Time to ruin, insolvency penalties and dividends in a Markov-modulated multi-risk model with common shocks

We consider a main insurance company with K subcompanies (or lines of busi- ness). The joint evolution of the surpluses of these lines of business is modeled by a Markov-modulated multivariate compound Poisson model with Poisson common shocks, modified by interactions between the lines of business and paiement of divi- dends. We assume that the financial situation of the subcompanies has an impact on the other companies, for example because they have part of their surplus invested in one another. If a line of business is in the red, the others have to pay a penalty, which is traduced by a decrease of the premium received by unit of time, or by a lost of dividends for the shareholders if the other line of business is "doing well". Conversely, a line of business with a high surplus level may increase the premium by unit of time of the others as they receive part of the dividends. In this paper, we focus on a particular line of business, and provide an approximation for expected time to ruin, and the expected amounts of dividends paid to the shareholders, and used to pay penalty due to insolvency of some subcompany. The method is to discretize claim amounts and to approximate the multidimensional surplus process of the subcompanies with a continuous time Markov process with finite state space. A technique of Frostig (2005) and Kella and Whitt (1992) enables us to get approximates, which are shown to converge to the desired values. It is possible to compare the behavior of the main company with and without the other subcompanies, which could provide a tool to help making consortium building decision.