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

**Mauricio Junca 1, Harold A. Moreno-Franco 2 and José Luis Pérez 3,\***


Received: 18 December 2018; Accepted: 29 January 2019; Published: 31 January 2019

**Abstract:** We consider the optimal bail-out dividend problem with fixed transaction cost for a Lévy risk model with a constraint on the expected present value of injected capital. To solve this problem, we first consider the optimal bail-out dividend problem with transaction cost and capital injection and show the optimality of reflected (*<sup>c</sup>*1, *<sup>c</sup>*2)-policies. We then find the optimal Lagrange multiplier, by showing that in the dual Lagrangian problem the complementary slackness conditions are met. Finally, we present some numerical examples to support our results.

**Keywords:** dividend payment; optimal control; capital injection constraint; spectrally negative Lévy processes; reflected Lévy processes; scale functions

**MSC:** 60G51; 91B30
