Abstract
Recently, Lefèvre and Picard (Insur Math Econ 49:512–519, 2011) revisited a non-standard risk model defined on a fixed time interval [0,t]. The key assumption is that, if n claims occur during [0,t], their arrival times are distributed as the order statistics of n i.i.d. random variables with distribution function F t (s), 0 ≤ s ≤ t. The present paper is concerned with two particular cases of that model, namely when F t (s) is of linear form (as for a (mixed) Poisson process), or of exponential form (as for a linear birth process with immigration or a linear death-counting process). Our main purpose is to obtain, in these cases, an expression for the non-ruin probabilities over [0,t]. This is done by exploiting properties of an underlying family of Appell polynomials. The ultimate non-ruin probabilities are then derived as a limit.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Asmussen S, Albrecher H (2010) Ruin probabilities. World Scientific, Singapore
Bühlmann H (1970) Mathematical methods in risk theory. Springer, Heidelberg
De Vylder FE (1999) Numerical finite-time ruin probabilities by the Picard-Lefèvre formula. Scand Actuar J 2:97–105
De Vylder FE, Goovaerts MJ (1999a) Explicit finite-time and infinite-time ruin probabilities in the continuous case. Insur Math Econ 24:155–172
De Vylder FE, Goovaerts MJ (1999b) Inequality extensions of Prabhu’s formula in ruin theory. Insur Math Econ 24:249–271
De Vylder FE, Goovaerts MJ (2000) Homogeneous risk models with equalized claim amounts. Insur Math Econ 26:223–238
Gerber HU (1988) Mathematical fun with ruin theory. Insur Math Econ 7:15–23
Grandell J (1997) Mixed poisson processes. Chapman and Hall, London
Ignatov ZG, Kaishev VK (2004) A finite-time ruin probability formula for continuous claim severities. J Appl Probab 41:570–578
Kaas R, Goovaerts MJ, Dhaene J, Denuit M (2009) Modern actuarial risk theory: using R, 2nd edn. Springer, Heidelberg
Kaz’min YA (2002) Appell polynomials. In: Hazewinkel M (ed) Encyclopaedia of mathematics. Springer, New York
Lefèvre, C, Loisel S (2009) Finite-time ruin probabilities for discrete, possibly dependent, claim severities. Methodol Comput Appl Probab 11:425–441
Lefèvre C, Picard P (2011) A new look at the homogeneous risk model. Insur Math Econ 49:512–519
Panjer HH, Willmot GE (1992) Insurance risk models. Society of Actuaries, Schaumburg
Picard P, Lefèvre C (1996) First crossing of basic counting processes with lower non-linear bundaries: a unified approach through pseudopolynomials (I). Adv Appl Probab 28:853–876
Picard P, Lefèvre C (1997) The probability of ruin in finite time with discrete claim size distribution. Scand Actuar J 1:58–69
Picard P, Lefèvre C (2003) On the first meeting or crossing of two independent trajectories for some counting processes. Stoch Process their Appl 104:217–242
Puri PS (1982) On the characterization of point processes with the order statistic property without the moment condition. J Appl Probab 19:39–51
Resnick SI (1992) Adventures in stochastic processes. Birkhäuser, Boston
Sangüesa C (2006) Approximations of ruin probabilities in mixed Poisson models with lattice claim amounts. Insur Math Econ 39:69–80
Shiu ESW (1988) Calculation of the probability of eventual ruin by Beekman’s convolution series. Insur Math Econ 7:41–47
Takács L (1967) Combinatorial methods in the theory of stochastic processe. Wiley, New York
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lefèvre, C., Picard, P. Ruin Probabilities for Risk Models with Ordered Claim Arrivals. Methodol Comput Appl Probab 16, 885–905 (2014). https://doi.org/10.1007/s11009-013-9334-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-013-9334-y
Keywords
- Order statistics
- (Mixed) Poisson process
- Linear birth process with immigration
- Linear death process
- Ruin probability
- Finite or infinite horizon
- Appell polynomials