The classical optimal dividend problem of de Finetti is about how an insurance company should dynamically pay out dividends such that the resulting payoff to shareholders is maximized. The catch here is that dividends are subtracted from the capital reserve and once the latter becomes negative, the company gets ruined. We look at the well-studied case where the cash flow of the insurance company is modeled by a particular class of stochastic processes exhibiting negative jumps that represent the claim payments. We give an overview of the main results derived for this stochastic optimal control problem and the techniques used for finding the optimal strategy. We also discuss several extensions so that the model becomes better equipped to serve as a proper risk management tool for insurance companies.
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