Smoothness of certain functions in two kinds of risk models with a barrier dividend strategy

In this paper, we study the smoothness of certain functions in two kinds of risk models with a barrier dividend strategy. Mainly using technique from the piecewise deterministic Markov processes theory, we prove that the function is continuously differentiable in the first risk model. Using the weak infinitesimal generator method of Markov processes, we prove that the function is twice continuously differentiable in the second risk model. Intego-differential equations satisfied by them are derived.