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The first passage event for sums of dependent Lévy processes with applications to insurance risk. (English) Zbl 1209.60029
This paper deals with first passage events and ruin probabilities for a Lévy process which is the sum of dependent Lévy processes. The dependence structure between the two processes is modeled by a Lévy copula. In particular, the authors consider a \(2\)-dimensional Lévy process \((X^1,X^2)\) and derive a quintuple law describing the first upwards passage event of \(X^1+X^2\) over a fixed barrier, by the joint distribution of five quantities: the time relative to the time of the previous maximum, the time of the previous maximum, the overshoot, the undershoot and the undershoot of the previous maximum.
The authors first provide the general expression for this quintuple law, and then consider two examples where explicit computations are carried out: when the jump parts of \((X^1,X^2)\) are spectrally positive compound Poisson processes, and when they are subordinators with negative drift. They also consider different Lévy copulas for modeling the dependence structure, in particular Clayton and (non-homogeneous) Archimedean Lévy copulas. Finally, they use their results to obtain asymptotic expressions for the ruin event of \(X^1+X^2\).

MSC:
60G51 Processes with independent increments; Lévy processes
60G50 Sums of independent random variables; random walks
60J75 Jump processes (MSC2010)
91B30 Risk theory, insurance (MSC2010)
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