On Regularities of Expected Discounted Penalty at Ruin in Two-Sided Jump-Diffusion Model

The expected discounted penalty with downside jumps has been extensively studied in Gerber and Shiu(1998), Gerber and Landry(1998), Tsai and Wilmott(2002) and others. In this paper, we study the expected discounted penalty in a perturbed compound Poisson model with two sided jumps. We show that it is always twice continuously differentiable provided that the jump size distribution has a bounded continuous(a.e.) density. Moreover, under some minor conditions, both its first and second derivatives vanish at infinity. Next, based on Boyarchenko and Levendorskǐi(2002), we derive an integro-differential equation for the expected discounted penalty. When the jump size distribution is exponential, as an application of the integro-differential equation, we obtain an explicit formula for the expected discounted penalty. We close this paper by giving a generalization of the renewal integral equation in Gerber and Landry(1998).