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 regionbounded 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  .
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 will consider two scenarios of controlled process : refraction and the impulse control . In first one the dividends are payed out when two-dimensional risk process exit fixed region . 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 . This paper is an attempt at understanding the effect of dependencies of two portfolios on the the join optimal strategy of paying dividends. For the proportional reinsurance we can observe for example the interesting phenomenon that is dependence of choice of the optimal barrier on the initial reserves. This is a contrast to the one-dimensional Cram\'{e .
D
consider different kinds of linear barriers. We
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: refraction and the impulse control
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the dividends are payed out
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two-dimensional risk process hits the barrier the minimal amount of dividends is payed out to keep the risk process within the regionbounded by the barrier
two-dimensional risk process exit fixed region
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. This paper is an attempt at understanding the effect of dependencies of two portfolios on the the join optimal strategy of paying dividends. For the proportional reinsurance we can observe for example the interesting phenomenon that is dependence of choice of the optimal barrier on the initial reserves. This is a contrast to the one-dimensional Cram\'{e