Abstract
In a variety of insurance risk models, ruin-related quantities in the class of expected discounted penalty function (EDPF) were known to satisfy defective renewal equations that lead to explicit solutions. Recent development in the ruin literature has shown that similar defective renewal equations exist for a more general class of quantities than that of EDPF. This paper further extends the analysis of this new class of functions in the context of a spectrally negative Lévy risk model. In particular, we present an operator-based approach as an alternative analytical tool in comparison with fluctuation theoretic methods used for similar quantities in the current literature. The paper also identifies a sufficient and necessary condition under which the classical results from defective renewal equation and those from fluctuation theory are interchangeable. As a by-product, we present a series representation of scale function as well as potential measure in terms of compound geometric distribution.
Similar content being viewed by others
References
Albrecher H, Constantinescu C, Pirsic C, Regensburger G, Rosenkranz M (2010) An algebraic operator approach to the analysis of Gerber-Shiu functions. Insurance Math Econom 46:42–51
Applebaum D (2004) Lévy processes and stochastic calculus. Cambridge University Press, Cambridge
Bening VE, Korolev VY (2002) Nonparametric estimation of the ruin probability for generalized risk processes. Theor Probab Appl 47(1):1–16
Biffis E, Morales M (2010) On a generalization of the Gerber-Shiu function to path-dependent penalties. Insurance Math Econom 46(1):92–97
Biffis E, Kyprianou AE (2010) A note on scale functions and the time value of ruin for Lévy insurance risk processes. Insurance Math Econom 46(1):85–91
Cai J, Feng R, Willmot GE (2009) On the expectation of total discounted operating costs up to default and its applications. Adv Appl Probab 41(2):495–522
Chauveau DE, van Rooij ACM, Ruymgaart FH (1994) Regularized inversion of noisy laplace transforms. Adv Appl Math 15:186–201
Cheung ECK, Feng R (2011) A unified analysis of claim costs up to default in a Markov arrival risk model. Preprint
Croux K, Veraverbeke N (1990) Nonparametric estimators for the probability of ruin. Insurance Math Econom 9(2):127–130
Dickson CMD, Hipp C (2001) On the time to ruin for Erlang(2) risk processes. Insurance Math Econom 29(3):333–344
Feng R (2009a) A matrix operator approach to the analysis of ruin-related quantities in the phase-type renewal risk model. Schweiz Aktuarver Mitt (1–2):71–87
Feng R (2009b) On the total operating costs up to default in a renewal risk model. Insurance Math Econom 34(2):305–314
Feng R (2011) An operator-based approach to the analysis of ruin-related quantities in jump diffusion risk models. Insurance Math Econom 48(2):304–313
Feng R, Shimizu Y (2012) A matrix operator approach to potential measure of a Markov additive process with applications in ruin theory (preprint)
Garrido J, Morales M (2006) On the expected discounted penalty function for Lévy risk processes. N Am Actuar J 10(4):196–217
Gerber HU, Landry B (1998) On the discounted penalty at ruin in a jump-diffusion and the perpetual put option. Insurance Math Econom 22(3):263–276
Gerber HU, Shiu ESW (1998) On the time value of ruin. N Am Actuar J 2(1):48–78
Gerber HU, Shiu ESW (2005) The time value of ruin in a Sparre Andersen model. N Am Actuar J 9(2):49–68
Gerber HU, Shiu ESW (2006) On optimal dividend strategies in the compound Poisson model. N Am Actuar J 10(2):76–93
Grandell J (1991) Aspects of risk theory. Springer, New York
Huzak M, Perman M, Šikić H, Vondraèk Z (2004) Ruin probabilities and decompositions for general perturbed risk processes. Ann Appl Probab 14(3):1378–1397
Kyprianou AE (2006) Introductory lecture notes on fluctuations of Lévy processes with applications. Springer, Berlin
Li S, Garrido J (2005) On a general class of renewal risk process: analysis of the Gerber-Shiu function. Adv Appl Probab 37(3):836–856
Mnatsakanov R, Ruymgaart LL, Ruymgaart FH (2008) Nonparametric estimation of ruin probabilities given a random sample of claims. Math Method of Statist 17(1):35–43
Morales M (2007) On the expected discounted penalty function for a perturbed risk process driven by a subordinator. Insurance Math Econom 40(2):293–301
Lin X, Willmot G (1999) Analysis of a defective renewal equation arising in ruin theory. Insurance Math Econom 25(1):63–84
Lin XS, Pavlova KP (2006) The compound Poisson risk model with a threshold dividend strategy. Insurance Math Econom 38(1):57–80
Øksendal B, Sulem A (2004) Applied stochastic control of jump diffusions. Springer, Berlin
Politis K (2003) Semiparametric estimation for non-ruin probabilities. Scand Actuar J (1):75–96
Shimizu Y (2011) Estimation of the expected discounted penalty function for Lévy insurance risks. Math Method of Statist 20(2):125–149
Shimizu Y (2012) Nonparametric estimation of the Gerber-Shiu function for the Wiener-Poisson risk model. Scand Actuar J (1):56–69
Shiu ESW (1988) Calculation of the probability of eventual ruin by Beekman’s convolution series. Insurance Math Econom 7(1):41–47
Tsai CC, Willmot GE (2002) A generalized defective renewal equation for the surplus process perturbed by diffusion. Insurance Math Econom 30(1):51–66
Willmot GE, Lin XS (2001) Lundberg approximations for compound distributions with insurance applications. Springer, New York
Yang H, Zhang L (2001) Spectrally negative Lévy processes with applications in risk theory. Adv Appl Probab 33, 281–291
Zhou X (2005) On a classical risk model with a constant dividend barrier. N Am Actuar J 9(4):95–108
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Feng, R., Shimizu, Y. On a Generalization from Ruin to Default in a Lévy Insurance Risk Model. Methodol Comput Appl Probab 15, 773–802 (2013). https://doi.org/10.1007/s11009-012-9282-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-012-9282-y
Keywords
- Expected discounted penalty function
- Costs up to default
- Defective renewal equation
- Compound geometric distribution
- Lévy risk model
- Scale function
- Potential measure
- Operator calculus