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Optimal loss-carry-forward taxation for the Lévy risk model

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  • Wang, Wenyuan
  • Hu, Yijun

Abstract

In the spirit of Albrecher and Hipp (2007), Albrecher et al. (2008b) and Kyprianou and Zhou (2009), we consider the reserve process of an insurance company which is governed by Rtπ=Xt−∫0tγπ(Sσ)dSσ, where X is a spectrally negative Lévy process with the usual exclusion of negative subordinator or deterministic drift, St:=max0≤σ≤tXσ the running supremum of X, and γπ(St) the rate of loss-carry-forward tax at time t which is subject to the taxation allocation policy π and is a function of St. The objective is to find the optimal policy which maximizes the total (discounted) taxation pay-out: Ex∫0τπe−ctγπ(St)dSt, where Ex and τπ refer to the expectation corresponding to the law of X such that X0=x, and the time of ruin, respectively. With the scale function of X denoted by Wc(x) and γπ(⋅) allowed to vary in [α,β](0≤α≤β<1), two situations are considered. (a)∫0∞(Wc(y))1−11−β(Wc)″(y)[(Wc)′(y)]2dy≥0. It is shown that the optimal strategy is to always pay tax at the maximum rate β.(b)∫0∞(Wc(y))1−11−β(Wc)″(y)[(Wc)′(y)]2dy<0. Then the optimal strategy prescribes to pay tax at the smallest rate α when the reserve is below some critical level u0, and to pay at the maximum rate β when the reserve is above u0.

Suggested Citation

  • Wang, Wenyuan & Hu, Yijun, 2012. "Optimal loss-carry-forward taxation for the Lévy risk model," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 121-130.
    • Handle: RePEc:eee:insuma:v:50:y:2012:i:1:p:121-130
    DOI: 10.1016/j.insmatheco.2011.10.011
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    References listed on IDEAS

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    1. Hao, Xuemiao & Tang, Qihe, 2009. "Asymptotic Ruin Probabilities of the Lévy Insurance Model under Periodic Taxation," ASTIN Bulletin, Cambridge University Press, vol. 39(2), pages 479-494, November.
    2. Wei, Li, 2009. "Ruin probability in the presence of interest earnings and tax payments," Insurance: Mathematics and Economics, Elsevier, vol. 45(1), pages 133-138, August.
    3. Renaud, Jean-François, 2009. "The distribution of tax payments in a Lévy insurance risk model with a surplus-dependent taxation structure," Insurance: Mathematics and Economics, Elsevier, vol. 45(2), pages 242-246, October.
    4. Wang, Wenyuan & Ming, Ruixing & Hu, Yijun, 2011. "On the expected discounted penalty function for risk process with tax," Statistics & Probability Letters, Elsevier, vol. 81(4), pages 489-501, April.
    5. Ming, Rui-Xing & Wang, Wen-Yuan & Xiao, Li-Qun, 2010. "On the time value of absolute ruin with tax," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 67-84, February.
    6. Albrecher, Hansjörg & Borst, Sem & Boxma, Onno & Resing, Jacques, 2009. "The tax identity in risk theory -- a simple proof and an extension," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 304-306, April.
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    Cited by:

    1. Wenyuan Wang & Yuebao Wang & Ping Chen & Xueyuan Wu, 2022. "Dividend and Capital Injection Optimization with Transaction Cost for Lévy Risk Processes," Journal of Optimization Theory and Applications, Springer, vol. 194(3), pages 924-965, September.
    2. Wenyuan Wang & Zhimin Zhang, 2019. "Optimal loss-carry-forward taxation for L\'{e}vy risk processes stopped at general draw-down time," Papers 1904.08029, arXiv.org.
    3. Bohan Li & Junyi Guo, 2021. "Optimal Investment and Reinsurance Under the Gamma Process," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 893-923, September.
    4. Dalal Al Ghanim & Ronnie Loeffen & Alex Watson, 2018. "The equivalence of two tax processes," Papers 1811.01664, arXiv.org, revised Oct 2019.
    5. Wenyuan Wang & Xueyuan Wu & Cheng Chi, 2019. "Optimal implementation delay of taxation with trade-off for L\'{e}vy risk Processes," Papers 1910.08158, arXiv.org.
    6. Al Ghanim, Dalal & Loeffen, Ronnie & Watson, Alexander R., 2020. "The equivalence of two tax processes," Insurance: Mathematics and Economics, Elsevier, vol. 90(C), pages 1-6.
    7. Ming, Ruixing & Wang, Wenyuan & Hu, Yijun, 2017. "On maximizing expected discounted taxation in a risk process with interest," Statistics & Probability Letters, Elsevier, vol. 122(C), pages 128-140.
    8. Wang, Wenyuan & Ming, Ruixing, 2018. "Two-side exit problems for taxed Lévy risk process involving the general draw-down time," Statistics & Probability Letters, Elsevier, vol. 138(C), pages 66-74.

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