Abstract
We investigate a control problem leading to the optimal payment of dividends in a Cramér-Lundberg-type insurance model in which capital injections incur proportional cost, and may be used or not, the latter resulting in bankruptcy. For general claims, we provide verification results, using the absolute continuity of 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 Løkka-Zervos (2008). 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.
Highlights
The easiest way to conceive such balance can roughly be stated as follows: when below low levels − a ≤ 0, reserve processes should be replenished by capital injections at some cost, and when above high levels b > 0, they should be taken out of the reserves as dividends–see for example [1,2] and the comprehensive book [3]
We have organized our paper around the classical guess and verify procedure for solving stochastic control problems: 1
Our paper shows that optimality is achieved here via “bounded buffer capital injections” (− a, 0, b) policies, consisting in stopping at the first time when the size of the overshoot below 0 exceeds the limit a defined in Theorem 12
Summary
/ {0, ∞} (i.e., de Finetti and Shreve, Lehoczky and Gaver policies fail to be optimal); the resulting value function is not of class C1 at 0 (in the sense that its derivative does not exist at x = 0 and, more, the right-hand derivative at 0 is not k as assumed, for Brownian claims, in [14], 5.2) This can be found in Remark 14 and it implies, in particular, that the verification Theorem 4 has to be given for absolutely-continuous functions. Our paper shows that optimality is achieved here via “bounded buffer capital injections” (− a, 0, b) policies, consisting in stopping at the first time when the size of the overshoot below 0 exceeds the limit a defined in Theorem 12 These policies turn out to work better than both the de Finetti and the Shreve, Lehoczky and Gaver policies, which are locally the worst possible choices!.
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