On differentiability of ruin functions under Markov-modulated models

This paper analyzes the continuity and differentiability of several classes of ruin functions under Markov-modulated insurance risk models with a barrier and threshold dividend strategy, respectively. Many ruin related functions in the literature, such as the expectation and the Laplace transform of the Gerber-Shiu discounted penalty function at ruin, of the total discounted dividends until ruin, and of the time-integrated discounted penalty and/or reward function of the risk process, etc, are special cases of the functions considered in this paper. Continuity and differentiability of these functions in the corresponding dual models are also studied.