Lévy insurance risk process with Poissonian taxation. (English) Zbl 1401.91216

Summary: The idea of taxation in risk process was first introduced by H. Albrecher and C. Hipp [Bl. DGVFM 28, No. 1, 13–28 (2007; Zbl 1119.62103)], who suggested that a certain proportion of the insurer’s income is paid immediately as tax whenever the surplus process is at its running maximum. In this paper, a spectrally negative Lévy insurance risk model under taxation is studied. Motivated by the concept of randomized observations proposed by H. Albrecher et al. [ASTIN Bull. 41, No. 2, 645–672 (2011; Zbl 1239.91072)], we assume that the insurer’s surplus level is only observed at a sequence of Poisson arrival times, at which the event of ruin is checked and tax may be collected from the tax authority. In particular, if the observed (pre-tax) level exceeds the maximum of the previously observed (post-tax) values, then a fraction of the excess will be paid as tax. Analytic expressions for the Gerber-Shiu expected discounted penalty function and the expected discounted tax payments until ruin are derived. The Cramér-Lundberg asymptotic formula is shown to hold true for the Gerber-Shiu function, and it differs from the case without tax by a multiplicative constant. Delayed start of tax payments will be discussed as well. We also take a look at the case where solvency is monitored continuously (while tax is still paid at Poissonian time points), as many of the above results can be derived in a similar manner. Some numerical examples will be given at the end.


91B30 Risk theory, insurance (MSC2010)
91B64 Macroeconomic theory (monetary models, models of taxation)
60G51 Processes with independent increments; Lévy processes
62P05 Applications of statistics to actuarial sciences and financial mathematics
60J75 Jump processes (MSC2010)
60K10 Applications of renewal theory (reliability, demand theory, etc.)
Full Text: DOI Link


[1] Albrecher, H., Avram, F., Constantinescu, C. & Ivanovs, J. (2014). The tax identity for Markov additive risk processes. Methodology and Computing in Applied Probability16(1), 245-258. · Zbl 1286.91062
[2] Albrecher, H., Badescu, A. L. & Landriault, D. (2008). On the dual risk model with tax payments. Insurance: Mathematics and Economics42(3), 1086-1094. · Zbl 1141.91481
[3] Albrecher, H., Borst, S., Boxma, O. & Resing, J. (2009). The tax identity in risk theory – a simple proof and an extension. Insurance: Mathematics and Economics44(2), 304-306. · Zbl 1163.91430
[4] Albrecher, H., Borst, S. C., Boxma, O. J. & Resing, J. (2011). Ruin excursions, the G/G/∞ queue, and tax payments in renewal risk models. Journal of Applied Probability48(A), 3-14. · Zbl 1223.91024
[5] Albrecher, H., Cheung, E. C. K. & Thonhauser, S. (2011). Randomized observation periods for the compound Poisson risk model: Dividends. ASTIN Bulletin41(2), 645-672. · Zbl 1239.91072
[6] Albrecher, H., Cheung, E. C. K. & Thonhauser, S. (2013). Randomized observation periods for the compound Poisson risk model: the discounted penalty function. Scandinavian Actuarial Journal2013(6), 424-452. · Zbl 1401.91089
[7] Albrecher, H., Gerber, H. U. & Shiu, E. S. W. (2011). The optimal dividend barrier in the Gamma–Omega model. European Actuarial Journal1(1), 43-55. · Zbl 1219.91062
[8] Albrecher, H. & Hipp, C. (2007). Lundberg’s risk process with tax. Blätter der DGVFM28(1), 13-28. · Zbl 1119.62103
[9] Albrecher, H. & Ivanovs, J. (2013). A risk model with an observer in a Markov environment. Risks1(3), 148-161.
[10] Albrecher, H. & Ivanovs, J. (2014). Power identities for Lévy risk models under taxation and capital injections. Stochastic Systems4(1), 157-172. · Zbl 1300.60067
[11] Albrecher, H., Ivanovs, J. & Zhou, X. (2015). Exit identities for L\’{e}vy processes observed at Poisson arrival times. Bernoulli. To appear. · Zbl 1338.60125
[12] Albrecher, H. & Lautscham, V. (2013). From ruin to bankruptcy for compound Poisson surplus processes. ASTIN Bulletin43(2), 213-243. · Zbl 1283.91084
[13] Albrecher, H., Renaud, J.-F. & Zhou, X. (2008). A Lévy insurance risk process with tax. Journal of Applied Probability45(2), 363-375. · Zbl 1144.60032
[14] Asmussen, S. & Albrecher, H. (2010). Ruin Probabilities, 2nd ed. New Jersey: World Scientific. · Zbl 1247.91080
[15] Avanzi, B., Cheung, E. C. K., Wong, B. & Woo, J.-K. (2013). On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency. Insurance: Mathematics and Economics52(1), 98-113. · Zbl 1291.91088
[16] Avanzi, B., Tu, V. & Wong, B. (2014). On optimal periodic dividend strategies in the dual model with diffusion. Insurance: Mathematics and Economics55, 210-224. · Zbl 1296.91143
[17] Biffis, E. & Kyprianou, A. E. (2010). A note on scale functions and the time value of ruin for Lévy insurance risk processes. Insurance: Mathematics and Economics46(1), 85-91. · Zbl 1231.91145
[18] Biffis, E. & Morales, M. (2010). On a generalization of the Gerber–Shiu function to path-dependent penalties. Insurance: Mathematics and Economics46(1), 92–97. · Zbl 1231.91146
[19] Cheung, E. C. K. & Landriault, D. (2012). On a risk model with surplus-dependent premium and tax rates. Methodology and Computing in Applied Probability14(2), 233-251. · Zbl 1260.91120
[20] Czarna, I. & Palmowski, Z. (2011). Ruin probability with Parisian delay for a spectrally negative Lévy risk process. Journal of Applied Probability48(4), 984-1002. · Zbl 1232.60036
[21] Czarna, I. & Palmowski, Z. (2014). Dividend problem with Parisian delay for a spectrally negative Lévy Risk process. Journal of Optimization Theory and Applications161(1), 239-256. · Zbl 1296.91150
[22] Dickson, D. C. M. & Hipp, C. (2001). On the time to ruin for Erlang(2) risk processes. Insurance: Mathematics and Economics29(3), 333-344. · Zbl 1074.91549
[23] Dickson, D. C. M. & Waters, H. R. (2004). Some optimal dividends problems. ASTIN Bulletin34(1), 49-74. · Zbl 1097.91040
[24] Dufresne, F., Gerber, H. U. & Shiu, E. S. W. (1991). Risk theory with the Gamma process. ASTIN Bulletin21(2), 177-192.
[25] Feng, R. & Shimizu, Y. (2013). On a generalization from ruin to default in a Lévy insurance risk model. Methodology and Computing in Applied Probability15(4), 773-802. · Zbl 1307.91096
[26] Furrer, H. (1998). Risk processes perturbed by α-stable Lévy motion. Scandinavian Actuarial Journal1998(1), 59-74. · Zbl 1026.60516
[27] Garrido, J. & Morales, M. (2006). On the expected discounted penalty function for Lévy risk processes. North American Actuarial Journal10(4), 196-216.
[28] Gerber, H. U. & Landry, B. (1998). On the discounted penalty at ruin in a jump-diffusion and the perpetual put option. Insurance: Mathematics and Economics22(3), 263-276. · Zbl 0924.60075
[29] Gerber, H. U., Lin, X. S. & Yang, H. (2006). A note on the dividends-penalty identity and the optimal dividend barrier. ASTIN Bulletin36(2), 489-503. · Zbl 1162.91374
[30] Gerber, H. U. & Shiu, E. S. W. (1998). On the time value of ruin. North American Actuarial Journal2(1), 48-72. · Zbl 1081.60550
[31] Gerber, H. U., Shiu, E. S. W. & Yang, H. (2012a). The Omega model: from bankruptcy to occupation times in the red. European Actuarial Journal2(2), 259-272. · Zbl 1256.91057
[32] Gerber, H. U., Shiu, E. S. W. & Yang, H. (2012b). Valuing equity-linked death benefits and other contingent options: a discounted density approach. Insurance: Mathematics and Economics51(1), 73-92. · Zbl 1284.91233
[33] Hao, X. & Tang, Q. (2009). Asymptotic ruin probabilities of the Lévy insurance model under periodic taxation. ASTIN Bulletin39(2), 479-494. · Zbl 1179.91104
[34] Klugman, S. A., Panjer, H. H. & Willmot, G. E. (2013). Loss Models: Further Topics. New Jersey: Wiley. · Zbl 1273.62008
[35] Kyprianou, A. E. (2013). Gerber–Shiu Risk Theory. Cham: Springer. · Zbl 1277.91003
[36] Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications: Introductory Lectures, 2nd ed. Springer-Verlag: Berlin Heidelberg. · Zbl 1384.60003
[37] Kyprianou, A. E. & Loeffen, R. L. (2010). Refracted Lévy processes. Annales de l’Institut Henri Poincaré – Probabilités et Statistiques46(1), 24-44.
[38] Kyprianou, A. E. & Palmowski, Z. (2007). Distributional study of De Finetti’s dividend problem for a general Lévy insurance risk process. Journal of Applied Probability44(2), 428-443. · Zbl 1137.60047
[39] Kyprianou, A. E. & Zhou, X. (2009). General tax structures and the Lévy insurance risk model. Journal of Applied Probability46(4), 1146-1156. · Zbl 1210.60098
[40] Lakshmikantham, V. & Rao, R. M. M. (1995). Theory of Integro-differential Equations. Lausanne: Gordon and Breach Science Publishers. · Zbl 0849.45004
[41] Landriault, D., Renaud, J.-F. & Zhou, X. (2014). An insurance risk model with Parisian implementation delays. Methodology and Computing in Applied Probability16(3), 583-607. · Zbl 1319.60098
[42] Li, S. & Garrido, J. (2004). On ruin for the Erlang(n) risk process. Insurance: Mathematics and Economics34(3), 391-408. · Zbl 1188.91089
[43] Li, B., Tang, Q. & Zhou, X. (2013). A time-homogeneous diffusion model with tax. Journal of Applied Probability50(1), 195-207. · Zbl 1271.62246
[44] Loeffen, R. L. (2008). On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes. Annals of Applied Probability18(5), 1669-1680. · Zbl 1152.60344
[45] Loeffen, R., Czarna, I. & Palmowski, Z. (2013). Parisian ruin probability for spectrally negative Lévy processes. Bernoulli19(2), 599-609. · Zbl 1267.60054
[46] Loeffen, R. L. & Renaud, J.-F. (2010). De Finetti’s optimal dividends problem with an affine penalty function at ruin. Insurance: Mathematics and Economics46(1), 98-108. · Zbl 1231.91212
[47] Ming, R.-X., Wang, W.-Y. & Xiao, L.-Q. (2010). On the time value of absolute ruin with tax. Insurance: Mathematics and Economics46(1), 67-84. · Zbl 1231.91218
[48] Morales, M. (2007). On the expected discounted penalty function for a perturbed risk process driven by a subordinator. Insurance: Mathematics and Economics40(2), 293-301. · Zbl 1130.91032
[49] Polyanin, A. D. & Manzhirov, A. V. (2008). Handbook of Integral Equations, 2nd ed. Boca Raton: Chapman & Hall/CRC. · Zbl 1154.45001
[50] Renaud, J.-F. (2009). The distribution of tax payments in a Lévy insurance risk model with a surplus-dependent taxation structure. Insurance: Mathematics and Economics45(2), 242-246. · Zbl 1231.91230
[51] Renaud, J.-F. & Zhou, X. (2007). Distribution of the present value of dividend payments in a Lévy risk model. Journal of Applied Probability44(2), 420-427. · Zbl 1132.60041
[52] Tang, Q. & Wei, L. (2010). Asymptotic aspects of the Gerber–Shiu function in the renewal risk model using Wiener–Hopf factorization and convolution equivalence. Insurance: Mathematics and Economics46(1), 19-31. · Zbl 1231.91243
[53] Tsai, C. C.-L. & Willmot, G. E. (2002). A generalized defective renewal equation for the surplus process perturbed by diffusion. Insurance: Mathematics and Economics30(1), 51-66. · Zbl 1074.91563
[54] Wang, S., Zhang, C. & Wang, G. (2010). A constant interest risk model with tax payments. Stochastic Models26(3), 384-398. · Zbl 1231.91246
[55] Wei, L. (2009). Ruin probability in the presence of interest earnings and tax payments. Insurance: Mathematics and Economics45(1), 133-138. · Zbl 1231.91249
[56] Wei, J., Yang, H. & Wang, R. (2010). On the Markov-modulated insurance risk model with tax. Blätter der DGVFM31(1), 65-78. · Zbl 1195.91071
[57] Willmot, G. E. & Lin, X. S. (2001). Lundberg Approximations for Compound Distributions with Applications. New York, NY: Springer-Verlag. · Zbl 0962.62099
[58] Zhang, Z. (2014). On a risk model with randomized dividend-decision times. Journal of Industrial and Management Optimization10(4), 1041-1058. · Zbl 1282.91164
[59] Zhang, Z. & Cheung, E. C. K. (2014a). The Markov additive risk process under an Erlangized dividend barrier strategy. Methodology and Computing in Applied Probability. In press. · Zbl 1338.91081
[60] Zhang, Z. & Cheung, E. C. K. (2014b). A note on a Lévy insurance risk model under periodic dividend decisions. Preprint.
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.