The policy iteration algorithm for a compound Poisson process applied to optimal dividend strategies under a Cramér–Lundberg risk model

Abstract

In this paper, we focus on the policy iteration algorithm (PIA) for the optimal dividend problem under the Cramér–Lundberg risk model. We conclude that the optimal value function is the minimum nonnegative solution of an optimization equation. Under any conditions, it can be approximated by iteration starting with the initial zero-valued policy, i.e., the policy of no dividend payment at all. An auxiliary optimization problem of maximizing dividend payment with terminal reward up to the first claim arrival time is introduced. This auxiliary optimization problem is solved completely in the sense that, no matter what the claim-size distribution is, the explicit formulae for both optimal strategy and value function are presented in general situation. These ensure that, at each iteration, we can get the explicit formulae of the improved strategy and its evaluation function. Finally, the PIA is illustrated with some numerical examples.