Absolute ruin problems in a compound Poisson risk model with constant dividend barrier and liquid reserves

In this paper, we consider a compound Poisson surplus model with constant dividend barrier and liquid reserves under absolute ruin. When the surplus is negative, the insurer is allowed to borrow money at a debit interest rate to continue the business; when the surplus is below a fixed level Delta, the surplus is kept as liquid reserves, which do not earn interest; when the surplus attains the level Delta, the excess of the surplus over the level receives interest at a constant rate; when the surplus reaches a higher level b, the excess of the surplus above b is all paid out as dividends to shareholders of the insurer. We first derive the integro-differential equations satisfied by the moment-generating function and moment of the discounted dividend payments until absolute ruin. Then, applying these results, we get explicit expressions of them for exponential claims and discuss the impact of the model parameters on the expected dividend payments by numerical examples.