Optimal Control Strategies Of Insurance Companies

In this paper we investigate two problems,minimization of absolute ruin proba-bility and maximization of the accumulated dividends for an insurance company.By studying the two problem,we exhibit the relation between risk and benefits of an in-surance company.In the problem of minimizing absolute ruin probability of an insurance company,we assume that the surplus of the company is modeled by a positively drifted Brownian motion,and the company can control its investment amount and risk exposure.To be more practical,we assume that there exists a negative correlation between liabilities and capital gains in financial market.By Bellman's dynamic programming principle,we get the associated HJB equation and verification theorem to show that a decreasing C~2(R)solution of the equation coincides with the minimal absolute ruin probability function.Under this negative correlation assumption,the two controls,investment amount and risk exposure are not independent.This leads an interesting result,the optimal strategies fail to be monotonic or continuous.From this phenomenon,we can find that the monotonicity between the mean growth rate and the volatility of the company's reserve fail in this model.In the optimal dividend problem,we introduce predictable injection processes and bankrupt time as two new controls in the optimization problem of dividend un-der Cram?er-Lundberg model.We assume that there are both fixed and proportional transaction costs when injections are made to be more realistic.By using Bellman's dynamic programming principle,we get the HJB equation which is not standard.We give the precise definition of viscosity solution for the HJB equation.And we prove that the value function is a viscosity solution of it.But the viscosity solution of the HJB equation is not necessarily the value function.We establish a sufficient condition to en-sure that a viscosity solution of the HJB equation is the value function.And also we prove that the value function satisfies the condition.Therefore,this condition is nec-essary and sufficient for the viscosity solution to be the value function.The condition constitutes a base of solving the optimal problem and reveals accurately the structure of the optimal strategy.Based on this condition,we present some examples to illus-trate peculiar properties of the value function and the optimal strategy:the surplus after injection could be over dividend bands;bankrupt may happen sooner when the initial surplus is higher;the optimal strategies may be very different when the corresponding value functions share the same form.