Optimal Ratcheting of Dividends in Insurance

Abstract

We address a long-standing open problem in risk theory, namely finding the optimal strategy to pay out dividends from an insurance surplus process if the dividends are paid according to a dividend rate that is not allowed to decrease. The optimality criterion here is to maximize the expected value of the aggregate discounted dividend payments up to the time of ruin. In the framework of the classical Cramér--Lundberg risk model, we solve the corresponding two-dimensional optimal control problem and show that the value function is the unique viscosity solution of the corresponding Hamilton--Jacobi--Bellman equation. We also show that the value function can be approximated arbitrarily closely by ratcheting strategies with only a finite number of possible dividend rates and identify the free boundary and the optimal strategies in several concrete examples. These implementations illustrate that the restriction of ratcheting does not lead to a large efficiency loss when compared to the classical unconstrained optimal dividend strategy.

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