Optimal Bail-Out Dividend Problem with Transaction Cost and Capital Injection Constraint

Abstract

We consider the bail-out optimal dividend problem under fixed transaction costs for a Lévy risk model with a constraint on the expected net present value of injected capital. In order to solve this problem, we first consider the bail-out optimal dividend problem under transaction costs and capital injection and show the optimality of reflected (c 1 , c 2 )-policies. We then find the optimal Lagrange multiplier, by showing that in the dual Laagrangian problem, the complementary slackness conditions are verified. Finally, we verify our results with some numerical examples.

Dividend and Capital Injection Optimization with Transaction Cost for Lévy Risk Processes

“…In our case, we employed an alternative mollifying technique (see Lemma 4.3 and 4.4 ) to deal with the difficulty of lack of sufficient differentiability. The mollifying arguments given in Lemma 4.3 and 4.4 are rigorous and differ from the approach adopted in [28] when proving their verification theorem.…”

“…Using the expected present value of dividends until ruin (the expected present value of the dividends subtracted by the discounted costs of capital injections) as the value function, [6] identified the condition under which the barrier strategy (respectively, the barrier dividend strategy together with capital injection strategy that reflects the reserve process at 0) is optimal among all admissible strategies. More results of non-impulse dividend optimization under the SNL risk processes can be found in [7,9,15,16,20,21,28,31,32,34,37,38,[45][46][47][48][49][50]53] and the references therein. The non-impulse dividends optimization under the SPL risk processes can be found in [4,5,12,13,53,55,57] and others.…”

“…We acknowledge that there is a parallel paper in the literature, [28], which was also finished independently around the same time. The first version of both papers were available on internet in the middle of 2018.…”

“…The first version of both papers were available on internet in the middle of 2018. The authors of [28] considered the bail-out optimal dividend problem under fixed transaction costs for a Lévy risk model with a constraint on the expected net present value of injected capital. While the main results in this paper and those in [28] appear to be very similar, the primary objectives of these two papers are notably different as well as the methods adopted in the proof of certain main results (for instance, the verification Lemma 4.3 and 4.4 in this paper vs Theorem 4.10 in [28]).…”

Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model

“…To our best knowledge, our paper hereafter seems to be the first to quantify the optimal buffer (severity of ruin) when bankruptcy should replace individual bailouts (to be fair, this possibility has been considered before, but without studying optimality-see for example [12]). For a different approach, quantifying the optimal policy under an expected total bailouts constraint, see [13].…”