The Gerber-Shiu discounted penalty function in the thinning risk model with refunding

In this paper, the thinning risk model with refunding is considered, where the counting process of the refund and the counting process of the claim are the thinning of the premium arrival counting process with a binomial process. The Gerber-Shiu discounted penalty function of this model is discussed, and the renewal equation and the asymptotic estimate satisfied by the Gerber-Shiu discounted penalty function are also derived. And according to the characteristic of the Gerber-Shiu discounted penalty function, we obtain the asymptotic estimate of the ruin probability.