Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results

Details Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results

2008-02-27

arxiv.org/abs/0802.4060v2

Annals of Applied Probability 2008, Vol. 18, No. 6, 2421-2449

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AAP529 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/08-AAP529

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.

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