Two-barrier optimality conjecture for the dividend control with fixed costs and regime switching

Establish that, for the optimal dividend control problem with fixed transaction cost β and discount rate q in which the surplus follows an endogenous regime-switching diffusion with two-valued drift (μ+, μ−) and two-valued volatility (σ+, σ−) switching at a threshold a, and where dividends are paid by impulses that reduce the surplus and incur the fixed cost β, the optimal admissible policy maximizing the expected discounted net dividends until ruin is a two-barrier impulsive dividend strategy: pay a lump-sum dividend to bring the surplus from an upper barrier z2 down to a lower barrier z1 whenever the surplus reaches z2, and otherwise pay nothing.

Background

The paper studies de Finetti’s optimal dividend problem under an SDE whose drift and volatility switch between (μ+, σ+) and (μ−, σ−) depending on whether the surplus is above or below a fixed threshold a. Dividend payments are made via impulses that incur a fixed transaction cost β, turning the problem into an impulse control problem with ruin at the first passage below zero.

In classical settings without regime switching or with regular coefficients, barrier-type policies are often optimal. Motivated by this, the authors posit that even in this nonhomogeneous diffusion with fixed costs, an optimal policy should be of two-barrier type: when the surplus hits an upper barrier z2, a lump sum is paid to bring it down to a lower barrier z1, and no payments are made otherwise.

The paper later verifies optimality under several sufficient conditions and characterizes the optimal (z1, z2) in many parameter regimes; however, the conjecture is posed in full generality before those results are established.