×

zbMATH — the first resource for mathematics

A direct approach to the discounted penalty function. (English) Zbl 1219.91063
Summary: This paper provides a new and accessible approach to establishing certain results concerning the discounted penalty function. The direct approach consists of two steps. In the first step, closedform expressions are obtained in the special case in which the claim amount distribution is a combination of exponential distributions. A rational function is useful in this context. For the second step, one observes that the family of combinations of exponential distributions is dense. Hence, it suffices to reformulate the results of the first step to obtain general results. The surplus process has downward and upward jumps, modeled by two independent compound Poisson processes. If the distribution of the upward jumps is exponential, a series of new results can be obtained with ease. Subsequently, certain results of H. U. Gerber and E. S.W. Shiu [N. Am. Actuar. J. 2, No. 1, 48–78 (1998; Zbl 1081.60550)] can be reproduced. The two-step approach is also applied when an independent Wiener process is added to the surplus process. Certain results are related to [Z. Zhang, H. Yang and S. Li, J. Comput. Appl. Math. 233, No. 8, 1773–1784 (2010; Zbl 1185.91198)], which uses different methods.

MSC:
91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
60K10 Applications of renewal theory (reliability, demand theory, etc.)
PDF BibTeX XML Cite
Full Text: DOI Link
References:
[1] Asmussen S., Journal of Computational Finance 11 pp 79– (2008) · doi:10.21314/JCF.2007.164
[2] Bao T.-H., Applied Mathematics and Computation 179 pp 559– (2006) · Zbl 1158.60374 · doi:10.1016/j.amc.2005.11.106
[3] BouCherie R. J., Probability in Engineering and Informational Systems 10 pp 261– (1996) · Zbl 1095.60510 · doi:10.1017/S0269964800004320
[4] BouCherie R. J., Probability in Engineering and Informational Systems 11 pp 305– (1997) · Zbl 1097.60509 · doi:10.1017/S0269964800004848
[5] Bowers N. L., Actuarial Mathematics, 2. ed. (1997)
[6] Burden R. L., Numerical Analysis, 4. ed. (1989)
[7] Cai J., Advances in Applied Probability 41 pp 495– (2009) · Zbl 1173.91023 · doi:10.1239/aap/1246886621
[8] Cramer H., Collective Risk Theory: A Survey from the Point of View of Stochastic Processes (1955)
[9] Dufresne D., Applied Stochastic Models in Business and Industry 23 (1) pp 23– (2007) · Zbl 1142.60321 · doi:10.1002/asmb.635
[10] Dufresne F., Probabilité et sévérité de la ruine: Modéle classique de la théorie du risque et une de ses extensions (1989)
[11] Dufresne F., Astin Bulletin 19 (1) pp 71– (1989) · doi:10.2143/AST.19.1.2014916
[12] Dufresne F., Insurance: Mathematics and Economics 10 pp 51– (1991) · Zbl 0723.62065 · doi:10.1016/0167-6687(91)90023-Q
[13] Feller W., An Introduction to Probability Theory and Its Applications (1971) · Zbl 0219.60003
[14] Gerber H. U., Insurance: Mathematics and Economics 22 (3) pp 263– (1998) · Zbl 0924.60075 · doi:10.1016/S0167-6687(98)00014-6
[15] Gerber H. U., Astin Bulletin 36 (2) pp 489– (2006) · Zbl 1162.91374 · doi:10.2143/AST.36.2.2017931
[16] Gerber H. U., Insurance: Mathematics and Economics 21 pp 129– (1997) · Zbl 0894.90047 · doi:10.1016/S0167-6687(97)00027-9
[17] Gerber H. U., North American Actuarial Journal 2 (1) pp 48– (1998) · Zbl 1081.60550 · doi:10.1080/10920277.1998.10595671
[18] Gerber H. U., Insurance: Mathematics and Economics 47 (2) pp 206– (2010) · Zbl 1231.91487 · doi:10.1016/j.insmatheco.2010.04.008
[19] JaCobsen M., Advances in Applied Probability 37 pp 963– (2005) · Zbl 1100.60021 · doi:10.1239/aap/1134587749
[20] Kou S., Advances in Applied Probability 35 pp 504– (2003) · Zbl 1037.60073 · doi:10.1239/aap/1051201658
[21] Labbé C., Applied Mathematics and Computation 215 pp 1852– (2009) · Zbl 1181.91100 · doi:10.1016/j.amc.2009.07.049
[22] Levendorskii S. Z., International Journal of Theoretical and Applied Finance 7 (3) pp 303– (2004) · Zbl 1107.91050 · doi:10.1142/S0219024904002463
[23] Lewis A. L., Journal of Applied Probability 45 (1) pp 118– (2008) · Zbl 1136.60330 · doi:10.1239/jap/1208358956
[24] Perry D., Stochastic Models 18 (1) pp 139– (2002) · Zbl 0998.60089 · doi:10.1081/STM-120002778
[25] Sarkar J., Insurance: Mathematics and Economics 36 (3) pp 421– (2005) · Zbl 1242.91097 · doi:10.1016/j.insmatheco.2005.02.007
[26] Schmidli H., Insurance: Mathematics and Economics 46 (1) pp 3– (2010) · Zbl 1231.91232 · doi:10.1016/j.insmatheco.2009.04.004
[27] Segerdahl C. O., On Homogeneous Random Processes and Collective Risk Theory (1939) · JFM 65.1371.01
[28] Temnov G., Journal of Mathematical Sciences 123 (1) pp 3780– (2004) · Zbl 1065.91040 · doi:10.1023/B:JOTH.0000036319.21285.22
[29] Xing X., Statistics and Probability Letters 78 pp 2692– (2008) · Zbl 1153.91024 · doi:10.1016/j.spl.2008.03.034
[30] Zhang Z., Statistics and Probability Letters 80 (7) pp 597– (2010) · Zbl 1202.91130 · doi:10.1016/j.spl.2009.12.016
[31] Zhang Z., Journal of Computational and Applied Mathematics 233 pp 1773– (2010) · Zbl 1185.91198 · doi:10.1016/j.cam.2009.09.014
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.