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Ruin probabilities and decompositions for general perturbed risk processes. (English) Zbl 1061.60075
Let \(C=(C(t),t\geq0)\) be a subordinator without a drift and with finite expectation and let the risk process \(R=(R(t),t\geq0)\) with premium rate \(c>0\) be defined by \(R(t)=ct-C(t)\). The authors consider a perturbed risk process \(X(t)=R(t)+Z(t)\) for \(t\geq0\), where the perturbation \(Z=(Z(t),t\geq0)\) is a spectrally negative, mean zero, Lévy process, which is independent of the process \(C\). This setting includes the Brownian perturbation and also the perturbation by \(\alpha\)-stable spectrally negative Lévy process for \(\alpha\in(1,2)\). The aim of the paper is to derive the Pollaczek-Khinchin formula for the survival probability \(\theta(x):=P(X(t)\geq -x\), \(\forall\,t\geq0)\) and to give an interpretation of the formula based on the decomposition of the dual risk process at modified ladder epochs.

MSC:
60J25 Continuous-time Markov processes on general state spaces
60G51 Processes with independent increments; Lévy processes
60J75 Jump processes (MSC2010)
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
91B30 Risk theory, insurance (MSC2010)
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