Equity Cost Induced Dichotomy for Optimal Dividends in the Cramér-Lundberg Model
Résumé
We investigate a control problem leading to the optimal payment of dividends in a Cramér-Lundberg-type insurance model in which capital injections are allowed at a certain cost. For general claims, we provide verification results arguing on absolutely continuous super-solutions of a convenient Hamilton-Jacobi variational inequality. As a by-product, for exponential claims, we prove the optimality of bounded buffer capital injections (−a, 0, b) policies. These policies consist in stopping at the first time when the size of the overshoot below 0 exceeds a certain limit a and only pay dividends when the reserve reaches an upper barrier b. An exhaustive and explicit characterization of optimal couples buffer/barrier is given via comprehensive structure equations. The optimal buffer is shown never to be of de Finetti (a = 0) or Shreve-Lehoczy-Gaver (a = ∞) type. The study results in a dichotomy between cheap and expensive equity, based on the cost-of-borrowing parameter, thus providing a non-trivial generalization of the Lokka-Zervos phase-transition [LZ08]. In the first case companies start paying dividends at the barrier b * = 0, while in the second they must wait for reserves to build up to some (fully determined) b * > 0 before paying dividends.
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