×

zbMATH — the first resource for mathematics

A generalized defective renewal equation for the surplus process perturbed by diffusion. (English) Zbl 1074.91563
The authors consider the surplus process \(V(t)= u+ ct- S(t)+\sigma W(t)\), \(t\geq 0\), where \(\sigma> 0\), \(\{W(t): t\geq 0\}\) is a standard Wiener process, that is independent of the compound Poisson process \(\{S(t): t\geq 0\}\). The defective renewal equation for the expected discounted function of a penalty at the time of ruin is generalized and its asymptotic formula is proposed.

MSC:
91B30 Risk theory, insurance (MSC2010)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alzaid, A., Aging concepts for items of unknown age, Communications in statistics: stochastic models, 10, 649-659, (1994) · Zbl 0817.60095
[2] Bondesson, L., On preservation of classes of life distributions under reliability operations: some complementary results, Naval research logistics quarterly, 30, 443-447, (1983) · Zbl 0525.62021
[3] Dufresne, F.; Gerber, H.U., Risk theory for the compound Poisson process that is perturbed by diffusion, Insurance: mathematics and economics, 10, 51-59, (1991) · Zbl 0723.62065
[4] Everitt, B., Hand, D., 1981. Finite Mixture Distributions. Chapman & Hall, New York. · Zbl 0466.62018
[5] Fagiuoli, E.; Pellerey, F., New partial orderings and applications, Naval research logistics, 40, 829-842, (1993) · Zbl 0795.62050
[6] Fagiuoli, E.; Pellerey, F., Preservation of certain classes of life distributions under Poisson shock models, Journal of applied probability, 31, 458-465, (1994) · Zbl 0806.60075
[7] Feller, W., 1971. An Introduction to Probability Theory and its Applications, Vol. 2, 2nd Edition. Wiley, New York. · Zbl 0219.60003
[8] Gerber, H.U., 1970. An extension of the renewal equation and its application in the collective theory of risk. Skandinavisk Aktuarietidskrift, 205-210. · Zbl 0229.60062
[9] Gerber, H.U.; Landry, B., On the discounted penalty at ruin in a jump-diffusion and the perpetual put option, Insurance: mathematics and economics, 22, 263-276, (1998) · Zbl 0924.60075
[10] Gerber, H.U.; Shiu, E.S.W., On the time value of ruin, North American actuarial journal, 2, 1, 48-78, (1998) · Zbl 1081.60550
[11] Lin, X.; Willmot, G.E., Analysis of a defective renewal equation arising in ruin theory, Insurance: mathematics and economics, 25, 63-84, (1999) · Zbl 1028.91556
[12] Tijms, H., 1994. Stochastic Models: An Algorithmic Approach. Wiley, Chichester, UK. · Zbl 0838.60075
[13] Wang, G., A decomposition of the ruin probability for the risk process perturbed by diffusion, Insurance: mathematics and economics, 28, 49-59, (2001) · Zbl 0993.60087
[14] Wang, G.; Wu, R., Some distributions for classical risk process that is perturbed by diffusion, Insurance: mathematics and economics, 26, 15-24, (2000) · Zbl 0961.62095
[15] Willmot, G.E., Bounds for compound distributions based on Mean residual lifetime and equilibrium distributions, Insurance: mathematics and economics, 21, 25-42, (1997) · Zbl 0924.62110
[16] Willmot, G.E.; Cai, J., On classes of lifetime distributions with unknown age, Probability in the engineering and informational sciences, 14, 473-484, (2000) · Zbl 0980.62009
[17] Willmot, G.E.; Lin, X., Exact and approximate properties of the distribution of the surplus before and after ruin, Insurance: mathematics and economics, 23, 91-110, (1998) · Zbl 0914.90074
[18] Willmot, G.E., Lin, X., 2000. Lundberg Approximations for Compound Distributions with Insurance Applications. Springer, New York. · Zbl 0962.62099
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.