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We consider the bail-out optimal dividend problem under fixed transactioncosts for a L\'evy risk model. Furthermore, we consider the version with aconstraint expected net present value of injected capital. To characterize thesolution to the aforementioned models, we first solve the bail-out optimaldividend problem under transaction costs and capital injection and show theoptimality of reflected (c1; c2)- policies. Next, we introduce the dualLagrangian problem and show that the complementary slackness conditions aresatisfied, characterizing the optimal Lagrange multiplier. Finally, weillustrate our findings with a series of numerical examples.

Optimal bail-out dividends problem with transaction cost and capital injection constraint