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Optimal dynamic premium control in non-life insurance. Maximizing dividend pay-outs. (English) Zbl 1039.91042
Let the controlled reserve \(\{R^{\beta,b}_{t}\}\) of a non-life insurance company be given by \[ R^{\beta,b}_{t}= x+\int_{0}^{t}f(\beta(s))\,ds- \sum_{i=1}^{N^{\beta}_{t}} U_{i}-\int_{0}^{t}\,dL^{\beta,b}(s),\text{ where } L^{\beta,b}(t)= (x-b)^{+}+ \int_{0}^{t}f(\beta(s)) I(R^{\beta,b}_{s}=b)\,ds; \] \(f(\beta)=(1+h(\beta))\beta\mu\); \(x\in R\) is the initial reserve; \(\{N_{t}^{\beta}\}\) is a Poisson process with intensity \(\beta\); the random variables \(U_{i},\;i=1,2,\ldots\) are i.i.d. with mean \(\mu\) claim sizes independent on \(N_{t}\). The company distributes dividends to shareholders according to a barrier strategy. The objective for the company is to find an optimal intensity process \(\{\beta^{*}(t)\}\) and a barrier \(b^{*}\) which maximizes the expected discounted future pay-out of dividend until the time of ruin. In this paper it is shown how to use a solution of the Bellman equation to construct the optimal premium policy and the optimal barrier. For exponential claim size distributions the closed form solution to the Bellman equation is found. For general claim size distributions the solution and optimal controls can be approximated numerically by a dynamic programming approach. The author also investigates the possibilities of the De Vylder approximation of the optimal control in the general case by using the closed form solution of an approximating problem with exponential claim size distributions.

MSC:
91B30 Risk theory, insurance (MSC2010)
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