The two-sided exit problem has been the subject of risk management analysis, used to better understand the dynamic of various insurance risk processes. In the two-sided exit setting, the discounted
aggregate claims are investigated under a dependent renewal process (also known as dependent Sparre Andersen risk process). Utilizing Lundberg's generalized equation and Laplace transform, we identify
the fundamental solutions to a given integral equation, which will be shown to play a role similar to the scale matrix for spectrally-negative Markov-additive processes. Explicit expressions and recursions are then identified for the two-sided probabilities and the moments of the aggregate claims respectively. A numerical example for the two-sided exit probabilities involving the Farlie-Gumbel-Morgenstern (FGM) copula is provided.