The expected time to ruin in a risk process with constant barrier via martingales

Two risk models with a constant dividend barrier are considered. In the two models claims arrive according to a Poisson process. In the first model the claim size has a phase type distribution. In the second model the claim size is exponentially distributed, but the arrival rate, the mean claim size, and the premium rate are governed by a random environment, which changes according to a Markov process. Kella and Whitt [Kella, O., Whitt, W., 1992. Useful martingales for stochastic storage processes with Lévy input. J. Appl. Probability 29, 396–403] martingale is applied in the first model. Asmussen and Kella [Asmussen, S., Kella, O., 2000. A multi-dimensional martingale for Markov additive processes and its applications. Adv. Appl. Probability 32, 376–393] multi-dimensional martingale is applied in the second model. The expected time to ruin and the amount of dividends paid until ruin occurs are obtained for both models.