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 of the corresponding two-dimensional risk process first
leaving 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 under a Cram\'{e}r light-tail assumption on the
claim size distribution.Comment: Published in at http://dx.doi.org/10.1214/08-AAP529 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
