The Annals of Applied Probability

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

Florin Avram, Zbigniew Palmowski, and Martijn R. Pistorius

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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 Cramér light-tail assumption on the claim size distribution.

Article information

Ann. Appl. Probab., Volume 18, Number 6 (2008), 2421-2449.

First available in Project Euclid: 26 November 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J15
Secondary: 60F10: Large deviations 60G50: Sums of independent random variables; random walks

First time passage problem Lévy process exponential asymptotics ruin probability


Avram, Florin; Palmowski, Zbigniew; Pistorius, Martijn R. Exit problem of a two-dimensional risk process from the quadrant: Exact and asymptotic results. Ann. Appl. Probab. 18 (2008), no. 6, 2421--2449. doi:10.1214/08-AAP529.

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