4 Rue de la Colline, 1480 Tubize, Belgium
Raimondo Manca*
Affiliation:
University of Rome La Sapienza, via del Castro Laurenziano 9, 00161 Roma, Italy
*
*Correspondence to: Raimondo Manca, University of Rome La Sapienza, via del Castro Laurenziano 9, 00161 Roma, Italy. Tel: +39 064 976 6507. Fax: +39 064 976 6765. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Non-homogeneous renewal processes are not yet well established. One of the tools necessary for studying these processes is the non-homogeneous time convolution. Renewal theory has great relevance in general in economics and in particular in actuarial science, however, most actuarial problems are connected with the age of the insured person. The introduction of non-homogeneity in the renewal processes brings actuarial applications closer to the real world. This paper will define the non-homogeneous time convolutions and try to give order to the non-homogeneous renewal processes. The numerical aspects of these processes are then dealt with and a real data application to an aspect of motorcar insurance is proposed.

Keywords

Non-homogeneous timeConvolutionRenewal processesNumerical solutionMotorcar insurance
Type
Papers
Information
Annals of Actuarial Science , Volume 9 , Issue 1 , March 2015 , pp. 36 - 57
Copyright
© Institute and Faculty of Actuaries 2014 

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

Baker, C.T.H. (1977). The Numerical Treatment of Integral Equations. Clarendon Press, New York, NY.Google Scholar
Chen, L.P. & Yang, X.Q. (2007). Pricing mortgage insurance with house price driven by a Poisson jump diffusion process. Chinese Journal of Applied Probability and Statistics, 23, 345–351.Google Scholar
Cheng, W.M., Feng, J., Zhou, J.L. & Sun, Q. (2006). Bayes reliability growth analysis for long-life products with small sample missing data. Mohu Xitong y.u Shuxue, 20, 149–153.Google Scholar
Corradi, G., Janssen, J. & Manca, R. (2004). Numerical treatment of homogeneous semi-Markov processes in transient case – a straightforward approach. Methodology and Computing in Applied Probability, 6, 233–246.CrossRefGoogle Scholar
Çynlar, E. (1975). Introduction to Stochastic Processes. Prentice Hall, New York, NY.Google Scholar
Daniel, J.W. (2008). Poisson processes (and mixture distributions). Available online at the address http://www.casact.org/library/studynotes/3_Poisson_2008.pdfGoogle Scholar
Dawid, A.P. (1979). Conditional independence in statistical theory. Journal of the Royal Statistical Society, Series B, 41, 1–31.Google Scholar
De Vylder, F. & Marceau, E. (1996). The numerical solution of the Schimitter problems: theory. Insurance Mathematics & Economics, 19, 1–18.CrossRefGoogle Scholar
Elkins, D.A. & Wortman, M.A. (2001). On numerical solution of the Markov renewal equation: tight upper and lower kernel bounds. Methodology and Computing in Applied Probability, 3, 239–253.CrossRefGoogle Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications vol. 1, 3rd edition, Wiley, London.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications vol. 2, 2nd edition, Wiley, London.Google Scholar
Freiberger, W. & Grenander, U. (1971). A Short Course in Computational Probability and Statistics. Springer Verlag, Berlin.CrossRefGoogle Scholar
Grandell, Jan (1971). A note on linear estimation of the intensity in a doubly stochastic Poisson field. Journal of Applied Probability, 8, 612–614.CrossRefGoogle Scholar
Hildebrand, F.B. (1987). Introduction to Numerical Analisys 2nd edition, Dover, New York.Google Scholar
Janssen, J. & Manca, R. (2001). Numerical solution of non homogeneous semi-Markov processes in transient case. Methodology and Computing in Applied Probability, 3, 271–293.CrossRefGoogle Scholar
Janssen, J. & Manca, R. (2006). Applied Semi-Markov Processes. Springer Verlag, New York, NY.Google Scholar
Kallenberg, O. (1975). Limits of compound and thinned point processes. Journal of Applied Probability, 12, 269–278.CrossRefGoogle Scholar
Lemaire, J. (1995). Bonus Malus Systems in Automobile Insurance. Kluwer, Dordrecht, Netherlands.CrossRefGoogle Scholar
Lu, Y. & Garrido, J. (2004). On double periodic non-homogeneous Poisson processes. Bulletin of the Swiss Association of Actuaries, 2, 195–212.Google Scholar
Martinez-Miranda, M.D., Nielsen, J.P., Sperlich, S. & Verrall, R. (2013). Continuous Chain Ladder: reformulating and generalising a classical insurance problem. Expert Systems with Applications, 40(14), 5588–5603.CrossRefGoogle Scholar
Martinez-Miranda, M.D., Nielsen, J.P. & Verrall, R. (2012). Double Chain Ladder. Astin Bulletin, 42(1), 59–76.Google Scholar
Mikosch, T. (2009). Non-Life Insurance Mathematics. Springer, Berlin.CrossRefGoogle Scholar
Mode, C.J. (1974). Applications of terminating nonhomogeneous renewal processes in family planning evaluation. Mathematical Biosciences, 22, 293–311.CrossRefGoogle Scholar
Moura, M. & Droguett, E.L. (2009). Mathematical formulation and numerical treatment based on transition frequency densities and quadrature methods for non-homogeneous semi-Markov processes. Reliability Engineering and System Safety, 94, 342–349.CrossRefGoogle Scholar
Moura, M. & Droguett, E.L. (2010). Numerical approach for assessing system dynamic availability via continuous time homogeneous semi-Markov processes. Methodology and Computing in Applied Probability, 12, 441–449.CrossRefGoogle Scholar
Norberg, R. (1993). Prediction of outstanding liabilities in non-life insurance. Astin Bulletin, 23, 95–115.CrossRefGoogle Scholar
Norberg, R. (1999). Prediction of outstanding liabilities II. Model variations and extensions. Astin Bulletin, 23, 5–25.CrossRefGoogle Scholar
Pfeifer, D. & Nešlehovathá, J. (2003). Modeling and generating dependent risk processes for IRM and DFA. Astin Colloquim Berlin, 1–30.Google Scholar
Riesz, F. (1913). Les systèmes d’equations lineaires a une infinite d’inconnues. Gauthier-Villars, Paris.Google Scholar
Wilson, S.P. & Costello, M.J. (2005). Predicting future discoveries of European marine species by using non-homogeneous renewal processes. Journal of the Royal Statistical Society: Series C, 54, 897–918.Google Scholar
Xie, M. (1989). On the solution of renewal-type integral equations. Communications in Statistics – Simulation and Computation, 18, 281–293.CrossRefGoogle Scholar