  We address a long-standing open problem in risk theory, namely the optimal
strategy to pay out dividends from an insurance surplus process, if the
dividend rate can never be decreased. 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 Cram\'{e}r-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.
