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Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results. (English) Zbl 1163.60010
The authors consider a two-dimensional risk model in which two companies split the amount in each claim and in each premium in some proportion. The eventual ruin probabilities are studied in case of the Lévy model and the Sparre Andersen-renewal risk model.
Two ruin problems are considered: the first time when at least one company is ruined and the first time when the companies experience simultaneous ruin.
The closed form expression for the ultimate ruin probability is obtained for the case when the claims arrive according to a Poisson process. The asymptotics of the ruin probability when the initial reserves of both companies tend to infinity under a Cramér light-tail assumption on the claim size distribution are analyzed.

MSC:
60F10 Large deviations
60G50 Sums of independent random variables; random walks
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
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