Optimal dividend problem with a nonlinear regular-singular stochastic control

In this paper, a problem with a nonlinear regular-singular stochastic control is studied for a big insurance portfolio. We assume that the reinsurance premium is calculated according to the exponential premium principle which makes the stochastic control problem nonlinear. Both non-cheap and cheap reinsurance are investigated. The objective of the insurer is to determine the optimal reinsurance and dividend policy so as to maximize the expected discounted dividends until ruin. Bounded dividend rates and unbounded dividend rates are considered. In both cases, explicit expressions for the value function and the corresponding optimal strategies are obtained. Finally, a numerical example is presented, which shows the impacts of risk aversion of the reinsurance company on the optimal value function and the retention level for reinsurance.

[1]  Tak Kuen Siu,et al.  Optimal Mixed Impulse-Equity Insurance Control Problem With Reinsurance , 2011, SIAM J. Control. Optim..

[2]  Thaleia Zariphopoulou,et al.  Pricing Dynamic Insurance Risks Using the Principle of Equivalent Utility , 2002 .

[3]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[4]  V. Young.  Equity-Indexed Life Insurance: Pricing and Reserving Using the Principle of Equivalent Utility , 2003 .

[5]  Mihail Zervos,et al.  Optimal dividend and issuance of equity policies in the presence of proportional costs , 2008 .

[6]  X. S. Lin,et al.  Stochastic Processes for Insurance and Finance. By T. Rolski, H. Schmidli, V. Schmidt and J. Teugels (John Wiley, Chichester, 1999) , 2000, British Actuarial Journal.

[7]  Virginia R. Young,et al.  Optimal insurance under Wang’s premium principle , 1999 .

[8]  Xun Yu Zhou,et al.  A Diffusion Model for Optimal Dividend Distribution for a Company with Constraints on Risk Control , 2002, SIAM J. Control. Optim..

[9]  A. Shiryaev,et al.  Optimization of the flow of dividends , 1995 .

[10]  Michael I. Taksar,et al.  Optimal proportional reinsurance policies for diffusion models with transaction costs , 1998 .

[11]  Junyi Guo,et al.  Optimal excess-of-loss reinsurance and dividend payments with both transaction costs and taxes , 2010 .

[12]  P. Lions,et al.  Stochastic differential equations with reflecting boundary conditions , 1984 .

[13]  Tomasz Rolski,et al.  Stochastic Processes for Insurance and Finance , 2001 .

[14]  Søren Asmussen,et al.  Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation , 2000, Finance Stochastics.

[15]  Tak Kuen Siu,et al.  On optimal reinsurance, dividend and reinvestment strategies , 2011 .

[16]  Lin He,et al.  Optimal financing and dividend control of the insurance company with proportional reinsurance policy , 2008 .

[17]  M. Kaluszka.  Optimal reinsurance under convex principles of premium calculation , 2005 .

[18]  Ming Zhou,et al.  Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle , 2012 .

[19]  Pricing equity-linked pure endowments via the principle of equivalent utility , 2003 .

[20]  Junyi Guo,et al.  Optimal dividend and equity issuance problem with proportional and fixed transaction costs , 2012 .

[21]  Michael I. Taksar,et al.  Controlling Risk Exposure and Dividends Payout Schemes:Insurance Company Example , 1999 .

[22]  Sid Browne,et al.  Optimal Investment Policies for a Firm With a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin , 1995, Math. Oper. Res..

[23]  Søren Asmussen,et al.  Controlled diffusion models for optimal dividend pay-out , 1997 .

[24]  Lin He,et al.  Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs , 2009 .

[25]  Marek Musiela,et al.  An example of indifference prices under exponential preferences , 2004, Finance Stochastics.

[26]  Xun Yu Zhou,et al.  Optimal risk and dividend control for a company with a debt liability , 1998 .

[27]  M. Kaluszka.  Optimal reinsurance under mean-variance premium principles , 2001 .

[28]  H. Schmidli,et al.  On minimizing the ruin probability by investment and reinsurance , 2002 .

[29]  Bjarne Højgaard,et al.  Optimal proportional reinsurance policies for diffusion models , 1998 .

[30]  Xun Yu Zhou,et al.  Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction , 2001 .

[31]  A constrained non-linear regular-singular stochastic control problem, with applications , 2004 .