Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-27T00:14:02.059Z Has data issue: false hasContentIssue false
  • English
  • Français

SOME ADVANCES ON THE ERLANG(n) DUAL RISK MODEL

Published online by Cambridge University Press:  27 August 2014

Eugenio V. Rodríguez-Martínez ,
Rui M. R. Cardoso  and
Alfredo D. Egídio dos Reis
Eugenio V. Rodríguez-Martínez
Affiliation:
ISEG and CEMAPRE, Department of Mathematics, Universidade de Lisboa, Rua do Quelhas 6, 1200-781 Lisboa, Portugal E-Mail: [email protected]
Rui M. R. Cardoso
Affiliation:
Centro de Matemática e Aplicações, Department of Mathematics, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Monte de Caparica, 2829-516 Caparica, Portugal E-Mail: [email protected]
Alfredo D. Egídio dos Reis*
Affiliation:
ISEG and CEMAPRE, Department of Management, Universidade de Lisboa, Rua do Quelhas 6, 1200-781 Lisboa, Portugal
*
E-Mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The dual risk model assumes that the surplus of a company decreases at a constant rate over time and grows by means of upward jumps, which occur at random times and sizes. It is said to have applications to companies with economical activities involved in research and development. This model is dual to the well-known Cramér-Lundberg risk model with applications to insurance. Most existing results on the study of the dual model assume that the random waiting times between consecutive gains follow an exponential distribution, as in the classical Cramér-Lundberg risk model. We generalize to other compound renewal risk models where such waiting times are Erlang(n) distributed. Using the roots of the fundamental and the generalized Lundberg's equations, we get expressions for the ruin probability and the Laplace transform of the time of ruin for an arbitrary single gain distribution. Furthermore, we compute expected discounted dividends, as well as higher moments, when the individual common gains follow a Phase-Type, PH(m), distribution. We also perform illustrations working some examples for some particular gain distributions and obtain numerical results.

Keywords

Dual risk modelErlang(n) interarrival timesPhase-Type distributiongeneralized Lundberg's equationruin probabilitytime of ruinexpected discounted dividends
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 45 , Issue 1 , January 2015 , pp. 127 - 150
Copyright
Copyright © ASTIN Bulletin 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

Afonso, L.B., Cardoso, R.M.R. and Egídio dos Reis, A.D. (2013) Dividend problems in the dual risk model. Insurance: Mathematics and Economics, 53 (3), 906–918.Google Scholar
Albrecher, H., Badescu, A. and Landriault, D. (2008) On the dual risk model with tax payments. Insurance: Mathematics and Economics, 42 (3), 1086–1094.Google Scholar
Avanzi, B. (2009) Strategies for dividend distribution: A review. North American Actuarial Journal, 13 (2), 217–251.Google Scholar
Avanzi, B. and Gerber, H.U. (2008) Optimal dividends in the dual model with diffusion. ASTIN Bulletin, 38 (2), 653–667.CrossRefGoogle Scholar
Avanzi, B., Gerber, H.U. and Shiu, E.S.W. (2007) Optimal dividends in the dual model. Insurance: Mathematics and Economics, 41 (1), 111–123.Google Scholar
Bayraktar, E. and Egami, M. (2008) Optimizing venture capital investment in a jump diffusion model. Mathematical Methods of Operations Research, 67 (1), 21–42.Google Scholar
Bühlmann, H. (1970) Mathematical Methods in Risk Theory. New York: Springer Verlag.Google Scholar
Cheung, E.C.K. (2012) A unifying approach to the analysis of business with random gains. Scandinavian Actuarial Journal, 2012 (3), 153–182.CrossRefGoogle Scholar
Cheung, E.C.K. and Drekic, S. (2008) Dividend moments in the dual risk model: Exact and approximate approaches. ASTIN Bulletin, 38 (2), 399–422.Google Scholar
Cramér, H. (1955) Collective Risk Theory: A Survey of the Theory from the Point of View of the Theory of Stochastic Process. Stockholm: Ab Nordiska Bokhandeln.Google Scholar
Dickson, D.C.M. and Li, S. (2013) The distributions of the time to reach a given level and the duration of negative surplus in the Erlang(2) risk model. Insurance: Mathematics and Economics, 52 (3), 490–497.Google Scholar
Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory. Philadelphia, PA: S.S. Huebner Foundation for Insurance Education.Google Scholar
Gerber, H.U. and Smith, N. (2008) Optimal dividends with incomplete information in the dual model. Insurance: Mathematics and Economics, 43 (2), 227–233.Google Scholar
Ji, L. and Zhang, C. (2012) Analysis of the multiple roots of the Lundberg fundamental equation in the PH(n) risk model. Applied Stochastic Models in Business and Industry, 28 (1), 73–90.Google Scholar
Landriault, D. and Willmot, G. (2008) On the Gerber–Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time distribution. Insurance: Mathematics and Economics, 42 (2), 600–608.Google Scholar
Lang, S. (2010) Linear Algebra (Undergraduate Texts in Mathematics). New York: Springer.Google Scholar
Li, S. (2008) The time of recovery and the maximum severity of ruin in a Sparre Andersen model. North American Actuarial Journal, 12 (4), 413–424.CrossRefGoogle Scholar
Li, S. and Garrido, J. (2004) On ruin for the Erlang(n) risk process. Insurance: Mathematics and Economics, 34 (3), 391–408.Google Scholar
Ng, A.C.Y. (2009) On a dual model with a dividend threshold. Insurance: Mathematics and Economics, 44 (2), 315–324.Google Scholar
Ng, A.C.Y. (2010) On the upcrossing and downcrossing probabilities of a dual risk model with phase-type gains. ASTIN Bulletin, 40 (1), 281–306.Google Scholar
Seal, H.L. (1969) Stochastic Theory of a Risk Business. New York: Wiley.Google Scholar
Song, M., Wu, R. and Zhang, X. (2008) Total duration of negative surplus for the dual model. Applied Stochastic Models in Business and Industry, 24 (6), 591–600.CrossRefGoogle Scholar
Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. New York: Wiley.Google Scholar
Yang, C. and Sendova, K.P. (2014) The ruin time under the Sparre-Andersen dual model. Insurance: Mathematics and Economics, 54 (1), 28–40.Google Scholar
Yang, H. and Zhu, J. (2008) Ruin probabilities of a dual Markov-modulated risk model. Communications in Statistics, Theory and Methods, 37, 3298–3307.Google Scholar
Zill, D.G. (2012) A First Course in Differential Equations with Modeling Applications, 10th ed.Boston: Cengage Learning.Google Scholar