The Optimal Dividend Strategy In The Compound Poisson Modelwith Debit Interest

We consider the optimal dividend problem in the compound Poisson model with debit interest and dividend restrict. The model assumes that the company is allowed to borrow money 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 some barrier. Our target is to maximize the expectation of the accumulated discounted dividends until absolute ruin. First of all,the HJB equation that V(u) satisfied and the verification of optimality is given. Then,we derive integro-differential equations and boundary conditions that ditermine the function V(u) under the threshold strategy. Finally,we obtain explicit solutions of V(u) when the claim amount distribution is exponential.