 
Summary: Exit problem of a twodimensional risk process
from a cone: exact and asymptotic results
Florin Avram
Zbigniew Palmowski
Martijn Pistorius
October 2, 2006
Abstract
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
twodimensional 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 lighttail and under subexponential assumptions
on the claim size distribution.
Key words: First time passage problem, L´evy process, fluctuation theory,
exponential asymptotics, subexponential distribution.
