On a risk model with debit interest and dividend payments

We consider the compound Poisson risk model with debit interest and dividend payments. The model assumes that the company is allowed to borrow at some debit interest rate when the surplus turns negative, and that the premium incomes are paid out as dividends to shareholders when the surplus reaches a horizontal barrier of level bb. We first derive integro-differential equations for the expected discounted value of all dividends until absolute ruin, Vb(u)Vb(u), which is twice continuously differentiable. In the case of exponential claim amounts, we obtain explicit expressions for Vb(u)Vb(u) and the optimal barrier b∗b∗ which maximizes Vb(u)Vb(u). We then perform a similar study for the Gerber–Shiu expected discounted penalty function. Again, when claims are exponentially distributed, we are able to find explicit expressions for the joint distribution of the surplus just prior to absolute ruin and the deficit at absolute ruin, which is a special case of the Gerber–Shiu function.