The Risk Model with Interest, Liquid Reserves and a Constant Dividend Barrier

In this paper, we consider the compound Poisson surplus model with interest, liquid reserves and a constant dividend barrier. When the surplus of an insurer is below a fixed level, the surplus is kept as liquid reserves, which does not earn interest. When the surplus attains the level, the surplus will receive interest at a constant rate. When the surplus hits another fixed higher lever, the excess of the surplus over this higher level will be distributed to the shareholders as dividends. We derive a system of integro-differential equations for the Gerber-Shiu discounted penalty function and obtain the solutions to these integro-differential equations. In the case where the claim sizes are exponential distributed, we get the exact solutions of zero discounted Gerber-Shiu function. We also get the integro-differential equation for the expectation of the discounted dividends until ruin which is the key to discuss the optimal dividend barrier. And we give the exact solution in the special case with exponential claim sizes.