# On the bail-out dividend problem for spectrally negative Markov additive models

## TL;DR

This paper establishes the optimality of a regime-modulated refraction-reflection dividend strategy for spectrally negative Markov additive processes under regime switching, confirming a barrier-type control conjecture through auxiliary problems and fluctuation identities.

## Contribution

It proves the optimality of a regime-dependent refraction-reflection strategy for dividend problems in Markov additive models, extending previous results to more general processes with regime switching.

## Key findings

- Optimal regime-modulated refraction-reflection strategy confirmed
- Explicit threshold characterization via fluctuation identities
- Transformation of regime-switching problem into local optimization

## Abstract

This paper studies the bail-out optimal dividend problem with regime switching under the constraint that the cumulative dividend strategy is absolutely continuous. We confirm the optimality of the regime-modulated refraction-reflection strategy when the underlying risk model follows a general spectrally negative Markov additive process. To verify the conjecture of a barrier type optimal control, we first introduce and study an auxiliary problem with the final payoff at an exponential terminal time and characterize the optimal threshold explicitly using fluctuation identities of the refracted-reflected Levy process. Second, we transform the problem with regime-switching into an equivalent local optimization problem with a final payoff up to the first regime switching time. The refraction-reflection strategy with regime-modulated thresholds can be shown as optimal by using results in the first step and some fixed point arguments for auxiliary recursive iterations.