Combined Optimal Stopping and Mixed Regular-Singular Control of Jump Diffusions
- 1 Department of Applied Mathematics; National University of Science and Technology, Bulawayo, Zimbabwe
- 2 Department of Mathematics and Computer Science, Great Zimbabwe University, Masvingo, Zimbabwe
- 3 Department of Mathematics and Computer Science, Great Zimbabwe University, Masvingo, Zimbabwe
- 4 Department of Mathematics and Computer Science, Great Zimbabwe University, Masvingo, Zimbabwe
- 5 Department of Mathematics and Computer Science, Great Zimbabwe University, Masvingo, Zimbabwe
- 6 Department of Mathematics and Computer Science, Great Zimbabwe University, Masvingo, Zimbabwe
Abstract
In this paper, we examine a model that maximises dividend payments for an insurance company with a debt liability. We assume that the company has a policy to reinvest a proportion of its surplus cash before paying dividends to shareholders. We model the dynamics of the cash reserves as a jump-diffusion process. Combined optimal stopping and mixed regular-singular control of the jump-diffusion process is presented and investigated. In the paper, we show that when the premium rate is less than the liability rate , then the company should not get into business and the optimal dividend policy is to immediately pay out the initial cash reserve as dividends to shareholders. For the case , we show that the optimal risk management depends on the current level of the cash reserves. We demonstrate that the company’s optimal dividend policy is to pay out as dividends surplus cash above a predetermined threshold. We also present numerical examples to illustrate the results obtained.
- Chikodza, E. (2008) Combined Singular and Impulse Control for Jump Diffusions. Ph.D. Thesis, University of Zimbabwe, Harare.
- Guo, X. (2003) A constrained non-linear regular singular control problem with application. Elsevier, Amsterdam.
- He, L., Liang, Z. and Wu, J.L. (2010) Optimal Dividend and Reisurance Strategy of a Property Insurance Company under Catastrophe Risk. http://arxiv.org/1009.1269
- Liu, B. (2004) Uncertain Theory. Springer-Verlag, Berlin.
- ksendal, B. (2010) Optimal Stopping and Stochastic Control of Differential Games for Jump Diffusions. Centre of Maths for Applications, Norway.
- Ksendal, B. (2009) Stochastic Differential Equations. Springer-Verlag, Berlin.
- Ksendal, B. and Sulem, A. (2009) Applied Stochastic Control of Jump-Diffusions. Springer, Berlin.
- Scheer, N. (2011) Optimal Stochastic Control of Dividends and Capital Injections. Ph.D. Thesis, Naturwissenschaftlichen Fakultat Der University, Koln.
- Taksar, I. and Zhou, X. (1998) Optimal Risk and Dividend Control for a Company with a Debt Liability. Insurance: Mathematics and Economics, 22, 105-122. https://doi.org/10.1016/S0167-6687(98)00012-2
- Zou, et al. (2009) Optimal Dividend Payments under Jump-Diffusion Risk Processes. Stochastic Models, 25, 332-347. https://doi.org/10.1080/15326340902870133
- Yang, R. et al. (2005) Optimal Impulse and Regular Control Strategies for Proportional Reinsurance. Journal of Applied Maths and Computing, 18, Article No. 145. https://doi.org/10.1007/BF02936561