Summary: Exit problem of a two-dimensional risk process
from a cone: exact and asymptotic results
October 2, 2006
Consider two insurance companies (or two branches of the same com-
pany) 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 infin-
ity, both under Cram´er light-tail and under subexponential assumptions
on the claim size distribution.
Key words: First time passage problem, L´evy process, fluctuation theory,
exponential asymptotics, subexponential distribution.