Hostname: page-component-89b8bd64d-ktprf Total loading time: 0 Render date: 2026-05-08T02:25:20.398Z Has data issue: false hasContentIssue false
  • English
  • Français

RISK MODELS IN INSURANCE AND EPIDEMICS: A BRIDGE THROUGH RANDOMIZED POLYNOMIALS

Published online by Cambridge University Press:  23 March 2015

Claude Lefèvre  and
Philippe Picard
Claude Lefèvre
Affiliation:
Université Libre de Bruxelles, Département de Mathématique, Campus de la Plaine C.P. 210, B-1050 Bruxelles, Belgium E-mail: clefevre@ulb.ac.be
Philippe Picard
Affiliation:
Université de Lyon, Institut de Science Financière et d'Assurances, 50 Avenue Tony Garnier, F-69007 Lyon, France E-mail: philippe.picard69@free.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

The purpose of this work is to construct a bridge between two classical topics in applied probability: the finite-time ruin probability in insurance and the final outcome distribution in epidemics. The two risk problems are reformulated in terms of the joint right-tail and left-tail distributions of order statistics for a sample of uniforms. This allows us to show that the hidden algebraic structures are of polynomial type, namely Appell in insurance and Abel–Gontcharoff in epidemics. These polynomials are defined with random parameters, which makes their mathematical study interesting in itself.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 29 , Issue 3 , July 2015 , pp. 399 - 420
Copyright
Copyright © Cambridge University Press 2015 

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.)

Article purchase

Temporarily unavailable

References

1. Andersson, H. & Britton, T. (2000). Stochastic epidemic models and their statistical analysis. New York: Springer, (LNS 151).Google Scholar
2. Asmussen, S. & Albrecher, H. (2010). Ruin probabilities. Singapore: World Scientific.Google Scholar
3. Bailey, N.T.J. (1975). The mathematical theory of infectious diseases and its applications. London: Griffin.Google Scholar
4. Ball, F.G. (1986). A unified approach to the distribution of total size and total area under the trajectory of infectives in epidemic models. Advances in Applied Probability 18: 289–310.Google Scholar
5. Ball, F.G. & O'Neill, P. (1999). The distribution of general final state random variables for stochastic epidemic models. Journal of Applied Probability 36: 473–491.Google Scholar
6. Ball, F.G., Sirl, D.J. & Trapman, P. (2014). Epidemics on random intersection graphs. Annals of Applied Probability 24: 1081–1128.Google Scholar
7. Caramellino, L. & Spizzichino, F. (1994). Dependence and aging properties of lifetimes with Schur-constant survival function. Probability in the Engineering and Informational Sciences 8: 103–111.Google Scholar
8. Chi, Y., Yang, J. & Qi, Y. (2009). Decomposition of a Schur-constant model and its applications. Insurance: Mathematics and Economics 44: 398–408.Google Scholar
9. Clancy, D. (1999). Outcomes of epidemic models with general infection and removal rate functions at certain stopping times. Journal of Applied Probability 36: 799–813.Google Scholar
10. Clancy, D. (2014). SIR epidemic models with general infectious period distribution. Statistics and Probability Letters 85: 1–5.Google Scholar
11. Consul, P.C. (1974). A simple urn model dependent upon predetermined strategy. Sankhyā B 36: 391–399.Google Scholar
12. Daniels, H.E. (1963). The Poisson process with a curved absorbing boundary. Bulletin of the International Statistical Institute 40: 994–1008.Google Scholar
13. Daniels, H.E. (2000). The first crossing-time density for Brownian motion with a perturbed linear boundary. Bernoulli 6: 571–580.Google Scholar
14. Daley, D. & Gani, J. (1999). Epidemic modelling: an introduction. Cambridge: Cambridge University Press.Google Scholar
15. Das, S. & Kratz, M. (2012). Alarm systems for insurance companies: A strategy for capital allocation. Insurance: Mathematics and Economics 51: 53–65.Google Scholar
16. Denuit, M., Lefèvre, C. & Picard, P. (2003). Polynomial structures in order statistics distributions. Journal of Statistical Planning and Inference 113: 151–178.Google Scholar
17. Di Bucchianico, A. (1997). Probabilistic and Analytical Aspects of the Umbral Calculus. CWI Tract 119. Amsterdam: CWI.Google Scholar
18. Dobson, I., Carreras, B.A. & Newman, D.E. (2005). A loading-dependent model of probabilistic cascading failure. Probability in the Engineering and Informational Sciences 19: 15–32.Google Scholar
19. Gani, J. & Jerwood, D. (1972). The cost of a general stochastic epidemic. Journal of Applied Probability 9: 257–269.Google Scholar
20. Giraitis, L. & Surgailis, D. (1986). Multivariate Appell polynomials and the central limit theorem. In Dependence in Probability and Statistics, Eberlein, E. & Taqqu, M.S. (eds.), New York: Birkhäuser, pp. 21–71.Google Scholar
21. Gontcharoff, W. (1937). Détermination des fonctions entières par interpolation. Paris: Hermann.Google Scholar
22. Ignatov, Z.G. & Kaishev, V.K. (2004). A finite-time ruin probability formula for continuous claim severities. Journal of Applied Probability 41: 570–578.Google Scholar
23. Islam, M., O'Shaughnessy, C. & Smith, B. (1996). A random graph model for the final-size distribution of household infections. Statistics in Medicine 15: 837–843.Google Scholar
24. Kaas, R., Goovaerts, M.J., Dhaene, J. & Denuit, M. (2003). Modern actuarial risk theory. Boston: Kluwer.Google Scholar
25. Kaz'min, Y.A. (2002). Appell polynomials. In Encyclopaedia of mathematics (Hazewinkel, M., Ed.), New York: Springer.Google Scholar
26. Khintchine, A.Y. (1938). On unimodal distributions. Izv. Nauchno- Issled. Inst. Mat. Mech. Tomsk. Gos. Univ. 2: 1–7 (in Russian).Google Scholar
27. Lefèvre, C. (2006). On the outcome of a cascading failure model. Probability in the Engineering and Informational Sciences 20: 413–427.Google Scholar
28. Lefèvre, C. & Picard, P. (1990). A non-standard family of polynomials and the final size distribution of Reed–Frost epidemic processes. Advances in Applied Probability 22: 25–48.Google Scholar
29. Lefèvre, C. & Picard, P. (2005). Nonstationarity and randomization in the Reed–Frost epidemic model. Journal of Applied Probability 42: 1–14.Google Scholar
30. Lefèvre, C. & Picard, P. (2011). A new look at the homogeneous risk model. Insurance: Mathematics and Economics 49: 512–519.Google Scholar
31. Lefèvre, C. & Picard, P. (2014). Ruin probabilities for risk models with ordered claim arrivals. Methodology and Computing in Applied Probability 16: 885–905.Google Scholar
32. Lefèvre, C. & Utev, S. (1996). Comparing sums of exchangeable Bernoulli random variables. Journal of Applied Probability 33: 285–310.Google Scholar
33. Martin-Löf, A. (1986). Symmetric sampling procedures, general epidemic processes and their threshold limit theorems. Journal of Applied Probability 23: 265–282.Google Scholar
34. Nelsen, R.B. (2005). Some properties of Schur-constant survival models and their copulas. Brazilian Journal of Probability and Statistics 19: 179–190.Google Scholar
35. O'Neill, P.D. (1997). An epidemic model with removal-dependent infection rate. Annals of Applied Probability 7: 90–109.Google Scholar
36. Picard, P. (1980). Applications of martingale theory to some epidemic models. Journal of Applied Probability 17: 583–599.Google Scholar
37. Picard, P. & Lefèvre, C. (1990). A unified analysis of the final size and severity distribution in collective Reed–Frost epidemic processes. Advances in Applied Probability 22: 269–294.Google Scholar
38. Picard, P. & Lefèvre, C. (1996). First crossing of basic counting processes with lower non-linear boundaries: a unified approach through pseudopolynomials (I). Advances in Applied Probability 28: 853–876.Google Scholar
39. Picard, P. & Lefèvre, C. (1997). The probability of ruin in finite time with discrete claim size distribution. Scandinavian Actuarial Journal 1: 58–69.Google Scholar
40. Picard, P. & Lefèvre, C. (2003). On the first meeting or crossing of two independent trajectories for some counting processes. Stochastic Processes and their Applications 104: 217–242.Google Scholar
41. Puri, P.S. (1982). On the characterization of point processes with the order statistic property without the moment condition. Journal of Applied Probability 19: 39–51.Google Scholar
42. Rolski, T., Schmidli, H., Schmidt, V. & Teugels, J.L. (1999). Stochastic processes for insurance and finance. Chichester: Wiley.Google Scholar
43. Salminen, P. (2011). Optimal stopping, Appell polynomials, and Wiener–Hopf factorization. Stochastics: An International Journal of Probability and Stochastic Processes 83: 611–622.Google Scholar
44. Schoutens, W. & Teugels, J.L. (1998). Lévy processes, polynomials and martingales. Stochastic Models 14: 335–349.Google Scholar
45. Sendova, K.P. & Zitikis, R. (2012). The order-statistic claim process with dependent claim frequencies and severities. Journal of Statistical Theory and Practice 6: 597–620.Google Scholar
46. Ta, B.Q. (2014). Probabilistic approach to Appell polynomials. Expositiones Mathematicae, in press.Google Scholar
47. Takács, L. (1967). Combinatorial methods in the theory of stochastic processes. New York: Wiley.Google Scholar