We consider an optimization problem of an insurance company in the diffusion setting, which controls the dividends payout as well as the capital injections. To maximize the cumulative expected discounted dividends minus the penalized discounted capital injections until the ruin time, there is a possibility of (cheap or non-cheap) proportional reinsurance. We solve the control problems by constructing two categories of suboptimal models, one without capital injections and one with no bankruptcy by capital injection. Then we derive the explicit solutions for the value function and totally characterize the optimal strategies. Particularly, for cheap reinsurance, they are the same as those in the model of no bankruptcy.
Key words
proportional reinsurance /
non-cheap /
optimal dividend /
capital injections
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References
[1] Asmussen, S., Taksar, M. Controlled diffusion models for optimal dividend pay-out. Insurance: Math. Econom., 20: 1-15 (1997)
[2] Asmussen, S., H?jgaard, B., Taksar, M. Optimal risk control and dividend distribution policies: Example of excess-of loss reinsurance for an insurance corporation. Finance Stoch., 4: 299-324 (2000)
[3] De Finetti B. Su un’impostazione alternativa dell teoria del rischio. Transactions of the 15th International Congress of Acturies, 2: 433-443 (1957)
[4] Fleming, W.H., Soner, M. Controlled Markov Processes and Viscosity Solutions. Springer-Verlag, 1993
[5] H?jgaard, B., Taksar, M. Controlling risk exposure and dividends pay-out schemes: Insurance company example. Math. Finance, 9: 153-182 (1999)
[6] H?jgaard, B., Taksar, M. Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution strategy. Quant. Finance, 4: 315-327 (2004)
[7] Kulenko, N., Schmidli, H. Optimal dividend strategies in a Cramér-Lunberg model with capital injections. Insurance: Math. Econom., 43: 270-278 (2008)
[8] L?kka, A., Zervos, M. Optimal dividend and issuance of equity policies in the presence of proportional costs. Insurance: Math. Econom., 42: 954-961 (2008)
[9] Paulsen, J., Gjessing, H.K. Optimal choice of dividend barriers for a risk process with stochastic return on investments. Insurance: Math. Econom., 20: 215-223 (1997)
[10] Sethi, S.P., Taksar, M. Optimal financing of a corporation subject to random returns. Math. Finance, 12: 155-172 (2002)
[11] Taksar, M., Markussen, C. Optimal dynamic reinsurance policies for large insurance portfolios. Finance Stoch., 7: 97-121 (2003)
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Footnotes
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Funding
Supported by the National Basic Research Program of China (973 Program) (Grant No. 2007CB814905) and the National Natural Science Foundation of China (NSFC Grant No. 10871102).
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