Abstract
We discuss the optimal dividend and capital injection strategies in the Cramér-Lundberg risk model. The value functionV(x)is defined by maximizing the discounted value of the dividend payment minus the penalized discounted capital injection until the time of ruin. It is shown thatV(x)can be characterized by the Hamilton-Jacobi-Bellman equation. We find the optimal dividend barrierb, the optimal upper capital injection barrier 0, and the optimal lower capital injection barrier-z*. In the case of exponential claim size especially, we give an explicit procedure to obtainb,-z*, and the value functionV(x).
Highlights
In the modern theory of risk, people tend to study the cost of postponing or avoiding outright ruin; that is, ruin does not mean the end of the game but only the necessity of raising additional money
It is shown that V(x) can be characterized by the Hamilton-Jacobi-Bellman equation
Borch [1] pointed out that it was a good investment to rescue an insolvent insurance company, provided that its deficit was not too large. He studied this problem for a random walk model and suggested that the company should be rescued only if the deficit was smaller than the expected profits from the rescue operation
Summary
In the modern theory of risk, people tend to study the cost of postponing or avoiding outright ruin; that is, ruin does not mean the end of the game but only the necessity of raising additional money. Sethi and Taksar [2] considered the problem of finding an optimal financing mix of retained earnings and external equity for maximizing the value of a corporation. They showed that the optimal policy can be characterized in terms of two threshold parameters. For the Cramer-Lundberg risk model without bankruptcy (i.e., the shareholders will inject capital to cover the deficit whatever serious it is) the optimal dividend problem was studied. We will discuss the optimal dividend payment and capital injection strategies in the Cramer-Lundberg risk model. We give an explicit procedure to obtain the optimal dividend barrier b, the optimal lower capital injection barrier −z∗, and the value function V(x) when the claim size is exponentially distributed
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