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 un-constrained optimal dividend strategy.