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}r-Lundberg model where the optimal choice of barrier among optimal barrier strategies is uniform for all initial reserves.
Consider two insurance companies (or two branches of the same company) that receive premiums at different rates and then split the amount they pay in fixed proportions for each claim (for simplicity we assume that they are equal) . We model the occurrence of claims according to a Poisson process. The ruin is achieved when the corresponding two-dimensional risk process first leaves the positive quadrant. We will consider two scenarios of the controlled process: refraction and impulse control. In the first case the dividends are payed out when the two-dimensional risk process exits the fixed region. In the second scenario , whenever the process hits the horizontal line, it is reduced by paying dividends to some fixed point in the positive quadrant where it waits for the next claim to arrive. In both models we calculate the discounted cumulative dividend payments until the ruin . This paper is the first attempt to understand the effect of dependencies of two portfolios on the joint optimal strategy of paying dividends. For example in case of proportional reinsurance one can observe the interesting phenomenon that choice of the optimal barrier depends on the initial reserves. This is in contrast with the one-dimensional Cram\'{e}r-Lundberg model where the optimal choice of the barrier is uniform for all initial reserves.
R
have the same claims and they divide premia in some specified proportions
receive premiums at different rates and then split the amount they pay in fixed proportions for each claim (for simplicity we assume that they are equal)
R
if
when
R
leave
leaves
A
the
D
the
R
first one
the first case
A
the
R
exit
exits the
R
whenever process hits
, whenever the process hits the
R
the risk process
it
R
dividend
dividends
R
and waits there for the first
where it waits for the next
A
the
D
time
R
an attempt at understanding
the first attempt to understand
R
the join
joint
R
the proportional reinsurance we can observe for example
example in case of proportional reinsurance one can observe
D
is dependence of
A
depends
R
a contrast to
in contrast with
R
barrier among optimal barrier strategies
the barrier