De Finetti’s dividend problem and impulse control for a two-dimensional insurance risk process

Consider two insurance companies (or two branches of the same company) that have the same claims and they divide premia in some specified proportions. We model the occurrence of claims according to a Poisson process. The ruin is achieved if the corresponding two-dimensional risk process first leave the positive quadrant. We consider different kinds of linear barriers. We will consider two scenarios of controlled process. In first one when two-dimensional risk process hits the barrier the minimal amount of dividends is payed out to keep the risk process within the region bounded by the barrier. In the second scenario whenever process hits horizontal line, the risk process is reduced by paying dividend to some fixed point in the positive quadrant and waits there for the first claim to arrive. In both models we calculate discounted cumulative dividend payments until the ruin time.