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Optimal singular dividend control with capital injection and affine penalty payment at ruin

Published online by Cambridge University Press:  12 August 2022

Ran Xu Open the ORCID record for Ran Xu [Opens in a new window]
Ran Xu*
Affiliation:
Department of Financial and Actuarial Mathematics, Xi'an Jiaotong-Liverpool University, Suzhou, China. E-mail: ran.xu@xjtlu.edu.cn
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Abstract

In this paper, we extend the optimal dividend and capital injection problem with affine penalty at ruin in (Xu, R. & Woo, J.K. (2020). Insurance: Mathematics and Economics 92: 1–16) to the case with singular dividend payments. The asymptotic relationships between our value function to the one with bounded dividend density are studied, which also help to verify that our value function is a viscosity solution to the associated Hamilton–Jacob–Bellman Quasi-Variational Inequality (HJBQVI). We also show that the value function is the smallest viscosity supersolution within certain functional class. A modified comparison principle is proved to guarantee the uniqueness of the value function as the viscosity solution within the same functional class. Finally, a band-type dividend and capital injection strategy is constructed based on four crucial sets; and the optimality of such band-type strategy is proved by using fixed point argument. Numerical examples of the optimal band-type strategies are provided at the end when the claim size follows exponential and gamma distribution, respectively.

Keywords

Capital injectionCramér–Lundberg modelHJBQVISingular dividendViscosity solution
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 37 , Special Issue 2: Probability and stochastic modeling in actuarial science and related fields , April 2023 , pp. 462 - 490
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

Albrecher, H. & Thonhauser, S. (2008). Optimal dividend strategies for a risk process under force of interest. Insurance: Mathematics and Economics 43(1): 134–149.Google Scholar
Azcue, P. & Muler, N. (2005). Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model. Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics 15(2): 261–308.CrossRefGoogle Scholar
Azcue, P. & Muler, N. (2012). Optimal dividend policies for compound Poisson processes: The case of bounded dividend rates. Insurance: Mathematics and Economics 51(1): 26–42.Google Scholar
Azcue, P. & Muler, N. (2014). Stochastic optimization in insurance: a dynamic programming approach. New York: Springer.CrossRefGoogle Scholar
Crandall, M.G & Lions, P.-L. (1983). Viscosity solutions of Hamilton-Jacobi equations. Transactions of the American Mathematical Society 277(1): 1–42.CrossRefGoogle Scholar
De Finetti, B. (1957). Su un'impostazione alternativa della teoria collettiva del rischio. In Transactions of the XVth International Congress of Actuaries, vol. 2. New York, pp. 433–443.Google Scholar
Dickson, D.C.M. & Waters, H.R. (2004). Some optimal dividends problems. ASTIN Bulletin: The Journal of the IAA 34(1): 49–74.CrossRefGoogle Scholar
Gerber, H.U. (1969). Entscheidungskriterien für den zusammengesetzten Poisson-Prozess. PhD thesis, ETH Zurich.Google Scholar
Gerber, H.U. (1972). Games of economic survival with discrete- and continuous-income processes. Operations Research 20(1): 37–45.CrossRefGoogle Scholar
Loeffen, R.L. (2009). An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density. Journal of Applied Probability 46(1): 85–98.CrossRefGoogle Scholar
Loeffen, R.L. & Renaud, J.-F. (2010). De Finetti's optimal dividends problem with an affine penalty function at ruin. Insurance: Mathematics and Economics 46(1): 98–108.Google Scholar
Nie, C., Dickson, D.C.M., & Li, S. (2015). The finite time ruin probability in a risk model with capital injections. Scandinavian Actuarial Journal 2015(4): 301–318.CrossRefGoogle Scholar
Scheer, N. & Schmidli, H. (2011). Optimal dividend strategies in a Cramer–Lundberg model with capital injections and administration costs. European Actuarial Journal 1(1): 57–92.CrossRefGoogle Scholar
Schmidli, H. (2008). Stochastic control in insurance. London: Springer Science & Business Media.Google Scholar
Seydel, R.C. (2009). Existence and uniqueness of viscosity solutions for QVI associated with impulse control of jump-diffusions. Stochastic Processes and their Applications 119(10): 3719–3748.CrossRefGoogle Scholar
Thonhauser, S. & Albrecher, H. (2007). Dividend maximization under consideration of the time value of ruin. Insurance: Mathematics and Economics 41(1): 163–184.Google Scholar
Vierkötter, M. & Schmidli, H. (2017). On optimal dividends with exponential and linear penalty payments. Insurance: Mathematics and Economics 72: 265–270.Google Scholar
Xu, R. & Woo, J.-K. (2020). Optimal dividend and capital injection strategy with a penalty payment at ruin: Restricted dividend payments. Insurance: Mathematics and Economics 92: 1–16.Google Scholar
Xu, R., Woo, J.-K., Han, X., & Yang, H. (2018). A plan of capital injections based on the claims frequency. Annals of Actuarial Science 12(2): 296–325.CrossRefGoogle Scholar
Zhang, Z., Cheung, E.C.K., & Yang, H. (2018). On the compound poisson risk model with periodic capital injections. ASTIN Bulletin: The Journal of the IAA 48(1): 435–477.CrossRefGoogle Scholar
Zhao, Y., Chen, P., & Yang, H. (2017). Optimal periodic dividend and capital injection problem for spectrally positive Lévy processes. Insurance: Mathematics and Economics 74: 135–146.Google Scholar
Zhu, J. & Yang, H. (2016). Optimal financing and dividend distribution in a general diffusion model with regime switching. Advances in Applied Probability 48(2): 406–422.CrossRefGoogle Scholar