×

On the expectations of the present values of the time of ruin perturbed by diffusion. (English) Zbl 1066.91062

The main purpose of this paper is to study the surplus process of the classical continuous time risk model containing an independent diffusion (Wiener) process. In previous papers [C.C.-L. Tsai, G.E. Wilmot (2002)], the function of the expected discount penalty is defined as depending (linearly) on a penalty scheme involving the penalty at ruin by oscillation, the penalty at ruin caused by a claim, and the corresponding expectations of the present values of the time of ruin due to oscillation and due to a claim. The expected discount penalty function is shown to depend on a compound geometric distribution tail whose properties are studied together with those of the expectations of the present values of the time of ruin due to oscillation and due to a claim (Section 2 of the paper). Recursive formulas and explicit expressions for the moments of the compound geometric distribution (tail) and for the considered expectations of the time of ruin are also derived (provided that appropriate mathematical conditions are fulfilled).
Section 3 exposes explicit analytical solutions for the same examined functions describing the present values of the time of ruin in certain specific cases: when the claim size distribution is a combination of exponentials (Example 1) or a mixture of Erlangs (Example 2).
Section 4 achieves the asymptotic formulas and the Tijms-type approximations of the compound distribution tail and of the time of ruin expectations.
The final Section 5 proposes a lower bound and an upper bound on the considered compound geometric distribution, provided that the associated claim size distribution belongs to some of the reliability-based classes.

MSC:

91B30 Risk theory, insurance (MSC2010)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
62E17 Approximations to statistical distributions (nonasymptotic)
62E20 Asymptotic distribution theory in statistics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dufresne, F.; Gerber, H. U., The probability and severity of ruin for combinations of exponential claim amount distributions and their translations, Insurance: Mathematics and Economics, 7, 75-80 (1988) · Zbl 0637.62101
[2] Dufresne, F.; Gerber, H. U., The surpluses immediately before and at ruin, and the amount of the claim causing ruin, Insurance: Mathematics and Economics, 7, 193-199 (1988) · Zbl 0674.62072
[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] Feller, W., 1971. An Introduction to Probability Theory and Its Applications, vol. 2, 2nd ed. Wiley, New York.; Feller, W., 1971. An Introduction to Probability Theory and Its Applications, vol. 2, 2nd ed. Wiley, New York. · Zbl 0219.60003
[5] Gerber, H.U., 1970. An extension of the renewal equation and its application in the collective theory of risk. Skandinavisk Aktuarietidskrift, 205-210.; 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
[6] 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
[7] Lin, X., Tail of compound distributions and excess time, Journal of Applied Probability, 33, 184-195 (1996) · Zbl 0848.60081
[8] 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
[9] Tijms, H., 1986. Stochastic Modeling and Analysis: A Computational Approach. Wiley, Chichester.; Tijms, H., 1986. Stochastic Modeling and Analysis: A Computational Approach. Wiley, Chichester.
[10] Tsai, C.C.L., 1999. On the surplus process of ruin theory when perturbed by a diffusion. Ph.D. Thesis. University of Waterloo, Ontario.; Tsai, C.C.L., 1999. On the surplus process of ruin theory when perturbed by a diffusion. Ph.D. Thesis. University of Waterloo, Ontario.
[11] Tsai, C. C.L., On the discounted distribution functions of the surplus process perturbed by diffusion, Insurance: Mathematics and Economics, 28, 401-419 (2001) · Zbl 1074.91562
[12] Tsai, C. C.L.; Willmot, G. E., A generalized defective renewal equation for the surplus process perturbed by diffusion, Insurance: Mathematics and Economics, 30, 51-66 (2002) · Zbl 1074.91563
[13] Tsai, C. C.L.; Willmot, G. E., On the moments of the surplus process perturbed by diffusion, Insurance: Mathematics and Economics, 31, 327-350 (2002) · Zbl 1063.91051
[14] 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
[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
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.