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RECURSIVE CALCULATION OF RUIN PROBABILITIES AT OR BEFORE CLAIM INSTANTS FOR NON-IDENTICALLY DISTRIBUTED CLAIMS

Published online by Cambridge University Press:  16 December 2014

Anisoara Maria Raducan ,
Raluca Vernic  and
Gheorghita Zbaganu
Anisoara Maria Raducan
Affiliation:
Institute for Mathematical Statistics and Applied Mathematics, Calea 13 Septembrie 13, 050711 Bucharest, Romania E-mail: anaraducan@yahoo.com
Raluca Vernic*
Affiliation:
Institute for Mathematical Statistics and Applied Mathematics, Calea 13 Septembrie 13, 050711 Bucharest, Romania and Faculty of Mathematics and Computer Science, Ovidius University of Constanta, 124 Mamaia Blvd., 900527 Constanta, Romania
Gheorghita Zbaganu
Affiliation:
Institute for Mathematical Statistics and Applied Mathematics, Calea 13 Septembrie 13, 050711 Bucharest, Romania and Faculty of Mathematics and Computer Science, University of Bucharest, and 14 Academiei St., 010014, Bucharest, Romania E-mail: zbagang@fmi.unibuc.ro
*
E-mail: rvernic@univ-ovidius.ro
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Abstract

In this paper, we present recursive formulae for the ruin probability at or before a certain claim arrival instant for some particular continuous time risk model. The claim number process underlying this risk model is a renewal process with either Erlang or a mixture of exponentials inter-claim times (ICTs). The claim sizes (CSs) are independent and distributed in Erlang's family, i.e., they can have different parameters, which yields a non-homogeneous risk process. We present the corresponding recursive algorithm used to evaluate the above mentioned ruin probability and we illustrate it on several numerical examples in which we vary the model's parameters to assess the impact of the non-homogeneity on the resulting ruin probability.

Keywords

Continuous time risk processruin probabilityrenewal processnon-homogeneous claim sizesErlang distributionmixture of exponentials distribution
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 45 , Issue 2 , May 2015 , pp. 421 - 443
Copyright
Copyright © ASTIN Bulletin 2014 

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References

Asmussen, S. and Albrecher, H. (2010) Ruin Probabilities. Singapore: World Scientific Publishing Co.Google Scholar
Blazevicius, K., Bieliauskiene, E. and Siaulys, J. (2010) Finite-time ruin probability in the inhomogeneous claim case. Lithuanian Mathematical Journal, 50 (3), 260–270.Google Scholar
Cai, J. (2002) Discrete time risk models under rates of interest. Probability in the Engineering and Informational Sciences, 16, 309–324.CrossRefGoogle Scholar
Cai, J. and Li, H. (2005) Multivariate risk model of phase type. Insurance: Mathematics and Economics, 36 (2), 137–152.Google Scholar
Castaner, A., Claramunt, M.M., Gathy, M., Lefevre, C. and Marmol, M. (2013) Ruin problems for a discrete time risk model with non-homogeneous conditions. Scandinavian Actuarial Journal 2013 (2), 83–102.Google Scholar
De Kok, T. G. (2003) Ruin probabilities with compounding assets for discrete time finite horizon problems, independent period claim sized and general premium structure. Insurance: Mathematics and Economics, 33, 645–658.Google Scholar
Dickson, D.C.M. (2005) Insurance Risk and Ruin. United Kingdom: Cambridge University Press.Google Scholar
Dickson, D.C.M. and Hipp, C. (2000) Ruin problems for phase-type (2) risk processes. Scandinavian Actuarial Journal 2000 (2), 147–167.Google Scholar
Drekic, S., Dickson, D.C.M., Stanford, D.A. and Willmot, G.E. (2004) On the distribution of the deficit at ruin when claims are phase-type. Scandinavian Actuarial Journal 2004 (2), 105–120.Google Scholar
Dufresne, D. (2007) Fitting combinations of exponentials to probability distributions. Applied Stochastic Models in Business and Industry, 23 (1), 23–48.Google Scholar
Lefevre, C. and Picard, P. (2006) A nonhomogeneous risk model for insurance. Computers and Mathematics with Applications, 51, 325–334.Google 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
Neuts, M.F. (1981) Matrix-Geometric Solutions in Stochastic Models. Baltimore: John Hopkins University Press.Google Scholar
Paulsen, J. (2008) Ruin models with investment income. Probability Surveys, 5, 416–434.CrossRefGoogle Scholar
Raducan, A.M., Lakatos, L. and Zbaganu, G. (2008) Computable Lindley processes in queueing and risk theories. Revue Roumaine de Mathé matiques Pures et Appliquées, 53 (2–3), 239–266.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999) Stochastic Processes for Insurance and Finance. England: Wiley.Google Scholar
Ross, S.M. (2011) Introduction to Probability Models. San Diego: Academic Press.Google Scholar
Stanford, D.A., Avram, F., Badescu, A.L., Breuer, L., Da Silva Soares, A. and Latouche, G. (2005) Phase-type approximations to finite-time ruin probabilities in the Sparre-Andersen and stationary renewal risk models. Astin Bulletin, 35 (1), 131–144.Google Scholar
Stanford, D.A. and Stroinski, K.J. (1994) Recursive methods for computing finite-time ruin probabilities for phase-distributed claim sizes. ASTIN Bulletin, 24 (2), 235–254.Google Scholar
Stanford, D.A., Stroinski, K.J. and Lee, K. (2000) Ruin probabilities based at claim instants for some non-Poisson claim processes. Insurance: Mathematics and Economics, 26, 251–267.Google Scholar
Tijms, H.C. (1994) Stochastic Models: An Algorithmic Approach. Chichester: Wiley.Google Scholar
Willmot, G.E. and Lin, X.S. (2011) Risk modelling with the mixed Erlang distribution. Applied Stochastic Models in Business and Industry, 27 (1), 2–16.CrossRefGoogle Scholar
Willmot, G.E. and Woo, J.K. (2007) On the class of Erlang mixtures with risk theoretic applications. North American Actuarial Journal, 11 (2), 99–115.Google Scholar
Zbaganu, G. (2007) Elements of Ruin Theory (in Romanian). Bucharest: Geometry Balkan Press.Google Scholar