On the expected discounted penalty function and optimal dividend strategy for a risk model with random incomes and interclaim-dependent claim sizes

In this paper, we consider a risk model with dependence between claim sizes and interclaim arrivals. In contrast with the classical risk model where the premium process is a linear function of time, we consider a dependent risk model where the aggregate premium process is a compound Poisson process, moreover, there is a constant barrier strategy in this model. The integral equations for the expected discounted penalty function and the expected discounted dividend payments until ruin are obtained. In particular, when the individual stochastic premium amount is exponentially distributed, it is proved that both the expected discounted penalty function and the expected discounted dividend payments until ruin satisfy the Volterra integral equations. Furthermore, the representations of the solutions are derived, respectively. In addition, when the individual stochastic premium amount and claim amount are exponentially distributed, we can get the explicit expressions for the Laplace transform of the ruin time and the expected discounted dividend payments until ruin. Finally, the optimal barrier is presented under the condition of maximizing the expectation of the difference between discounted dividends until ruin and the deficit at ruin.