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Analysis of some ruin-related quantities in a Markov-modulated risk model. (English) Zbl 1344.60075
Summary: In this paper, we study the joint Laplace transform and probability generating function of some random quantities that occur in each environment state by the time of ruin in a Markov-modulated risk process. These quantities include the duration spent in each state, the number of claims and the aggregate amount of claims that occurred in each state by the time of ruin. Explicit formulae for the joint transforms, given the initial surplus, and the initial and terminal environment states, are expressed in terms of a matrix version of the scale function. Moments and covariances of these ruin-related quantities are obtained and numerical illustrations are presented. The joint transform of the duration spent in each state, the number of claims, and the aggregate amount of claims that occurred in each state by the time the surplus attains a certain level are also investigated.

60J28 Applications of continuous-time Markov processes on discrete state spaces
60J27 Continuous-time Markov processes on discrete state spaces
60K37 Processes in random environments
91B30 Risk theory, insurance (MSC2010)
44A10 Laplace transform
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