Exit problem of a two-dimensional risk process from a cone: exact and asymptotic results

Consider two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions. We model the occurrence of claims according to a renewal process. One ruin problem considered is that when the corresponding two-dimensional risk process first leaves the positive quadrant; another is that of entering the negative quadrant. When the claims arrive according to a Poisson process we obtain a closed form expression for the ultimate ruin probability. In the general case we analyze the asymptotics of the ruin probability when the initial reserves of both companies tend to infinity, both under Cramér light-tail and under subexponential assumptions on the claim size distribution.