Optimal ratcheting of dividends in insurance

TL;DR

This paper solves a long-standing problem in risk theory by determining the optimal dividend payout strategy in an insurance model where the dividend rate can only stay the same or increase, showing minimal efficiency loss compared to unconstrained strategies.

Contribution

It provides a solution to the optimal control problem with ratcheting dividend rates in the classical Cramér-Lundberg model, including the characterization of the value function and strategies.

Findings

The value function is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation.

Ratcheting strategies with finite rates approximate the optimal solution arbitrarily closely.

Restricting dividend rates to ratcheting does not significantly reduce efficiency.

Abstract

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.