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Joint moments of the total discounted gains and losses in the renewal risk model with two-sided jumps. (English) Zbl 1427.91077

Summary: This paper considers a renewal insurance risk model with two-sided jumps (e.g. [C. Labbé et al., ibid. 218, No. 7, 3035–3056 (2011; Zbl 1239.91081)]), where downward and upward jumps typically represent claim amounts and random gains respectively. A generalization of the Gerber-Shiu expected discounted penalty function [H. U. Gerber and E. S. W. Shiu, N. Am. Actuar. J. 2, No. 1, 48–78 (1998; Zbl 1081.60550)] is proposed and analyzed for sample paths leading to ruin. In particular, we shall incorporate the joint moments of the total discounted costs associated with claims and gains until ruin into the Gerber-Shiu function. Because ruin may not occur, the joint moments of the total discounted claim costs and gain costs are also studied upon ultimate survival of the process. General recursive integral equations satisfied by these functions are derived, and our analysis relies on the concept of “moment-based discounted densities” introduced by the first author [Insur. Math. Econ. 53, No. 2, 343–354 (2013; Zbl 1304.91095)]. Some explicit solutions are obtained in two examples under different cost functions when the distribution of each claim is exponential or a combination of exponentials (while keeping the distributions of the gains and the inter-arrival times between successive jumps arbitrary). The first example looks at the joint moments of the total discounted amounts of claims and gains whereas the second focuses on the joint moments of the numbers of downward and upward jumps until ruin. Numerical examples including the calculations of covariances between the afore-mentioned quantities are given at the end along with some interpretations.

MSC:

91B05 Risk models (general)
60K10 Applications of renewal theory (reliability, demand theory, etc.)
91G05 Actuarial mathematics
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[1] Albrecher, H.; Gerber, H. U.; Yang, H., A direct approach to the discounted penalty function, North Am. Actuar. J., 14, 4, 420-434 (2010) · Zbl 1219.91063
[2] Asmussen, S.; Albrecher, H., Ruin Probabilities (2010), World Scientific: World Scientific New Jersey · Zbl 1247.91080
[3] Asmussen, S.; Avram, F.; Pistorius, M. R., Russian and american put options under exponential phase-type Lévy models, Stoch. Process. Appl., 109, 1, 79-111 (2004) · Zbl 1075.60037
[4] Biffis, E.; Morales, M., On a generalization of the Gerber-Shiu function to path-dependent penalties, Insur. Math. Econ., 46, 1, 92-97 (2010) · Zbl 1231.91146
[5] Breuer, L., First passage times for Markov additive processes with positive jumps of phase type, J. Appl. Probab., 45, 3, 779-799 (2008) · Zbl 1156.60059
[6] Cai, J.; Feng, R.; Willmot, G. E., On the expectation of total discounted operating costs up to default and its applications, Adv. Appl. Probab., 41, 2, 495-522 (2009) · Zbl 1173.91023
[7] Cai, N., On first passage times of a hyper-exponential jump diffusion process, Oper. Res. Lett., 37, 2, 127-134 (2009) · Zbl 1163.60039
[8] Cheung, E. C.K., On a class of stochastic models with two-sided jumps, Queueing Syst., 69, 1, 1-28 (2011) · Zbl 1235.60126
[9] Cheung, E. C.K., Moments of discounted aggregate claim costs until ruin in a Sparre Andersen risk model with general interclaim times, Insur. Math. Econ., 53, 2, 343-354 (2013) · Zbl 1304.91095
[10] Cheung, E. C.K.; Landriault, D., A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model, Insur. Math. Econ., 46, 1, 127-134 (2010) · Zbl 1231.91156
[11] Cheung, E. C.K.; Landriault, D.; Willmot, G. E.; Woo, J.-K., Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models, Insur. Math. Econ., 46, 1, 117-126 (2010) · Zbl 1231.91157
[12] Cheung, E. C.K.; Liu, H., On the joint analysis of the total discounted payments to policyholders and shareholders: threshold dividend strategy, Ann. Actuar. Sci., 10, 2, 236-269 (2016)
[13] Cheung, E. C.K.; Liu, H.; Woo, J.-K., On the joint analysis of the total discounted payments to policyholders and shareholders: dividend barrier strategy, Risks, 3, 4, 491-514 (2015)
[14] Cheung, E. C.K.; Woo, J.-K., On the discounted aggregate claim costs until ruin in dependent Sparre Andersen risk processes, Scand. Actuar. J., 2016, 1, 63-91 (2016) · Zbl 1401.91109
[15] Dickson, D. C.M.; Hipp, C., On the time to ruin for Erlang(2) risk processes, Insur. Math. Econ., 29, 3, 333-344 (2001) · Zbl 1074.91549
[16] Doney, R. A.; Kyprianou, A. E., Overshoots and undershoots of Lévy processes, Ann. Appl. Probab., 16, 1, 91-106 (2006) · Zbl 1101.60029
[17] Dufresne, D., Fitting combinations of exponentials to probability distributions, Appl. Stoch. Models Bus. Ind., 23, 1, 23-48 (2007) · Zbl 1142.60321
[18] Feng, R., On the total operating costs up to default in a renewal risk model, Insur. Math. Econ., 45, 2, 305-314 (2009) · Zbl 1231.91183
[19] Feng, R., A matrix operator approach to the analysis of ruin-related quantities in the phase-type renewal risk model, Bull. Swiss Assoc. Actuar., 2009, 1-2, 71-87 (2009) · Zbl 1333.91025
[20] Frostig, E.; Pitts, S. M.; Politis, K., The time to ruin and the number of claims until ruin for phase-type claims, Insur. Math. Econ., 51, 1, 19-25 (2012) · Zbl 1284.91232
[21] Gerber, H. U.; Shiu, E. S.W., On the time value of ruin, North Am. Actuar. J., 2, 1, 48-72 (1998) · Zbl 1081.60550
[22] Kou, S. G.; Wang, H., First passage times of a jump diffusion process, Adv. Appl. Probab., 35, 2, 504-531 (2003) · Zbl 1037.60073
[23] Kyprianou, A. E.; Zhou, X., General tax structures and the Lévy insurance risk model, J. Appl. Probab., 46, 4, 1146-1156 (2009) · Zbl 1210.60098
[24] Labbé, C.; Sendov, H. S.; Sendova, K. P., The Gerber-Shiu function and the generalized Cramér-Lundberg model, Appl. Math. Comput., 218, 7, 3035-3056 (2011) · Zbl 1239.91081
[25] Labbé, C.; Sendova, K. P., The expected discounted penalty function under a risk model with stochastic income, Appl. Math. Comput., 215, 5, 1852-1867 (2009) · Zbl 1181.91100
[26] Landriault, D.; Shi, T.; Willmot, G. E., Joint densities involving the time to ruin in the Sparre Andersen risk model under exponential assumptions, Insur. Math. Econ., 49, 3, 371-379 (2011) · Zbl 1229.91161
[27] Li, J.; Dickson, D. C.M.; Li, S., Analysis of some ruin-related quantities in a Markov-modulated risk model, Stoch. Models, 32, 3, 351-365 (2016) · Zbl 1344.60075
[28] Li, S.; Garrido, J., On ruin for the Erlang \((n)\) risk process, Insur. Math. Econ., 34, 3, 391-408 (2004) · Zbl 1188.91089
[29] Perry, D.; Stadje, W.; Zacks, S., First-exit times for compound poisson processes for some types of positive and negative jumps, Commun. Stat. Stoch. Models, 18, 1, 139-157 (2002) · Zbl 0998.60089
[30] Seal, H. L., Stochastic Theory of a Risk Business (1969), Wiley: Wiley New York · Zbl 0196.23501
[31] Willmot, G. E., On the discounted penalty function in the renewal risk model with general interclaim times, Insur. Math. Econ., 41, 1, 17-31 (2007) · Zbl 1119.91058
[32] Woo, J.-K., Some remarks on delayed renewal risk models, ASTIN Bull., 40, 1, 199-219 (2010) · Zbl 1230.91083
[33] Woo, J.-K., A generalized penalty function for a class of discrete renewal processes, Scand. Actuar. J., 2012, 2, 130-152 (2012) · Zbl 1277.60146
[34] Zhang, Z.; Yang, H., A generalized penalty function in the Sparre Andersen risk model with two-sided jumps, Stat. Probab. Lett., 80, 7-8, 597-607 (2010) · Zbl 1202.91130
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