راه حل ویسکوزیته و کنترل تکانه از مدل انتشار با بیمه های اتکایی و هزینه های معاملات ثابت

We consider an optimal impulse control problem on reinsurance, dividend and reinvestment of an insurance company. To close reality, we add fixed and proportional transaction costs to this problem. The value of the company is associated with expected present value of net dividends pay out minus the net reinvestment capitals until ruin time. We focus on non-cheap proportional reinsurance. We prove that the value function is a unique solution to associated Hamilton–Jacobi–Bellman equation, and establish the regularity property of the viscosity solution under a weak assumption. We solve the non-uniformly elliptic equation associated with the impulse control problem. Finally, we derive the value function and the optimal strategy of the control problem.

Optimal dividend strategy, as one major public concern to assess the stability of companies that take on risks, has been a long standing problem and has also become an increasingly popular topic in actuarial research. Its origin can be traced as early as the work of Finetti (1957), where a discrete-time model for optimal dividend was introduced. Finetti stated that the optimal strategy is a barrier strategy, and determined the optimal level of the barrier. The work laid the foundation of study of dividend strategies. Since then the optimal strategies related to ruin problems have received renewed interests in the literature. Some early works include Borch, 1967 and Borch, 1969 and Gerber, 1972 and Gerber, 1979, and a survey paper Avanzi (2009) and references therein. Some recent works are He and Liang, 2008 and He and Liang, 2009, Liang and Huang (2011), Alvarez and Lempa (2008), Avanzi and Gerber (2008) and Albrecher and Gerber (2009), Paulsen and Gjessing (1997), Asmussen and Taksar (1997), Højgaard and Taksar (2004), Gerber and Shiu, 1998, Gerber and Shiu, 2003a, Gerber and Shiu, 2003b and Gerber and Shiu, 2004 and references therein.

In this paper, we consider the optimal control problem of the insurance company with fixed and proportional transaction costs. The management of the company controls the reinsurance rate, times and amounts of dividends payout and reinvestment to maximize the expected present value of the net dividends payout minus the net reinvestment capitals until the ruin time. To be more realistic, we consider the fixed and proportional transaction costs. The former are generated by the advisory and consulting as well as the latter are generated by the tax. It is the first time to study non-cheap proportional reinsurance policy in the impulse control problem. We prove that the value function of the impulse control problem is a unique solution to the associated HJB equation, and establish the regularity property of the viscosity solution under a weak assumption. We solve the non-uniformly elliptic equation associated with the impulse control problem. We identify the value function and optimal strategy with the corresponding parameters. The insurance company will not adopt reinsurance policy if the reinsurance premium is too expensive, not adopt reinvestment if the reinvestment transaction cost is too expensive and only have one time dividend payout if the dividend transaction cost is too expensive. So the optimal strategies are tightly connected with the relationships among the parameters.